A numerical approach to solve hyperbolic telegraph equations via Pell–Lucas polynomials

Abstract

In this article, a collocation approximation is investigated for approximate solutions of hyperbolic telegraph partial differential equations (HTPDEs). The method is based on evenly spaced collocation points and Pell–Lucas polynomials (PLPs). The form of solution, derivatives of unknown function in equation and conditions are expressed in matrix forms which depend on PLMs. By the help of these matrix forms and collocation points, problem is reduced to a system of linear algebraic equations. In addition, error analysis is performed for method. Thus, errors are bound by an upper bound. By making the applications of these techniques, the computed outcomes are offered in tables and graphs. Also the obtained outcomes by method are also compared with outcomes of other methods in the literature. These comparisons show that our method is more influential than other methods. All results have been computed by the aid of a code generated in MATLAB.

Keywords:

• Collocation method
• error analysis
• hyperbolic equations
• partial differential equations
• Pell–Lucas polynomials

AMS:

• 35L20
• 42C05
• 65M15
• 65M70
• 65N35

1. Introduction

Mathematical models in fields such as physics, engineering and economics are often expressed by using differential equations. Hyperbolic telegraph partial differential equations (HTPDEs), which are differential equation, are also often used in wave spread of electrical signals. Nowadays, many researchers have studied on HTPDEs. Sharifi and Rashidinia, Nazir et al., Jebreen et al., Bicer and Yalcinbas, Zarebnia and Parvaz, Adebayo et al., Dugassa et al., Nagaveni, Bahsi and Yalcinbas, Wang et al. have presented, respectively, a collocation approach based on cubic B-splines [1], new cubic trigonometric B-splines approach [2], a numerical algorithm used MWGM (multi-wavelet Galerkin method) [3], Bernoulli polynomial approach [4], a collocation approach based on quadratic B-spline functions [5,6], Laplace transform collocation method [7], matrix inverse method [8], Haar wavelet collocation method [9], a collocation scheme based on Fibonacci polynomials [10], a reduced order extrapolating method [11]. Also, many methods such as a collocation technique used radial basis function [12], Legendre MWGM [13], Chebyshev–Tau method [14], DRBIE method (a numerical technique used dual reciprocity method (DRM) with boundary integral equation) [15], homotopy perturbation method [16], variational iteration method [17], variational iteration and homotopy perturbation methods [18], differential quadrature method [19], finite difference scheme [20], a spectral collocation scheme based on Chebyshev polynomials [21], Haar wavelet method [22] and DGJ method [23] have been studied for solving HTPDEs in literature. Yüzbaşı et al. made a great contribution to the literature by doing many studies on partial differential equations [24–29]. Sezer et al. presented many studies on partial differential equations [30–33]. Partial differential equations are very popular today and there are numerous works on partial differential equations in the literature [34–75].

In this paper, it is introduced a collocation method using PLPs to get approximate solutions of hyperbolic equation [13,14,27,30,76] (1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1)

under initial conditions (2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) and boundary conditions (3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) The basic aim of this paper is to compute the truncated Pell–Lucas polynomial solutions of (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ) as follows: (4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4)

The representations of parameters/variables in (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) )–(4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ) are described in Table 1.

Table 1. The representations of the parameters/variables in the (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) )–(4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ).

Parameters/Variables	Definitions
$z(\alpha ,\beta )$	Unknown function
$z_{2N}(\alpha ,\beta )$	Approximate solution
$g(\alpha ,\beta ),f_0(\alpha ),f_1(\alpha ),h_0(\alpha ),h_1(\alpha )$	Known functions
$Q_i(\alpha )$	PLPs
$a_{i,j}$	Unknown Pell–Lucas coefficients

Also, Pell–Lucas polynomials are defined in [77,78] $Q_r(\alpha )=\sum _{s=0}^{\text{⟦}r/2\text{⟧}}{2}^{r-2s}\frac{r}{r-s}(\genfrac{}{}{0ex}{}{r-s}{s}){\alpha }^{r-2s}.$Now, let's give two important features of PLPs that will be used in the next section. One of them is regarding the recurrence relation of PLPs defined by $Q_i(\alpha )=2\alpha Q_{i-1}(\alpha )+Q_{i-2}(\alpha ),i\ge 2$when $Q_0(\alpha )=2$ and $Q_1(\alpha )=2\alpha$. Other one is regarding derivatives of them as follows: (5) $Q_i^{\mathrm{\prime }}(\alpha )=2\alpha Q_{i-1}^{\mathrm{\prime }}(\alpha )+Q_{i-2}^{\mathrm{\prime }}(\alpha )+2Q_{i-1}(\alpha ),i\ge 2$(5) when $Q_0^{\mathrm{\prime }}(\alpha )=0$ and $Q_1^{\mathrm{\prime }}(\alpha )=2$.

2. Matrix relations for solution method

In this section, partial differential equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) and conditions (2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )-(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ) are expressed in matrix forms by aid of PLPs.

Lemma 2.1

Pell–Lucas polynomial solutions (4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ) are given in matrix form [29] (6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) where $\begin{array}{rl}{\overline{\mathbf{A}}}_N& ={\left[\begin{array}{cccc}{\mathbf{A}}_0& {\mathbf{A}}_1& \cdots & {\mathbf{A}}_N\end{array}\right]}^T,\\ {\mathbf{A}}_t& ={\left[\begin{array}{cccc}{a}_{t,0}& a_{t,1}& \cdots & a_{t,N}\end{array}\right]}^T,t=0,1,\dots ,N\\ {\mathbf{Q}}_N(\alpha )& =\left[\begin{array}{cccc}{Q}_0(\alpha )& Q_1(\alpha )& \cdots & Q_N(\alpha )\end{array}\right],\\ \mathbf{0}& =\mathrm{zeros}(1\times (N+1)),\\ {\overline{\mathbf{Q}}}_N(\beta )& ={\left[\begin{array}{cccc}{\mathbf{Q}}_N(\beta )& \mathbf{0}& \cdots & \mathbf{0}\\ \mathbf{0}& {\mathbf{Q}}_N(\beta )& \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \mathbf{0}& \cdots & {\mathbf{Q}}_N(\beta )\end{array}\right]}_{(N+1)\times (N+1{)}^2}.\end{array}$

Proof.

We write expression $Q_{N,N}(\alpha ,\beta )=Q_N(\alpha )Q_N(\beta )$ in (4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ) in matrix form as ${\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )$. Next, we multiply vector ${\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )$ by vector ${\overline{\mathbf{A}}}_N$ from right. Hence, we get relation (6(6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) ).

Lemma 2.2

The second derivative with respect to α of Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) is expressed in matrix form [29] (7) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\alpha \alpha }}={\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(7)

Proof.

To proof it, we can use Lemma 2.1. We take derivative of solution form (6(6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) ) with respect to α and so we get (8) $(z_{2N}(\alpha ,\beta ){)}_{\alpha }={\mathbf{Q}}_N^{\mathrm{\prime }}(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(8) Also, we have relation ${\mathbf{Q}}_N^{\mathrm{\prime }}(\alpha )={\mathbf{Q}}_N(\alpha )\mathbf{H}$ by using (5(5) $Q_i^{\mathrm{\prime }}(\alpha )=2\alpha Q_{i-1}^{\mathrm{\prime }}(\alpha )+Q_{i-2}^{\mathrm{\prime }}(\alpha )+2Q_{i-1}(\alpha ),i\ge 2$(5) ). Thus, we can express (8(8) $(z_{2N}(\alpha ,\beta ){)}_{\alpha }={\mathbf{Q}}_N^{\mathrm{\prime }}(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(8) ) as (9) $(z_{2N}(\alpha ,\beta ){)}_{\alpha }={\mathbf{Q}}_N(\alpha )\mathbf{H}{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(9) where if N is odd [79,80] $\mathbf{H}=\left[\begin{array}{cccccccc}0& 1& 0& -3& 0& 5& \cdots & (-1{)}^{\frac{N-1}{2}}N\\ 0& 0& 4& 0& -8& 0& \cdots & 0\\ 0& 0& 0& 6& 0& -10& \cdots & (-1{)}^{\frac{N-3}{2}}2N\\ 0& 0& 0& 0& 8& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& 0& 0& \cdots & 0\end{array}\right]$and if N is even [79,80] $\mathbf{H}=\left[\begin{array}{cccccccc}0& 1& 0& -3& 0& 5& \cdots & 0\\ 0& 0& 4& 0& -8& 0& \cdots & (-1{)}^{\frac{N-2}{2}}2N\\ 0& 0& 0& 6& 0& -10& \cdots & 0\\ 0& 0& 0& 0& 8& 0& \cdots & (-1{)}^{\frac{N-4}{2}}2N\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& 0& 0& \cdots & 0\end{array}\right].$The size of the matrix $\mathbf{H}$ is $(N+1)\times (N+1)$. Similarly, we take derivative of (8(8) $(z_{2N}(\alpha ,\beta ){)}_{\alpha }={\mathbf{Q}}_N^{\mathrm{\prime }}(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(8) ) with respect to α and we obtain (10) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\alpha \alpha }}={\mathbf{Q}}_N^{{}^{″}}(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(10) Finally, we write obviously relation ${\mathbf{Q}}_N^{{}^{″}}(\alpha )$ in form ${\mathbf{Q}}_N^{{}^{″}}(\alpha )={\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2$. Hence, by substituting this expression in Equation (10(10) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\alpha \alpha }}={\mathbf{Q}}_N^{{}^{″}}(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(10) ), we get (7(7) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\alpha \alpha }}={\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(7) ).

Lemma 2.3

The matrix representations of the first derivative and the second derivative with respect to β in Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) respectively are represented by (11) $(z_{2N}(\alpha ,\beta ){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N$(11) and (12) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\beta \beta }}={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N.$(12)

Proof.

Similarly to Lemma 2.2, the first derivative according to β of Equation (6(6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) ) is taken and thus we gain (13) $(z_{2N}(\alpha ,\beta ){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N^{\mathrm{\prime }}(\beta ){\overline{\mathbf{A}}}_N.$(13) Then, by taking the derivative of (13(13) $(z_{2N}(\alpha ,\beta ){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N^{\mathrm{\prime }}(\beta ){\overline{\mathbf{A}}}_N.$(13) ), we get (14) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\beta \beta }}={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N^{{}^{″}}(\beta ){\overline{\mathbf{A}}}_N.$(14) Similar to previous lemma, let's express obviously derivative ${\overline{\mathbf{Q}}}_N^{\mathrm{\prime }}(\beta )$ and ${\overline{\mathbf{Q}}}_N^{{}^{″}}(\beta )$. Thus we can write, respectively ${\overline{\mathbf{Q}}}_N^{\mathrm{\prime }}(\beta )={\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}$ and ${\overline{\mathbf{Q}}}_N^{{}^{″}}(\beta )={\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2.$ Then, by substituting these obtained relations in (13(13) $(z_{2N}(\alpha ,\beta ){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N^{\mathrm{\prime }}(\beta ){\overline{\mathbf{A}}}_N.$(13) )–(14(14) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\beta \beta }}={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N^{{}^{″}}(\beta ){\overline{\mathbf{A}}}_N.$(14) ), respectively, we complete the proof of theorem.

Theorem 2.1

We assume that Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) has an approximate solution as in form (4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ). In that case, we give matrix relation (15) $\begin{array}{rl}& {\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ & +c{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=g(\alpha ,\beta )\end{array}$(15)

Proof.

If we use Lemma 2.1–2.3 in Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ), then we obtain Equation (15(15) $\begin{array}{rl}& {\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ & +c{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=g(\alpha ,\beta )\end{array}$(15) ). More clearly, matrix relations in Equations (6(6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) ), (7(7) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\alpha \alpha }}={\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(7) ), (11(11) $(z_{2N}(\alpha ,\beta ){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N$(11) ) and (12(12) $(z_{2N}(\alpha ,\beta ){)}_{\mathrm{\beta \beta }}={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N.$(12) ) are written in Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ). As a result, we gain the desired result.

Lemma 2.4

We obtain matrix representation of conditions (2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ) as (16) $\begin{array}{rl}& z_{2N}(\alpha ,0)={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0(\alpha )\\ & (z_{2N}(\alpha ,0){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1(\alpha )\end{array}$(16) and (17) $\begin{array}{rl}& z_{2N}(0,\beta )={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=h_0(\beta )\\ & z_{2N}(K,\beta )={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=h_1(\beta ).\end{array}$(17)

Proof.

For the proof of this lemma, Lemmas 2.1 and 2.3 are used. First, we write 0 instead of β, respectively, in Lemmas 2.1 and 2.3. So, we find (18) $\begin{array}{rl}& z_{2N}(\alpha ,0)={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N\\ & (z_{2N}(\alpha ,0){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N.\end{array}$(18) Then, we write, respectively, 0 and K instead of α in Lemma 2.1. Hence, we get (19) $\begin{array}{rl}& z_{2N}(0,\beta )={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N\\ & z_{2N}(K,\beta )={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.\end{array}$(19) Finally, by utilizing right side of conditions (2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ), (16(16) $\begin{array}{rl}& z_{2N}(\alpha ,0)={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0(\alpha )\\ & (z_{2N}(\alpha ,0){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1(\alpha )\end{array}$(16) ) and (17(17) $\begin{array}{rl}& z_{2N}(0,\beta )={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=h_0(\beta )\\ & z_{2N}(K,\beta )={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=h_1(\beta ).\end{array}$(17) ) are obtained.

3. The method of solution

In this section, we present Pell–Lucas collocation method. First, we define the collocation points. Then, we use the collocation points and matrix relations in Section 2. Thus we constitute the method of solution.

Theorem 3.1

The collocation points are described as [29] (20) ${\alpha }_r=\frac{K}{N}r,{\beta }_s=\frac{M}{N}s,r,s=0,1,\dots ,N.$(20) When we investigate the solution of Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) in form (4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ), by using (20(20) ${\alpha }_r=\frac{K}{N}r,{\beta }_s=\frac{M}{N}s,r,s=0,1,\dots ,N.$(20) ) we reduce Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) to the following system of algebraic equations: (21) $\mathbf{W}{\overline{\mathbf{A}}}_N=\mathbf{G}.$(21) Here, $\begin{array}{rl}& \mathbf{W}=\left[\begin{array}{c}{\mathbf{W}}_0\\ {\mathbf{W}}_1\\ ⋮\\ {\mathbf{W}}_N\end{array}\right],{\mathbf{W}}_i=\left[\begin{array}{c}\mathbf{W}({\alpha }_i,{\beta }_0)\\ \mathbf{W}({\alpha }_i,{\beta }_1)\\ ⋮\\ \mathbf{W}({\alpha }_i,{\beta }_N)\end{array}\right],\\ \mathbf{G}& =\left[\begin{array}{c}{\mathbf{G}}_0\\ {\mathbf{G}}_1\\ ⋮\\ {\mathbf{G}}_N\end{array}\right],{\mathbf{G}}_j=\left[\begin{array}{c}g({\alpha }_j,{\beta }_0)\\ g({\alpha }_j,{\beta }_1)\\ ⋮\\ g({\alpha }_j,{\beta }_N)\end{array}\right].\end{array}$

Proof.

According to Theorem 2.1, we have (22) $\begin{array}{rl}& {\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ & +c{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=g(\alpha ,\beta ).\end{array}$(22) Then, by using the collocation points (20(20) ${\alpha }_r=\frac{K}{N}r,{\beta }_s=\frac{M}{N}s,r,s=0,1,\dots ,N.$(20) ) instead of α and β in Equation (22(22) $\begin{array}{rl}& {\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta )\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ & +c{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N(\alpha ){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N=g(\alpha ,\beta ).\end{array}$(22) ), we obtain system of matrix equations (23) $\{\begin{array}{c}{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=g({\alpha }_0,{\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=g({\alpha }_0,{\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=g({\alpha }_0,{\beta }_N)\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=g({\alpha }_1,{\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=g({\alpha }_1,{\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=g({\alpha }_1,{\beta }_N)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=g({\alpha }_N,{\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=g({\alpha }_N,{\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=g({\alpha }_N,{\beta }_N)\end{array}$(23) This system (23(23) $\{\begin{array}{c}{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=g({\alpha }_0,{\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=g({\alpha }_0,{\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=g({\alpha }_0,{\beta }_N)\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=g({\alpha }_1,{\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=g({\alpha }_1,{\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=g({\alpha }_1,{\beta }_N)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=g({\alpha }_N,{\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=g({\alpha }_N,{\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2{\overline{\mathbf{A}}}_N+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=g({\alpha }_N,{\beta }_N)\end{array}$(23) ) shortly can also be shown as (24) $\mathbf{W}{\overline{\mathbf{A}}}_N=\mathbf{G}$(24) where $\begin{array}{rl}\mathbf{W}& =\left[\begin{array}{c}{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_0)-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_1)-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N({\beta }_N)-{\mathbf{Q}}_N({\alpha }_0){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N)\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_0)-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_1)-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N({\beta }_N)-{\mathbf{Q}}_N({\alpha }_1){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_0)-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_0)\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_1)-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_1)\\ ⋮\\ {\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{H}}}^2+b{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N)\overline{\mathbf{H}}\\ +c{\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N({\beta }_N)-{\mathbf{Q}}_N({\alpha }_N){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_N)\end{array}\right],\\ \mathbf{G}& =\left[\begin{array}{c}g({\alpha }_0,{\beta }_0)\\ g({\alpha }_0,{\beta }_1)\\ ⋮\\ g({\alpha }_0,{\beta }_N)\\ g({\alpha }_1,{\beta }_0)\\ g({\alpha }_1,{\beta }_1)\\ ⋮\\ g({\alpha }_1,{\beta }_N)\\ ⋮\\ g({\alpha }_N,{\beta }_0)\\ g({\alpha }_N,{\beta }_1)\\ ⋮\\ g({\alpha }_N,{\beta }_N)\end{array}\right].\end{array}$Hence, the proof is completed.

Lemma 3.1

The condition (2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) ) and the condition (3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ) by using the collocation point (20(20) ${\alpha }_r=\frac{K}{N}r,{\beta }_s=\frac{M}{N}s,r,s=0,1,\dots ,N.$(20) ) are expressed in matrix form as follows: (25) $\mathbf{U}{\overline{\mathbf{A}}}_N=\lambda ,\overline{\mathbf{U}}{\overline{\mathbf{A}}}_N=\gamma ,\mathbf{V}{\overline{\mathbf{A}}}_N=\mu ,\overline{\mathbf{V}}{\overline{\mathbf{A}}}_N=\psi .$(25)

Proof.

To begin with proof, we use the matrix relation in Lemma 2.4 and the collocation point (20(20) ${\alpha }_r=\frac{K}{N}r,{\beta }_s=\frac{M}{N}s,r,s=0,1,\dots ,N.$(20) ). Namely, by substituting the collocation points (20(20) ${\alpha }_r=\frac{K}{N}r,{\beta }_s=\frac{M}{N}s,r,s=0,1,\dots ,N.$(20) ) instead of α and β in the Equations (18(18) $\begin{array}{rl}& z_{2N}(\alpha ,0)={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N\\ & (z_{2N}(\alpha ,0){)}_{\beta }={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N.\end{array}$(18) )–(19(19) $\begin{array}{rl}& z_{2N}(0,\beta )={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N\\ & z_{2N}(K,\beta )={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.\end{array}$(19) ), we obtain (26) $\{\begin{array}{c}{z}_{2N}({\alpha }_0,0)={\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0({\alpha }_0)={\lambda }_0\\ z_{2N}({\alpha }_1,0)={\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0({\alpha }_1)={\lambda }_1\\ ⋮\\ z_{2N}({\alpha }_N,0)={\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0({\alpha }_N)={\lambda }_N\\ (z_{2N}({\alpha }_0,0){)}_{\beta }={\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1({\alpha }_0)={\gamma }_0\\ (z_{2N}({\alpha }_1,0){)}_{\beta }={\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1({\alpha }_1)={\gamma }_1\\ ⋮\\ (z_{2N}({\alpha }_N,0){)}_{\beta }={\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1({\alpha }_N)={\gamma }_N\end{array}$(26) and (27) $\{\begin{array}{c}{z}_{2N}(0,{\beta }_0)={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=h_0({\beta }_0)={\mu }_0\\ z_{2N}(0,{\beta }_1)={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=h_0({\beta }_1)={\mu }_1\\ ⋮\\ z_{2N}(0,{\beta }_N)={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=h_0({\beta }_N)={\mu }_N\\ z_{2N}(K,{\beta }_0)={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=h_1({\beta }_0)={\psi }_0\\ z_{2N}(K,{\beta }_1)={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=h_1({\beta }_1)={\psi }_1\\ ⋮\\ z_{2N}(K,{\beta }_N)={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=h_1({\beta }_N)={\psi }_N.\end{array}$(27) Briefly, we can write the Equations (26(26) $\{\begin{array}{c}{z}_{2N}({\alpha }_0,0)={\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0({\alpha }_0)={\lambda }_0\\ z_{2N}({\alpha }_1,0)={\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0({\alpha }_1)={\lambda }_1\\ ⋮\\ z_{2N}({\alpha }_N,0)={\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N(0){\overline{\mathbf{A}}}_N=f_0({\alpha }_N)={\lambda }_N\\ (z_{2N}({\alpha }_0,0){)}_{\beta }={\mathbf{Q}}_N({\alpha }_0){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1({\alpha }_0)={\gamma }_0\\ (z_{2N}({\alpha }_1,0){)}_{\beta }={\mathbf{Q}}_N({\alpha }_1){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1({\alpha }_1)={\gamma }_1\\ ⋮\\ (z_{2N}({\alpha }_N,0){)}_{\beta }={\mathbf{Q}}_N({\alpha }_N){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}}{\overline{\mathbf{A}}}_N=f_1({\alpha }_N)={\gamma }_N\end{array}$(26) ) and (27(27) $\{\begin{array}{c}{z}_{2N}(0,{\beta }_0)={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=h_0({\beta }_0)={\mu }_0\\ z_{2N}(0,{\beta }_1)={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=h_0({\beta }_1)={\mu }_1\\ ⋮\\ z_{2N}(0,{\beta }_N)={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=h_0({\beta }_N)={\mu }_N\\ z_{2N}(K,{\beta }_0)={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_0){\overline{\mathbf{A}}}_N=h_1({\beta }_0)={\psi }_0\\ z_{2N}(K,{\beta }_1)={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_1){\overline{\mathbf{A}}}_N=h_1({\beta }_1)={\psi }_1\\ ⋮\\ z_{2N}(K,{\beta }_N)={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_N){\overline{\mathbf{A}}}_N=h_1({\beta }_N)={\psi }_N.\end{array}$(27) ) as (28) $\mathbf{U}{\overline{\mathbf{A}}}_N=\lambda ,\overline{\mathbf{U}}{\overline{\mathbf{A}}}_N=\gamma ,\mathbf{V}{\overline{\mathbf{A}}}_N=\mu ,\overline{\mathbf{V}}{\overline{\mathbf{A}}}_N=\psi ,$(28) where $\begin{array}{rl}\lambda & ={\left[\begin{array}{cccc}{\lambda }_0& {\lambda }_1& \cdots & {\lambda }_N\end{array}\right]}^T,\gamma ={\left[\begin{array}{cccc}{\gamma }_0& {\gamma }_1& \cdots & {\gamma }_N\end{array}\right]}^T,\\ \mu & ={\left[\begin{array}{cccc}{\mu }_0& {\mu }_1& \cdots & {\mu }_N\end{array}\right]}^T,\psi ={\left[\begin{array}{cccc}{\psi }_0& {\psi }_1& \cdots & {\psi }_N\end{array}\right]}^T,\\ \mathbf{U}& ={\left[\begin{array}{cccc}{\mathbf{U}}_0& {\mathbf{U}}_1& \cdots & {\mathbf{U}}_N\end{array}\right]}^T,\overline{\mathbf{U}}={\left[\begin{array}{cccc}{\overline{\mathbf{U}}}_0& {\overline{\mathbf{U}}}_1& \cdots & {\overline{\mathbf{U}}}_N\end{array}\right]}^T,\\ \mathbf{V}& ={\left[\begin{array}{cccc}{\mathbf{V}}_0& {\mathbf{V}}_1& \cdots & {\mathbf{V}}_N\end{array}\right]}^T,\overline{\mathbf{V}}={\left[\begin{array}{cccc}{\overline{\mathbf{V}}}_0& {\overline{\mathbf{V}}}_1& \cdots & {\overline{\mathbf{V}}}_N\end{array}\right]}^T,\\ {\mathbf{U}}_i& ={\mathbf{Q}}_N({\alpha }_i){\overline{\mathbf{Q}}}_N(0),{\overline{\mathbf{U}}}_i={\mathbf{Q}}_N({\alpha }_i){\overline{\mathbf{Q}}}_N(0)\overline{\mathbf{H}},\\ {\mathbf{V}}_j& ={\mathbf{Q}}_N(0){\overline{\mathbf{Q}}}_N({\beta }_j),{\overline{\mathbf{V}}}_j={\mathbf{Q}}_N(K){\overline{\mathbf{Q}}}_N({\beta }_j).\end{array}$

Finally, we complete the proof of theorem.

Theorem 3.2

We assume that we seek the solution of problem (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ) in form (4(4) $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{i=0}^N\sum _{j=0}^N{a}_{i,j}{Q}_{i,j}(\alpha ,\beta ),Q_{i,j}(\alpha ,\beta )=Q_i(\alpha )Q_j(\beta ).\end{array}$(4) ). Then, it is formed the augmented matrix [29] (29) $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}].$(29)

Proof.

The matrix systems (21(21) $\mathbf{W}{\overline{\mathbf{A}}}_N=\mathbf{G}.$(21) ) and (25(25) $\mathbf{U}{\overline{\mathbf{A}}}_N=\lambda ,\overline{\mathbf{U}}{\overline{\mathbf{A}}}_N=\gamma ,\mathbf{V}{\overline{\mathbf{A}}}_N=\mu ,\overline{\mathbf{V}}{\overline{\mathbf{A}}}_N=\psi .$(25) ) are combined as a single system. We represent this new system as $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}]$. Namely, we obtain (30) $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}]=\left[\begin{array}{cc}\mathbf{W};& \mathbf{G}\\ \mathbf{U};& \lambda \\ \overline{\mathbf{U}};& \gamma \\ \mathbf{V};& \mu \\ \overline{\mathbf{V}};& \psi \end{array}\right].$(30) Thus, we obtain the desired result.

Corollary 3.1

The system in Theorem 3.2 is solved. Solution of system (29(29) $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}].$(29) ) is Pell–Lucas coefficient matrix ${\overline{\mathbf{A}}}_N$ in Equation (6(6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) ). Then, the obtained matrix ${\overline{\mathbf{A}}}_N$ is substituted in Equation (6(6) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N$(6) ). Finally, by computing the matrix multiplication, it is obtained Pell–Lucas polynomial solution.

4. Error analysis

The two theorems about error analysis for the presented method in Section 3 are given in this part. The first theorem concerns the upper bound of errors in method. The second theorem concerns an error estimation technique to estimate errors with the help of residual function.

Theorem 4.1

Upper Bound for Errors

Let's show $z(\alpha ,\beta )$ as exact solution and $z_{2N}(\alpha ,\beta )$ as Pell–Lucas polynomial solution (PLPS). Also $z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )$ shows Maclaurin expansion with $2N$th degree of $z(\alpha ,\beta )$ in domain $[0,K]\times [0,M]$. In that case, errors of PLPS $z_{2N}(\alpha ,\beta )$ for $0\le \alpha \le K,0\le \beta \le M$ are bounded by Yüzbaşı and Yıldırım [29] (31) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\le k_N\Vert {\widetilde{\stackrel{~}{\mathbf{A}}}}_N{\Vert }_{\mathrm{\infty }}-{\mathbf{q}}_N{\overline{\mathbf{q}}}_N\Vert {\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}\\ & +\frac{1}{(2N+1)!}{\Vert {(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2N+1}u(c_{\alpha },c_{\beta })\Vert }_{\mathrm{\infty }}\end{array}$(31) Here, $k_N=\Vert {\mathbf{X}}_N(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}$, ${\widetilde{\stackrel{~}{\mathbf{A}}}}_N$ shows coefficient matrix of $z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )$, ${\mathbf{q}}_N=\Vert {\mathbf{Q}}_N(\alpha ){\Vert }_{\mathrm{\infty }}$ and ${\overline{\mathbf{q}}}_N=\Vert {\overline{\mathbf{Q}}}_N(\beta ){\Vert }_{\mathrm{\infty }}$.

Proof.

First, let's write the expansion of $2N$th degree Maclaurin series of $z(\alpha ,\beta )$ more explicitly. The expansion of Maclaurin series is expressed as (32) $\begin{array}{rl}{z}_{2N}^{\mathrm{Mac}}(\alpha ,\beta )& =\sum _{n=0}^{2N}\frac{1}{n!}(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta }{)}^{n}z(\alpha ,\beta ){|}_{(\alpha ,\beta )=(0,0)}\\ & =\sum _{i=0}^{2N}\sum _{j=0}^{2N-i}\frac{1}{i!j!}\frac{{\mathrm{\partial }}^{i+j}}{\mathrm{\partial }{\alpha }^i\mathrm{\partial }{\beta }^j}z(\alpha ,\beta ){|}_{(\alpha ,\beta )=(0,0)}{\alpha }^i{\beta }^j\end{array}$(32) which becomes (33) $z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )={\mathbf{X}}_N(\alpha ,\beta ){\widetilde{\stackrel{~}{\mathbf{A}}}}_N$(33) in matrix form. Also, here ${\widetilde{\stackrel{~}{\mathbf{A}}}}_N=[{\tilde{\mathbf{A}}}_0{\tilde{\mathbf{A}}}_1\cdots {\tilde{\mathbf{A}}}_N{]}^T$, ${\tilde{\mathbf{A}}}_j=[{\tilde{a}}_{i,0}{\tilde{a}}_{i,1}\cdots {\tilde{a}}_{i,N}{]}^T$ and ${\mathbf{X}}_N(\alpha ,\beta )$ is matrix in dimensional $1\times (N+1{)}^2$ as follows: (34) $\begin{array}{rl}& {\mathbf{X}}_N(s,t)=[X_{0,0}(\alpha ,\beta )\cdots X_{0,N}(\alpha ,\beta )X_{1,0}(\alpha ,\beta )\\ & \cdots X_{1,N}(\alpha ,\beta )\cdots X_{N,0}(\alpha ,\beta )\cdots X_{N,N}(\alpha ,\beta )]\end{array}$(34) where $X_{m,n}(\alpha ,\beta )={\alpha }^m{\beta }^n$.

Second, according to Section 2, PLPS $z_{2N}(\alpha ,\beta )$ are expressed by (35) $z_{2N}(\alpha ,\beta )={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(35) As a next step, error $\Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}$ of PLPS can be written as (36) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )+z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & \le \Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & +\Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}.\end{array}$(36) Note that we are making use of triangle inequality and Maclaurin expansion $z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )$ here.

Then, let's write more clearly terms on right-side of inequality (36(36) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )+z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & \le \Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & +\Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}.\end{array}$(36) ). For this purpose, by using Equations (33(33) $z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )={\mathbf{X}}_N(\alpha ,\beta ){\widetilde{\stackrel{~}{\mathbf{A}}}}_N$(33) ) and (35(35) $z_{2N}(\alpha ,\beta )={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(35) ), it is obtained (37) $\begin{array}{rl}& \Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\Vert {\mathbf{X}}_N(\alpha ,\beta ){\widetilde{\stackrel{~}{\mathbf{A}}}}_N-{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}\\ & \le \Vert {\mathbf{X}}_N(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\Vert {\widetilde{\stackrel{~}{\mathbf{A}}}}_N{\Vert }_{\mathrm{\infty }}\\ & -\Vert {\mathbf{Q}}_N(\alpha ){\Vert }_{\mathrm{\infty }}\Vert {\overline{\mathbf{Q}}}_N(\beta ){\Vert }_{\mathrm{\infty }}\Vert {\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}.\end{array}$(37) Here, ${\mathbf{q}}_N$ and ${\overline{\mathbf{q}}}_N$ represent, respectively, $\Vert {\mathbf{Q}}_N(\alpha ){\Vert }_{\mathrm{\infty }}$ in $[0,K]$ and $\Vert {\overline{\mathbf{Q}}}_N(\beta ){\Vert }_{\mathrm{\infty }}$ in $[0,M]$. $\Vert {\mathbf{X}}_N(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}$ is (38) $\Vert {\mathbf{X}}_N(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}=k_N:=\{\begin{array}{ll}{K}^N\times M^N,& \mathrm{if}K>1\mathrm{and}M>1\\ M^N,& \mathrm{if}K\le 1\mathrm{and}M>1\\ 1,& \mathrm{if}K\le 1\mathrm{and}M\le 1\end{array}$(38)

in the domain $[0,K]\times [0,M]$.

Thus, we can express inequality (37(37) $\begin{array}{rl}& \Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\Vert {\mathbf{X}}_N(\alpha ,\beta ){\widetilde{\stackrel{~}{\mathbf{A}}}}_N-{\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}\\ & \le \Vert {\mathbf{X}}_N(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\Vert {\widetilde{\stackrel{~}{\mathbf{A}}}}_N{\Vert }_{\mathrm{\infty }}\\ & -\Vert {\mathbf{Q}}_N(\alpha ){\Vert }_{\mathrm{\infty }}\Vert {\overline{\mathbf{Q}}}_N(\beta ){\Vert }_{\mathrm{\infty }}\Vert {\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}.\end{array}$(37) ) as (39) $\Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\le k_N\Vert {\widetilde{\stackrel{~}{\mathbf{A}}}}_N{\Vert }_{\mathrm{\infty }}-{\mathbf{q}}_N{\overline{\mathbf{q}}}_N\Vert {\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}.$(39) We know that $2N$th degree Maclaurin series of $z(\alpha ,\beta )$ and its remainder term are respectively as follows: (40) $\begin{array}{rl}z(\alpha ,\beta )& =z(0,0)+\alpha z_{\alpha }+\beta z_{\beta }\\ & +\frac{1}{2!}{(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2}z(\alpha ,\beta )\\ & +\cdots +\frac{1}{(2n)!}{(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2n}z(\alpha ,\beta )\end{array}$(40) and (41) $\begin{array}{rl}& \frac{1}{(2N+1)!}{(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2N+1}u(c_{\alpha },c_{\beta }),\\ & 0\le \alpha \le K,0\le \beta \le M.\end{array}$(41)

As a result, we write term $\Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}$ in inequality (36(36) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )+z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & \le \Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & +\Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}.\end{array}$(36) ) as follows: (42) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\frac{1}{(2N+1)!}{\Vert {(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2N+1}u(c_{\alpha },c_{\beta })\Vert }_{\mathrm{\infty }},\\ & 0\le \alpha \le K,0\le \beta \le M.\end{array}$(42) By substituting Equations (39(39) $\Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\le k_N\Vert {\widetilde{\stackrel{~}{\mathbf{A}}}}_N{\Vert }_{\mathrm{\infty }}-{\mathbf{q}}_N{\overline{\mathbf{q}}}_N\Vert {\overline{\mathbf{A}}}_N{\Vert }_{\mathrm{\infty }}.$(39) ) and (42(42) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\frac{1}{(2N+1)!}{\Vert {(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2N+1}u(c_{\alpha },c_{\beta })\Vert }_{\mathrm{\infty }},\\ & 0\le \alpha \le K,0\le \beta \le M.\end{array}$(42) ) in inequality (36(36) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =\Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )+z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & \le \Vert z(\alpha ,\beta )-z_{2N}^{\mathrm{Mac}}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & +\Vert z_{2N}^{\mathrm{Mac}}(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}.\end{array}$(36) ), for $0\le \alpha \le K,0\le \beta \le M$ it is obtained (43) $\begin{array}{rl}& \Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\le k_N\Vert {\widetilde{\stackrel{~}{\mathbf{A}}}}_N{\Vert }_{\mathrm{\infty }}-{\mathbf{q}}_N{\overline{\mathbf{q}}}_N\Vert \overline{{\mathbf{A}}_N}{\Vert }_{\mathrm{\infty }}\\ & +\frac{1}{(2N+1)!}\times {\Vert {(\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+\beta \frac{\mathrm{\partial }}{\mathrm{\partial }\beta })}^{2N+1}u(c_{\alpha },c_{\beta })\Vert }_{\mathrm{\infty }}.\end{array}$(43) and thus the proof is completed.

Theorem 4.2

Estimating of Error

We suppose that $z(\alpha ,\beta )$ and $z_{2N}(\alpha ,\beta )$ are, respectively, exact solution and Pell–Lucas polynomial solution of problem (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ). Also, residual function $R_{2N}(\alpha ,\beta )$ is described by Yüzbaşı and Yıldırım [29] (44) $R_{2N}(\alpha ,\beta )=L[z_{2N}(\alpha ,\beta )]-g(\alpha ,\beta ).$(44) Accordingly, to estimate error $e_{2N}(\alpha ,\beta )$, the following error problem is obtained (45) $\{\begin{array}{l}(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+b(e_{2N}{)}_{\beta }(\alpha ,\beta )+c{e}_{2N}(\alpha ,\beta )\\ -(e_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )=-R_{2N}(\alpha ,\beta )\\ e_{2N}(\alpha ,0)=0\\ (e_{2N}{)}_{\beta }(\alpha ,0)=0\\ e_{2N}(0,\beta )=0\\ e_{2N}(K,\beta )=0.\end{array}$(45)

Proof.

Let $z(\alpha ,\beta )$ and $z_{2N}(\alpha ,\beta )$ be exact solution and PLPS of Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ), respectively. Therefore, Equation (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) ) is written in the form (46) $\begin{array}{rl}& L[z(\alpha ,\beta )]=g(\alpha ,\beta ),\\ & L[z(\alpha ,\beta )]=z_{\mathrm{\beta \beta }}(\alpha ,\beta )+b{z}_{\beta }(\alpha ,\beta )+\mathrm{cz}(\alpha ,\beta )-z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ).\end{array}$(46)

Now, since PLPS $z_{2N}(\alpha ,\beta )$ satisfies Equations (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ), we get (47) $\begin{array}{rl}L[z_{2N}(\alpha ,\beta )]& =(z_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+b(z_{2N}{)}_{\beta }(\alpha ,\beta )\\ & +c{z}_{2N}(\alpha ,\beta )-(z_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )\\ & =g(\alpha ,\beta )+R_{2N}(\alpha ,\beta )\end{array}$(47) and (48) $\begin{array}{ll}(z_{2N})(\alpha ,0)=f_0(\alpha ),& s\in [0,K]\\ (z_{2N}{)}_{\beta }(\alpha ,0)=f_1(\alpha ),& s\in [0,K]\\ (z_{2N})(0,\beta )=h_0(\beta ),& \beta \in [0,M]\\ (z_{2N})(K,\beta )=h_1(\beta ),& \beta \in [0,M].\end{array}$(48) Moreover, error function is defined by (49) $e_{2N}(\alpha ,\beta )=z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ).$(49) After all, we subtract Equations (47(47) $\begin{array}{rl}L[z_{2N}(\alpha ,\beta )]& =(z_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+b(z_{2N}{)}_{\beta }(\alpha ,\beta )\\ & +c{z}_{2N}(\alpha ,\beta )-(z_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )\\ & =g(\alpha ,\beta )+R_{2N}(\alpha ,\beta )\end{array}$(47) )–(48(48) $\begin{array}{ll}(z_{2N})(\alpha ,0)=f_0(\alpha ),& s\in [0,K]\\ (z_{2N}{)}_{\beta }(\alpha ,0)=f_1(\alpha ),& s\in [0,K]\\ (z_{2N})(0,\beta )=h_0(\beta ),& \beta \in [0,M]\\ (z_{2N})(K,\beta )=h_1(\beta ),& \beta \in [0,M].\end{array}$(48) ) from Equations (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\beta }^2}+b\frac{\mathrm{\partial }z}{\mathrm{\partial }\beta }+\mathrm{cz}=\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{\alpha }^2}+g(\alpha ,\beta ),0\le \alpha \le K,0\le \beta \le M\end{array}$(1) )–(2(2) $z(\alpha ,0)=f_0(\alpha ),z_{\beta }(\alpha ,0)=f_1(\alpha ),0\le \alpha \le K,$(2) )–(3(3) $z(0,\beta )=h_0(\beta ),z(K,\beta )=h_1(\beta ),0\le \beta \le M.$(3) ). Hence, it becomes (50) $\{\begin{array}{l}(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+b(e_{2N}{)}_{\beta }(\alpha ,\beta )+c{e}_{2N}(\alpha ,\beta )\\ -(e_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )=-R_{2N}(\alpha ,\beta ),\\ e_{2N}(\alpha ,0)=f_0(\alpha ),\\ (e_{2N}{)}_{\beta }(\alpha ,0)=f_1(\alpha ),e_{2N}(0,\beta )=h_0(\beta ),\\ e_{2N}(K,\beta )=h_1(\beta ).\end{array}$(50) Thus, (45(45) $\{\begin{array}{l}(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+b(e_{2N}{)}_{\beta }(\alpha ,\beta )+c{e}_{2N}(\alpha ,\beta )\\ -(e_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )=-R_{2N}(\alpha ,\beta )\\ e_{2N}(\alpha ,0)=0\\ (e_{2N}{)}_{\beta }(\alpha ,0)=0\\ e_{2N}(0,\beta )=0\\ e_{2N}(K,\beta )=0.\end{array}$(45) ) is obtained.

Corollary 4.1

By solving the error problem (50(50) $\{\begin{array}{l}(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+b(e_{2N}{)}_{\beta }(\alpha ,\beta )+c{e}_{2N}(\alpha ,\beta )\\ -(e_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )=-R_{2N}(\alpha ,\beta ),\\ e_{2N}(\alpha ,0)=f_0(\alpha ),\\ (e_{2N}{)}_{\beta }(\alpha ,0)=f_1(\alpha ),e_{2N}(0,\beta )=h_0(\beta ),\\ e_{2N}(K,\beta )=h_1(\beta ).\end{array}$(50) ), the estimated error function is computed by (51) $e_{2N,2M}(\alpha ,\beta )\cong \sum _{m=0}^N\sum _{n=0}^N{a}_{m,n}^{\ast }{Q}_{m,n}(\alpha ,\beta ).$(51) Hence, the errors can be estimated by using (51(51) $e_{2N,2M}(\alpha ,\beta )\cong \sum _{m=0}^N\sum _{n=0}^N{a}_{m,n}^{\ast }{Q}_{m,n}(\alpha ,\beta ).$(51) ) even if it is not known exact solution of problem.

Furthermore, it is calculated $L_2$ error and $L_{\mathrm{\infty }}$ error by using $\begin{array}{rl}{L}_2& =\Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_2\\ & ={({\int }_c^d{\int }_a^b(z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){)}^2\mathrm{d}\alpha \mathrm{d}\beta )}^{1/2}\end{array}$and $\begin{array}{rl}{L}_{\mathrm{\infty }}& =\Vert z(\alpha ,\beta )-z_{2N}(\alpha ,\beta ){\Vert }_{\mathrm{\infty }}\\ & =max\{|z(\alpha ,\beta )-z_{2N}(\alpha ,\beta )|,a\le \alpha \le b,c\le \beta \le d\}.\end{array}$In Section 5, errors $L_2$ and $L_{\mathrm{\infty }}$ are used to make comparisons between the presented method and other methods.

5. Numerical applications

In this part, examples are made for methods presented in Sections 3 and 4. The numerical results obtained as a result of applications are explained on tables and supported visually with graphics. In addition, some comparisons are performed between the results of our method and the results of other methods in the literature. Let's note that numerical results are calculated by using MATLAB and graphics are created with the help of MATLAB. The definitions of some of variables in this section are given in Table 2.

Table 2. The representations of some variables in the Section 5.

Variable	Definition
$z(\alpha ,\beta )$	Exact solution
$z_{2N}(\alpha ,\beta )$	Approximate solution
$e_{2N}(\alpha ,\beta )$	Actual error
$e_{2N,2M}(\alpha ,\beta )$	The estimated error

Example 5.1

Let us consider the telegraph equation [13,30] (52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) with initial conditions (53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) and boundary conditions (54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) Here, the exact solution is $z(\alpha ,\beta )=e^{-\beta }\sin (\alpha ).$

Now, we investigate approximate solution of problem (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) )–(53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ) as follows: $\begin{array}{rl}{z}_{2N}(\alpha ,\beta )& \cong \sum _{m=0}^N\sum _{n=0}^N{a}_{m,n}{Q}_{m,n}(\alpha ,\beta ),\\ Q_{m,n}(\alpha ,\beta )& =Q_m(\alpha )Q_n(\beta )\end{array}$or in matrix form according to Lemma 2.1 it can be expressed by (55) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(55) According to Section 3 for N = 3, we write set of collocation points as (56) $\begin{array}{rl}& \{{\alpha }_0=0,{\alpha }_1=\frac{1}{3},{\alpha }_2=\frac{2}{3},{\alpha }_3=1\},\\ & \{{\beta }_0=0,{\beta }_1=\frac{1}{3},{\beta }_2=\frac{2}{3},{\beta }_3=1\}.\end{array}$(56) Applying method in Section 3, we obtain basic matrix equation as (57) $\mathbf{W}{\overline{\mathbf{A}}}_N=\mathbf{G}$(57) where (58) $\begin{array}{rl}& \mathbf{W}=\left[\begin{array}{c}{\mathbf{W}}_0\\ {\mathbf{W}}_1\\ ⋮\\ {\mathbf{W}}_N\end{array}\right],{\mathbf{W}}_i=\left[\begin{array}{c}\mathbf{W}({\alpha }_i,{\beta }_0)\\ \mathbf{W}({\alpha }_i,{\beta }_1)\\ ⋮\\ \mathbf{W}({\alpha }_i,{\beta }_N)\end{array}\right],\\ & \mathbf{G}=\left[\begin{array}{c}{\mathbf{G}}_0\\ {\mathbf{G}}_1\\ ⋮\\ {\mathbf{G}}_N\end{array}\right],{\mathbf{G}}_j=\left[\begin{array}{c}g({\alpha }_j,{\beta }_0)\\ g({\alpha }_j,{\beta }_1)\\ ⋮\\ g({\alpha }_j,{\beta }_N)\end{array}\right],\\ & \mathbf{W}({\alpha }_i,{\beta }_j)={\mathbf{Q}}_N({\alpha }_i){\overline{\mathbf{Q}}}_N({\beta }_j){\overline{\mathbf{H}}}^2+4{\mathbf{Q}}_N({\alpha }_i){\overline{\mathbf{Q}}}_N({\beta }_j)\overline{\mathbf{H}}\\ & +2{\mathbf{Q}}_N({\alpha }_i){\overline{\mathbf{Q}}}_N({\beta }_j)-{\mathbf{Q}}_N({\alpha }_i){\mathbf{H}}^2{\overline{\mathbf{Q}}}_N({\beta }_j),\\ & {\overline{\mathbf{Q}}}_N({\beta }_j)=[\begin{array}{cccc}2& 2{\beta }_j& 4{\beta }_j^2+2& 8{\beta }_j^3+6{\beta }_j\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\\ & \begin{array}{cccc}0& 0& 0& 0\\ 2& 2{\beta }_j& 4{\beta }_j^2+2& 8{\beta }_j^3+6{\beta }_j\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\\ & \begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 2& 2{\beta }_j& 4{\beta }_j^2+2& 8{\beta }_j^3+6{\beta }_j\\ 0& 0& 0& 0\end{array}\\ & \begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 2& 2{\beta }_j& 4{\beta }_j^2+2& 8{\beta }_j^3+6{\beta }_j\end{array}],\\ & \mathbf{H}=\left[\begin{array}{cccc}0& 1& 0& -3\\ 0& 0& 4& 0\\ 0& 0& 0& 6\\ 0& 0& 0& 0\end{array}\right],{\mathbf{Q}}_N({\alpha }_i)=\left[\begin{array}{c}2\\ 2{\alpha }_i\\ 4{\alpha }_i^2+2\\ 8{\alpha }_i^3+6{\alpha }_i\end{array}\right],\\ & \overline{\mathbf{H}}=\left[\begin{array}{cccc}\mathbf{H}& 0& 0& 0\\ 0& \mathbf{H}& 0& 0\\ 0& 0& \mathbf{H}& 0\\ 0& 0& 0& \mathbf{H}\end{array}\right],\\ & {\overline{\mathbf{A}}}_N=\left[\begin{array}{cccc}{\mathbf{A}}_0& {\mathbf{A}}_1& {\mathbf{A}}_2& {\mathbf{A}}_3\end{array}\right],\\ & {\mathbf{A}}_i=\left[\begin{array}{cccc}{a}_{i,0}& a_{i,1}& a_{i,2}& a_{i,3}\end{array}\right].\end{array}$(58) According to Lemma 2.4, matrix form of conditions (53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ) is also created.

Then, system (57(57) $\mathbf{W}{\overline{\mathbf{A}}}_N=\mathbf{G}$(57) ) and matrix form of conditions (53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ) are written as a single system called $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}]$. Note that dimension of system $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}]$ is $4((N+1{)}^2\times (N+1{)}^2)$. By solving system $[\tilde{\mathbf{W}};\tilde{\mathbf{G}}]$, we calculate coefficient matrix as (59) $\begin{array}{rl}{\overline{\mathbf{A}}}_N& =[21/8093257/49119-35/364227\\ & -25/36133330/1559-525/2062481/6923\\ & -83/103788/84321-274/47027\\ & -9/4274111/103258\\ & -39/5572113/11113-22/884711/82153{]}^T.\end{array}$(59)

Finally, we compute approximate solution by replacing the calculated coefficient matrix ${\overline{\mathbf{A}}}_N$ in solution form (55(55) $[z_{2N}(\alpha ,\beta )]={\mathbf{Q}}_N(\alpha ){\overline{\mathbf{Q}}}_N(\beta ){\overline{\mathbf{A}}}_N.$(55) ). Thus, method in Section 3 is applied. The error function is calculated by (60) $e_{2N}(\alpha ,\beta )=z(\alpha ,\beta )-z_{2N}(\alpha ,\beta )$(60) and with the help of this function in (60(60) $e_{2N}(\alpha ,\beta )=z(\alpha ,\beta )-z_{2N}(\alpha ,\beta )$(60) ), results of method can be interpreted. On the other hand, to apply method in Theorem (4.2), we define residual function as follows: (61) $R_{2N}(\alpha ,\beta )=L[z_{2N}(\alpha ,\beta )]-g(\alpha ,\beta ).$(61) The obtained Pell–Lucas polynomial solutions $z_{2N}(\alpha ,\beta )$ satisfy Equation (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) ) and conditions (53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ). From here, by using residual function in Equation (61(61) $R_{2N}(\alpha ,\beta )=L[z_{2N}(\alpha ,\beta )]-g(\alpha ,\beta ).$(61) ), it can be written as (62) $L[e_{2N}(\alpha ,\beta )]=L[z(\alpha ,\beta )]-L[z_{2N}(\alpha ,\beta )]=-R_{2N}(\alpha ,\beta )$(62) and (63) $\begin{array}{rl}& e_{2N}(\alpha ,0)=0,\\ & (e_{2N}{)}_{\beta }(\alpha ,0)=0,\\ & e_{2N}(1,\beta )=0,\\ & e_{2N}(0,\beta )=0.\end{array}$(63) That is, error problem is determined as (64) $\{\begin{array}{l}(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+4(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+2e_{2N}(\alpha ,\beta )\\ -(e_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )=-R_{2N}(\alpha ,\beta )\\ e_{2N}(\alpha ,0)=0,\\ (e_{2N}{)}_{\beta }(\alpha ,0)=0,\\ e_{2N}(1,\beta )=0,\\ e_{2N}(0,\beta )=0.\end{array}$(64) The solution of this error problem (64(64) $\{\begin{array}{l}(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+4(e_{2N}{)}_{\mathrm{\beta \beta }}(\alpha ,\beta )+2e_{2N}(\alpha ,\beta )\\ -(e_{2N}{)}_{\mathrm{\alpha \alpha }}(\alpha ,\beta )=-R_{2N}(\alpha ,\beta )\\ e_{2N}(\alpha ,0)=0,\\ (e_{2N}{)}_{\beta }(\alpha ,0)=0,\\ e_{2N}(1,\beta )=0,\\ e_{2N}(0,\beta )=0.\end{array}$(64) ) gives the estimated error function $e_{2N,2M}(\alpha ,\beta )$.

The tables and the graphs for problem (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) )–(53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ) show the following:

• The exact solution and Pell–Lucas polynomial solution for N = 7 are depicted, respectively, in Figures 1(a) and (b).

• Figure 2(a) shows the actual error $e_{2N}(\alpha ,\beta )$ for N = 7 while Figure 2(b) shows the estimated error $e_{2N,2M}(\alpha ,\beta )$ for $(N,M)=(7,8)$.

• Table 3 compares the actual errors for $N=5,N=7,N=8$ and the estimated errors for $(N,M)=(5,6),(N,M)=(7,8),(N,M)=(8,9)$.

• Table 4 compares the results of the presented method with the results of the methods TM [30] and LMM [13] in the literature. Let's note that conditions $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$ are used in these two methods and in order to make a comparison, results are obtained according to conditions in these two methods in Table 4.

Figure 1. Plots of solutions (exact and Pell-Lucas polynomial) of problem (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) )–(53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ).

Figure 2. Plots of error functions (actual and estimation) of problem (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) )–(53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ).

Table 3. Comparison of the actual errors for N = 5, N = 7, N = 8 and the estimated errors for $(N,M)=(5,6),(N,M)=(7,8),(N,M)=(8,9)$ for problem (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) )–(53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ).

	$e_{2N}(\alpha ,\beta )$	$e_{2N,2M}(\alpha ,\beta )$	$e_{2N}(\alpha ,\beta )$	$e_{2N,2M}(\alpha ,\beta )$	$e_{2N}(\alpha ,\beta )$	$e_{2N,2M}(\alpha ,\beta )$
$({\alpha }_i,{\beta }_i)$	N = 5	$(N,M)=(5,6)$	N = 7	$(N,M)=(7,8)$	N = 8	$(N,M)=(8,9)$
(0.1,0.1)	2.3861e−005	1.7787e−006	2.7544e−009	3.7871e−010	8.1124e−011	1.7881e−011
(0.2,0.2)	9.9515e−006	1.5279e−006	3.5651e−009	2.9252e−010	3.1929e−011	4.7241e−011
(0.3,0.3)	2.2910e−006	8.6731e−007	2.6156e−009	1.7031e−009	1.2133e−010	5.5359e−011
(0.4,0.4)	5.4566e−006	1.3987e−006	1.8732e−009	2.3013e−009	1.3252e−010	1.6336e−010
(0.5,0.5)	1.2024e−005	1.9720e−006	1.7353e−009	1.5669e−010	3.5916e−011	3.0813e−013
(0.6,0.6)	1.0770e−005	1.1663e−006	3.9958e−009	1.9776e−009	4.0594e−010	9.8569e−011
(0.7,0.7)	4.3108e−006	3.2156e−007	5.1365e−009	2.8250e−009	2.8691e−010	9.0418e−011
(0.8,0.8)	2.6002e−005	6.6829e−007	4.1188e−010	2.0440e−009	7.8652e−010	2.5852e−010
(0.9,0.9)	2.7589e−005	5.0380e−008	8.8956e−009	3.4017e−009	6.0005e−010	3.4386e−010

Table 4. Comparison of absolute errors (actual and estimation) with other methods for problem (52(52) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+4z_{\beta }(\alpha ,\beta )+2z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(52) )–(53(53) $z(\alpha ,0)=\sin (\alpha ),z_{\beta }(\alpha ,0)=-\sin (\alpha )$(53) )–(54(54) $z(0,\beta )=0,z(1,\beta )=e^{-\beta }\sin (1).$(54) ).

	TM [30]		PM	PM	PM	PM
$({\alpha }_i,{\beta }_i)$	N = 7	LMM [13]	N = 7	$(N,M)=(7,8)$	N = 8	$(N,M)=(8,9)$
(0.2,0.2)	4.68e−10	2.00e−10	3.2063e−010	5.1923e−010	2.4982e−010	1.5430e−011
(0.3,0.3)	2.04e−8	1.00e−10	4.7356e−009	1.0932e−009	3.6164e−011	1.3110e−011
(0.4,0.4)	3.07e−7	4.00e−9	1.6891e−009	1.8774e−010	2.4562e−010	3.4498e−011
(0.5,0.5)	2.57e−6	1.11e−9	2.6548e−009	8.4714e−010	7.7205e−011	5.9681e−012
(0.7,0.7)	6.68e−5	3.36e−8	4.3872e−006	1.7619e−007	5.9320e−009	5.6671e−008
(0.8,0.8)	2.480e−4	8.32e−7	1.8440e−005	3.8953e−007	3.8880e−007	3.4063e−007
(0.9,0.9)	7.96e−4	2.80e−6	5.6926e−005	4.9865e−006	2.5034e−006	1.5041e−006

According to results obtained from tables and graphics, the following results can be said:

• The higher the value of N, the closer to exact solution results are obtained. That is, if the value of N is selected big enough, the made error is reduced.

• The results obtained from error estimation method are similar to results obtained from actual errors. This reveals the success of the error estimation method.

• The results of the suggested method have fewer errors at more points compared to the available literature works.

• In summary, it is seen that methods in Sections 3–4 are effective and more successful results are obtained compared to other methods.

Example 5.2

Our next example is HPDE [14,27,76] (65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) with initial conditions (66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) and boundary conditions (67) $z(0,\beta )=0,z(1,\beta )=0.$(67) Here, $g(\alpha ,\beta )=(\alpha -{\alpha }^2)e^{-\beta }(2-2\beta +{\beta }^2)+2{\beta }^2{e}^{-\beta }$and the exact solution of (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) ) is $z(\alpha ,\beta )=(\alpha -{\alpha }^2){\beta }^2{e}^{-\beta }.$

The Pell–Lucas polynomial solution (PLPS) of problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ) is searched in matrix form $z_{2N}(\alpha ,\beta )\cong \sum _{m=0}^N\sum _{n=0}^N{a}_{m,n}{Q}_{m,n}(\alpha ,\beta ).$The numerical results obtained from problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ) are described below:

• The exact solution and the PLPS for N = 7 are shown, respectively, in Figures 3(a) and (b).

• Figure 4(a) shows the actual error $e_{2N}(\alpha ,\beta )$ for N = 7 while Figure 4(b) shows the estimated error $e_{2N,2M}(\alpha ,\beta )$ for $(N,M)=(7,8)$.

• Table 5 compares the exact solution, the actual errors for $N=5,N=7,N=9$, the PLPS and the estimated errors for $(N,M)=(5,6),(N,M)=(7,8),(N,M)=(9,10)$.

• Table 6 compares the actual errors for the value of N = 7 with the results obtained from methods RWM [76], CTM [14] and BCM [27] in the literature.

• Table 7 compares the norms $\Vert L{\Vert }_2$ and $\Vert L{\Vert }_{\mathrm{\infty }}$ between PM and BCM [27].

• Table 7 presents the CPU time(s). This represents the time passed while calculating in MATLAB.

Figure 3. Comparison of solutions (exact and approximate) for problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ).

Figure 4. Comparison of error function (actual and estimation) for problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ).

Table 5. Comparison of results of the presented method for problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ).

	Exact Sol.	Approximate Sol.	Error (Actual)	Error (Estimation)
$({\alpha }_i,{\beta }_i)$		N = 5	N = 5	$(N,M)=(5,6)$
(0,0)	0	0.00010835	1.0835e−004	1.0707e−004
(0.1,0.1)	0.00081435	0.00087037	5.6011e−005	5.8098e−005
(0.2,0.2)	0.0052399	0.0052429	2.9988e−006	4.3986e−006
(0.3,0.3)	0.014001	0.014026	2.4491e−005	2.8214e−006
(0.4,0.4)	0.02574	0.025823	8.2359e−005	5.1105e−005
(0.5,0.5)	0.037908	0.038017	1.0892e−004	7.5034e−005
(0.6,0.6)	0.047417	0.04749	7.2511e−005	4.4213e−005
(0.7,0.7)	0.051099	0.051101	2.2389e−006	1.1446e−005
(0.8,0.8)	0.046011	0.045976	3.5376e−005	3.5338e−005
(0.9,0.9)	0.029639	0.02964	1.5631e−006	3.1980e−006
(1,1)	0	5.7732e−05	5.7732e−005	5.9214e−005
	Error (Actual )	Error (Estimation)	Error (Actual )	Error (Estimation)
$({\alpha }_i,{\beta }_i)$	N = 7	$(N,M)=(7,8)$	N = 9	$(N,M)=(9,10)$
(0,0)	5.1149e−008	4.8539e−008	1.1873e−009	2.0444e−009
(0.1,0.1)	2.2866e−007	2.4259e−007	1.1257e−009	1.0802e−009
(0.2,0.2)	8.6103e−008	4.8237e−008	1.2237e−009	1.3751e−009
(0.3,0.3)	1.3002e−007	5.0805e−008	1.6090e−009	1.8142e−009
(0.4,0.4)	7.5958e−008	1.7826e−007	5.3173e−010	8.3557e−010
(0.5,0.5)	2.5588e−009	8.7722e−008	7.6891e−010	1.2076e−009
(0.6,0.6)	3.0364e−007	2.3091e−007	2.2659e−009	1.7565e−009
(0.7,0.7)	2.1389e−007	1.3050e−007	5.2873e−010	6.4856e−010
(0.8,0.8)	2.8353e−007	3.2177e−007	2.1988e−009	8.8163e−010
(0.9,0.9)	1.7216e−008	2.8317e−008	4.2334e−010	2.3741e−009
(1,1)	3.0322e−008	2.7231e−008	3.5662e−011	9.0589e−011

Table 6. Comparison of errors between the presented method and other methods for problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ) for $u(s_i,1)$.

	RWM [76]	CTM [14]	BCM [27]	PM
${\alpha }_i$	n = 6, m = 9	n = m = 7	N = 7	N = 7
0.125	1.2e−6	5.8e−8	1.25e−7	5.1933e−07
0.250	1.9e−6	1.0e−7	1.06e−8	9.7951e−08
0.375	2.4e−6	1.2e−7	3.72e−7	8.5250e−08
0.500	2.6e−6	1.3e−7	6.21e−7	9.5191e−08
0.625	2.4e−6	1.2e−7	3.71e−7	8.5250e−08
0.750	1.9e−6	1.0e−7	1.08e−8	9.7951e−08
0.875	1.2e−6	5.8e−7	1.25e−7	5.1933e−07

Table 7. Comparison of norms ($\Vert L{\Vert }_2$, $\Vert L{\Vert }_{\mathrm{\infty }}$) for problem (65(65) $\begin{array}{rl}& z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=g(\alpha ,\beta )+z_{\mathrm{\alpha \alpha }}(\alpha ,\beta ),\\ & 0\le \alpha \le 1,0\le \beta \le 1\end{array}$(65) )–(66(66) $z(\alpha ,0)=0,z_{\beta }(\alpha ,0)=0$(66) )–(67(67) $z(0,\beta )=0,z(1,\beta )=0.$(67) ).

	5	7	9	5	7	9
N	for BCM [27]	for BCM [27]	for BCM [27]	for PM	for PM	for PM
$\Vert L{\Vert }_2$	1.5000e−04	6.4000e−7	1.99000e−9	1e−04	6e−07	8e−09
$\Vert L{\Vert }_{\mathrm{\infty }}$	3.9000e−4	1.4000e−6	4.1000e−9	1e−04	6e−07	3e−09
CPU time(s)				0.03125	0.046875	0.359375

According to tables and graphs, the following comments can be made:

• As the value of N raises, the error in the method decreases. According to the obtained results, when value N = 9 is selected, satisfactory results are achieved.

• The fact that the estimated error function $e_{2N,2M}(\alpha ,\beta )$ is resemblance to the actual error function $e_{2N}(\alpha ,\beta )$ reveals success of error estimation method.

• More appropriate results are found by using the presented method for both actual errors and $\Vert L{\Vert }_2$ and $\Vert L{\Vert }_{\mathrm{\infty }}$ errors.

• According to Table 7, we can obtained the numerical results in a short time. This is an advantage of the presented method.

• The methods in Sections 3 and 4 can be said to be applicable.

Example 5.3

Finally, let's solve HPDE (68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) with the mixed conditions (69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) and the boundary conditions (70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) The problem has the exact solution $z(\alpha ,\beta )=e^{\alpha -\beta }.$

We consider the PLPS of problem (68(68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) )–(69(69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) )–(70(70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) ) in form $\begin{array}{rl}& z_{2N}(\alpha ,\beta )\cong \sum _{m=0}^N\sum _{n=0}^N{a}_{m,n}{Q}_{m,n}(\alpha ,\beta ),\\ & Q_{m,n}(\alpha ,\beta )=Q_m(\alpha )Q_n(\beta ).\end{array}$Figures 5(a) and (b) show, respectively, the exact solution and PLPS of problem (68(68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) )–(69(69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) )–(70(70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) ). Figure 6(a) shows the actual error $e_{2N}(\alpha ,\beta )$ for N = 5 while Figure 6(b) shows the estimated error $e_{2N,2M}(\alpha ,\beta )$ for $(N,M)=(5,6)$. Table 8 presents the actual error functions for many values of N and the estimated error functions for many values of $(N,M)$ to problem (68(68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) )–(69(69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) )–(70(70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) ). According to all these results, it is observed that the made error  decreases as values of N raise. It can be said that the error estimation method is quite influential. That is, the results obtained from the presented methods are quite successful.

Figure 5. Comparison of solutions (exact and approximate) for problem (68(68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) )–(69(69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) )–(70(70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) ).

Figure 6. Comparison of error functions (actual and estimation) of problem (68(68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) )–(69(69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) )–(70(70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) ).

Table 8. The solutions (exact and approximate) and errors (actual and estimation) of problem (68(68) $z_{\mathrm{\beta \beta }}(\alpha ,\beta )+z_{\beta }(\alpha ,\beta )+z(\alpha ,\beta )=z_{\mathrm{\alpha \alpha }}(\alpha ,\beta )$(68) )–(69(69) $z(\alpha ,0)=e^{\alpha },z_{\beta }(\alpha ,0)=-e^{\alpha }$(69) )–(70(70) $z(0,\beta )=e^{-\beta },u(1,\beta )=e^{4-\beta }.$(70) ).

		Approximate solution	Actual error	Estimation error
$({\alpha }_i,{\beta }_i)$	Exact solution	N = 5	N = 5	$(N,M)=(5,6)$
(0,0)	1	1.000019346556285	1.9347e−005	5.0242e−007
(0.1,0.1)	1	1.000009331894920	9.3319e−006	1.3225e−008
(0.2,0.2)	1	0.999990205671150	9.7943e−006	3.9905e−007
(0.3,0.3)	1	1.000001251019173	1.2510e−006	1.9397e−006
(0.4,0.4)	1	1.000017158477937	1.7158e−005	6.6843e−006
(0.5,0.5)	1	1.000011328570584	1.1329e−005	8.3942e−006
(0.6,0.6)	1	0.999985023382220	1.4977e−005	4.8325e−006
(0.7,0.7)	1	0.999968278807151	3.1721e−005	1.2550e−006
(0.8,0.8)	1	0.999994025676992	5.9743e−006	1.6356e−006
(0.9,0.9)	1	1.000045040696352	4.5041e−005	1.0298e−006
(1,1)	1	0.999974721563533	2.5278e−005	6.1766e−007
	Actual error	Estimation error	Actual error	Estimation error
$({\alpha }_i,{\beta }_i)$	N = 7	$(N,M)=(7,8)$	N = 9	$(N,M)=(9,10)$
(0,0)	1.6895e−008	1.4391e−009	4.5614e−009	1.9612e−009
(0.1,0.1)	9.5063e−008	3.0093e−009	3.5769e−011	6.9660e−010
(0.2,0.2)	3.8182e−008	4.2036e−009	4.8219e−010	4.3978e−010
(0.3,0.3)	4.0027e−008	9.1975e−009	3.3301e−009	1.6844e−009
(0.4,0.4)	5.3743e−008	8.3174e−009	2.6448e−009	5.1903e−010
(0.5,0.5)	1.6156e−008	5.4707e−009	1.5127e−009	8.4217e−010
(0.6,0.6)	1.0152e−007	6.6101e−009	1.7501e−009	9.3675e−010
(0.7,0.7)	4.7013e−008	6.7747e−009	2.4936e−009	5.2310e−010
(0.8,0.8)	1.4429e−007	1.7964e−010	3.8782e−009	1.3427e−009
(0.9,0.9)	5.5272e−008	8.9314e−010	2.5941e−009	2.0775e−009
(1,1)	4.2588e−008	9.2207e−011	3.3825e−010	7.1315e−011

6. Conclusion

In this paper, a collocation method is studied by using the PLPs for HTPDEs. Then, error analysis is performed for method. The three different examples are made for suggested methods. The outcomes of our methods are presented in tables and graphs. In addition, comparisons are made for these examples with the results of other methods available in the literature. The made comparisons are also shown in tables. Moreover, there are also the made comparisons for $\Vert L{\Vert }_2$ and $\Vert L{\Vert }_{\mathrm{\infty }}$ errors in this study. According to all the outcomes, it is said that the suggested methods offer more convenient outcomes than other methods. When the results of method are examined, it is observed that Pell–Lucas collocation method and error estimation method are quite successful. On the other hand, it is inferred that There is a decrease in the actual error and the estimated error with raise in N. However, if the number N is chosen too large, the matrix size also increases. This may also cause an increase in error when calculating in MATLAB. For this reason, the number N should be chosen large enough. From all the obtained outcomes, it is observed that methods are quite effective and successful. It should be noted that applications and graphics are obtained by using MATLAB. On the other hand, the numerical results are obtained in a short time. This is an advantage of the presented method. Furthermore, the presented method can be adapted to different type problems of partial differential equations after essential matrix forms are created.

Acknowledgments

The authors would like to thank the reviewers for all helpful comments to improve their manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).