Unsteady Casson fluid flow on MHD with an internal heat source

Abstract

The present paper investigates the heat transfer features of a viscous fluid in an unsteady incompressible Casson fluid flow through a porous medium with an internal heat source under the influence of a magnetic field. A Casson fluid passes through the given porous medium using Darcy model. The study involves simulating the flow using mathematical models based on conservation laws and similarity transformations. The resulting highly nonlinear coupled ordinary differential equations are solved using the Bernoulli wavelet numerical approach. The study utilizes graphical tools to illustrate the flow behaviour under different physical factors, including the effect of the internal heat source. The normalization of the flow behaviour by the magnetic field suggests that it could potentially be used to control various flows. The study finds that various physical parameters which affects the velocity and temperature profiles of the fluid.

KEYWORDS:

• Numerical Bernoulli wavelet
• internal heat source
• MHD
• Casson fluid

Nomenclature

$B$	=	magnetic field (Tesla)
$C_p$	=	specific heat (Joules per kilogram per degree Celsius (J/kg°C))
$\mathrm{Ec}$	=	Eckert number
$\mathrm{Hs}$	=	internal heat source
$\mathrm{Mg}$	=	MHD parameter
$\mathrm{Mp}$	=	porous medium parameter
$p$	=	pressure $\left(\mathrm{Pa}\right)$
Q*	=	coefficient of heat source $W/m^3$
$\mathrm{Pr}$	=	Prandtl number
$S$	=	squeeze number
$T$	=	temperature $\left(\text{Kelvin scale }K\right)$
u and v	=	velocities in x- and y-directions $\left(m/s\right)$
$x,y$	=	Cartesian coordinates

Greek Symbols

$\text{β}$	=	Casson fluid parameter
${\kappa }_T$	=	thermal diffusivity $\left(m^2/s\right)$
$K$	=	permeability $\left(m^2\right)$
$\rho$	=	density $\left(\mathrm{kg}/m^3\right)$
$\nu$	=	kinematic viscosity $\left(m^2/s\right)$
$\mu$	=	viscosity $\left(\text{kg}/\text{m}\text{s}\right)$
$\sigma$	=	electrical conductivity (Siemens per Metre $S/m$)

1. Introduction

Unsteady flow refers to the motion of a fluid that changes with time. In other words, it is the flow of a fluid that does not remain constant or steady over time. This type of flow can occur in a variety of situations, such as when a valve is suddenly opened or closed, when a pump is turned on or off, when there are changes in the flow rate or direction of the fluid. In unsteady flow, the velocity and pressure of the fluid vary with time and location within the fluid. This can result in complex and unpredictable behaviour, such as turbulence, vortices, and eddies. Chanson [1] provides an overview of unsteady flow in hydraulic structures and discusses its effects on structure design and performance. Garcia et al. [2] reviews the current state of knowledge of unsteady flow and sediment transport in rivers and estuaries, and highlights directions for further research. Ol and Buning [3] present a numerical study of unsteady flow over a pitching and plunging air foil, and investigates the effects of various parameters on the aerodynamic performance. Gallardo Bravo et al. [4] discussed the effects of unsteady flow on the performance of turbo machinery and presents numerical methods for simulating unsteady flow in these systems. Prospathopoulos and Hansen [5] reviews the current state of knowledge of unsteady flow in wind turbines, and highlights directions for further research to improve the efficiency and reliability of wind energy technology.

Internal heat source refers to the heat generated within a material or a system due to various physical and chemical processes, such as nuclear reactions [6], chemical reactions [7], and electrical resistance [8]. The presence of an internal heat source can significantly affect the temperature distribution and thermal behaviour of the system. The significance of internal heat source in the real world lies in its many practical applications [6,8]. Thus, understanding and modelling the thermal behaviour of systems with internal heat source is important for optimizing the design and operation of processes in these industries. Ciampi and Tuoni [9] are studied the temperature distribution in turbulent flow between parallel plates with heat flux and internal heat source. Sugavanam et al. [10] are considered conjugate heat transfer from a flush heat source on a conductive board in laminar channel flow.

Magnetohydrodynamics (MHD) is the study of the interaction between electrically conducting fluids and magnetic fields. In MHD flow, the presence of a magnetic field can induce an electric current in the fluid, which in turn generates a Lorentz force that affects the flow behaviour. MHD flow has many practical applications [11–16], including in power generation, materials processing, and space exploration. MHD generators [17] are used in some power plants to convert thermal energy into electricity. MHD pumps and mixers [18] are used in materials processing to control the flow behaviour of fluids. MHD thrusters [19] are used in space exploration to control the motion of space craft. Understanding and modelling the flow behaviour of MHD systems is therefore essential for optimizing the design and operation of processes in these industries. Recent research on MHD flow has focused on the development of new numerical methods and experimental techniques for studying the complex behaviour of MHD systems. For example, one study investigated the effects of Hall current on MHD flow in a channel with porous walls, and found that the Hall current can significantly alter the flow behaviour [20]. Another study developed a numerical model to simulate MHD flow in a liquid metal battery, and demonstrated the importance of accurate modelling of the magnetic field on the flow behaviour [21].

Fluid dynamics offers tools to study the evolution of planets, ocean tides, weather patterns, plate tectonics, and also blood circulation. Some of the important technological applications of fluid dynamics include rocket engines, wind turbines, oil pipelines, and air conditioning systems. In reality, many fluids exhibit non-Newtonian [22] behaviour, which can significantly affect the flow behaviour and heat transfer characteristics of the system. Casson fluid [23–30] is a type of non-Newtonian fluid that exhibits both shear thinning and yield stress behaviour. In other words, its viscosity decreases as the shear rate increases, and it requires a certain amount of stress to start flowing. The Casson fluid model is commonly used to describe the flow behaviour of complex fluids, such as food products [31] and biomedical fluids [32]. The significance of Casson fluid flow in the real world lies in its many practical applications. Therefore, understanding and modelling the flow behaviour of Casson fluids is essential for optimizing the design and operation of processes in these industries. Recently, researchers has focused on the understanding and modelling of Casson fluid flow in various applications. Chamkha et al. [33] investigated the effects of magnetic field on Casson fluid flow in a channel with porous walls, and found that the magnetic field can significantly alter the flow behaviour. Li and Zhang [34] developed a numerical model to simulate Casson fluid flow in a microchannel, and demonstrated the importance of accurate modelling of the wall effects on the flow behaviour.

The combination of unsteady heat transfer in MHD Casson fluid flow with an internal heat source are contributed to the development of more efficient and effective engineering designs and processes. Unsteady Casson fluid flow with MHD and internal heat source is a complex flow phenomenon that has been studied by many researchers in recent years. A literature survey on this topic reveals that several authors have investigated the effects of different parameters on the flow behaviour and heat transfer characteristics of this system. Kumar et al. [35] investigated the effects of internal heat generation on unsteady Casson flow in a square cavity. The authors found that the internal heat source can significantly enhance the heat transfer rate and affect the flow behaviour. Rashidi et al. [36]analysed the unsteady Casson flow with MHD and an internal heat source in a porous medium. The authors found that the magnetic field and the internal heat generation can significantly affect the flow behaviour and heat transfer characteristics. Srinivasacharya and Gireesha [37] investigated the unsteady MHD Casson fluid flow with heat source and thermal radiation. They found that increasing the radiation parameter and the magnetic field strength can significantly reduce the heat transfer rate and increase the skin friction coefficient. Obaidat et al. [38] studied the unsteady MHD free convective flow of Casson fluid with internal heat source and chemical reaction. They found that increasing the chemical reaction parameter can significantly increase the heat transfer rate and reduce the velocity of the fluid. Muthuraj and Kandasamy [39] analysed the effects of chemical reaction on unsteady MHD Casson fluid flow with heat source and thermal radiation through porous media. They found that increasing the chemical reaction parameter can significantly increase the heat transfer rate and reduce the velocity of the fluid. They also found that the thermal radiation parameter has a significant effect on the heat transfer rate and the velocity of the fluid.

Most engineering problems and phenomena are inherently in the form of highly nonlinearity. Apart from a few problems, most of them do not have exact solution. Consequently, these nonlinear equations should be solved using different numerical methods. There are different numerical methods have recently introduced some ways to obtain numerical solution for such highly nonlinear fluid flow problems, such as the homotopy perturbation method, the RK-fourth order method, the finite-difference methods, the perturbation method, the differential transformation method, optimal homotopy analysis method and numerical wavelet methods.

Currently, in applied mathematics, the wavelet theory plays a crucial role in computer engineering, analysis of signals, image processing, and mathematical modelling [40–43]. Many mathematician’s contributions towards wavelet based numerical methods are as follows: Laguerre wavelets method for Lane–Emden equation [44], Hermite wavelet method [45], cardinal B-spline [46], Laguerre wavelets collocation method [47], new generalized Hermite wavelet method [48], Bernoulli wavelet method [49–51] are solved two heat transfer problems by using Hermite wavelet technique. Kumbinarasaiah and Raghunatha [52] explained a new method called the Hermite wavelet method to solve the highly nonlinear Jeffery–Hamel flow problem. Raghunatha and Kumbinarasaiah [53] studied the variation of nonlinear temperature in a permeable moving fin of the rectangular domain by using the Hermite wavelet method and the Differential transformation method. Vinod and Raghunatha [54] are investigated the flow and heat transfer for non-Newtonian viscoelastic fluid in an axisymmetric channel with a porous wall by using Hermite wavelet technique. Recently, Kumbinarasaiah et al. [55] are analysed Jeffery–Hamel flow and heat transfer in Eyring–Powell fluid through Bernoulli wavelet collocation method. Raghunatha et al. [56] are solve triple-diffusive convection in Jeffery–Hamel flow by application of Bernoulli wavelet method. Raghunatha and Vinod [57] are examined effects of heat transfer on MHD suction–injection model of viscous fluid flow through differential transformation and Bernoulli wavelet techniques.

To the best of our knowledge, the current problem with the numerical Bernoulli wavelet technique has not received any attention in the literature. The intent of the present study is to solve highly non-linear coupled differential equations with the help of the Bernoulli wavelet approach. The implementation of the method was thoroughly discussed in our earlier work Raghunatha and Vinod [57]. The analysis is conducted using the Bernoulli wavelet approach, the accuracy of this method is verified by correlating it with the other numerical methods, and the result is pretty satisfactory.

2. Mathematical formulation

The physical configuration is as shown in Figure 1. We consider an unsteady incompressible electrically conducting viscous Casson fluid flow through porous medium with an internal heat source between two parallel plates that are spaced apart by $h\left(t\right)=l(1-\mathrm{\alpha t}{)}^{0.5}$ where $l$ is the initial space between the plates. For $\alpha <0$ the two plates are separated and for $\alpha >0$ the two plates are squeezed until they touch $t={\alpha }^{-1}$. The symmetric nature of the flow is adopted. The viscous dissipation effect, the generation of heat due to friction caused by shear in the fluid flow, is engaged. This effect is moderately imperative in the case when the fluid is largely viscous or flowing at a high speed. The basic governing equations are [24–30,58]

Figure 1. Physical configuration.

The equation of continuity: (1) $\begin{array}{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0\end{array}$(1) The $x$-axis momentum equation: (2) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }x}\\ & +\nu \left(1+\frac{1}{\text{β}}\right)\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\right)-\frac{\sigma B^2}{\rho }u-\frac{\mu }{\mathrm{\rho K}}u\end{array}$(2)

The $y$-axis momentum equation: (3) $\begin{array}{rl}& \frac{\mathrm{\partial }v}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }y}\\ & +\nu \left(1+\frac{1}{\text{β}}\right)\left(\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{y}^2}\right)-\frac{\mu }{\mathrm{\rho K}}v\end{array}$(3) The energy equation (Noor et al. [59]) (4) $\begin{array}{rl}& \frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}={\kappa }_T\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\\ & +\frac{\nu }{C_p}\left(4{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\right)+\frac{Q^{\ast }}{\rho C_p}T\end{array}$(4) Where $u$ and $v$ are the velocities in the $x$ and $y$ directions, respectively, $p$ is the pressure, $\rho$ is the fluid density, $\mu$ is the fluid viscosity, $\nu$ is the kinematic viscosity, $\sigma$ is electrical conductivity, $\text{β}$ is the Casson fluid parameter, $T$ is the temperature, ${\kappa }_T$ is the thermal diffusivity, $K$ is the permeability of the medium, $C_p$ is the specific heat, $Q^{\ast }$ is the coefficient internal heat source and $B=\frac{B_0}{\sqrt{1-\mathrm{\alpha t}}}$ is the magnetic field.

The boundary conditions of our model are [24–30,58] (5) $\begin{array}{rl}& T=T_H,u=0,v=v_w=\frac{\mathrm{\partial }h}{\mathrm{\partial }t}\text{at}y=h\left(t\right)\text{and}\\ & v=\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=\frac{\mathrm{\partial }T}{\mathrm{\partial }y}=0\text{at}y=0\end{array}$(5) Eliminating the pressure term from Equations (2) and (3) by cross differentiation and simplifying the quantities using similarity transformations $u=\frac{\mathrm{\alpha x}}{2(1-\mathrm{\alpha t})}\frac{\mathrm{df}\left(\eta \right)}{\mathrm{d\eta }},v=\frac{-\mathrm{\alpha l}}{2{(1-\mathrm{\alpha t})}^{\frac{1}{2}}}f\left(\eta \right),\theta =\frac{T}{T_H},\eta =\frac{y}{l{(1-\mathrm{\alpha t})}^{\frac{1}{2}}}$ the resulting governing Equations (2) and (3) can be written in the form (6) $\begin{array}{rl}& \left(1+\frac{1}{\text{β}}\right)\frac{d^{4}f}{d{\eta }^4}-S\left(\eta \frac{d^{3}f}{d{\eta }^3}-f\frac{d^{3}f}{d{\eta }^3}+\frac{\mathrm{df}}{\mathrm{d\eta }}\frac{d^{2}f}{d{\eta }^2}+3\frac{d^{2}f}{d{\eta }^2}\right)\\ & =\left(\mathrm{Mp}+\mathrm{Mg}\right)\frac{d^{2}f}{d{\eta }^2}\end{array}$(6) Where $\mathrm{Mp}=\frac{l^2}{K}\left(1-\mathrm{\alpha t}\right)$ is the porous medium parameter, $\mathrm{Mg}=\frac{\sigma B_0^2{l}^2}{\mathrm{\rho \nu }}$ is the MHD parameter, $S=\frac{\alpha l^2}{2\nu }$ is the squeeze number. Substitute similarity transformations in Equation (4)(4) $\begin{array}{rl}& \frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}={\kappa }_T\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\\ & +\frac{\nu }{C_p}\left(4{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\right)+\frac{Q^{\ast }}{\rho C_p}T\end{array}$(4) and simplify we get (7) $\begin{array}{rl}& \frac{d^2\theta }{d{\eta }^2}+PrS\left(f\frac{\mathrm{d\theta }}{\mathrm{d\eta }}-\eta \frac{\mathrm{d\theta }}{\mathrm{d\eta }}\right)\\ & +Pr\mathrm{Ec}\left({\left(\frac{d^{2}f}{d{\eta }^2}\right)}^2+4{\delta }^2{\left(\frac{\mathrm{df}}{\mathrm{d\eta }}\right)}^2\right)\\ & +\mathrm{Hs}\theta =0.\end{array}$(7) Where $Pr=\frac{\mu C_p}{{\kappa }_T}$ is the Prandtl number, $\mathrm{Ec}=\frac{1}{C_p{T}_H}{\left(\frac{\mathrm{\alpha x}}{2\left(1-\mathrm{\alpha t}\right)}\right)}^2$ is the Eckert number, $\mathrm{Hs}=\frac{Q_0^{\ast }{l}^2}{{\kappa }_T}$ is the internal heat source.

The respective boundary conditions are (8) $\begin{array}{rl}& f\left(0\right)=0,\frac{\mathrm{df}\left(1\right)}{\mathrm{d\eta }}=0,\frac{d^{2}f\left(0\right)}{d{\eta }^2}=0,f\left(1\right)=1,\\ & \frac{\mathrm{d\theta }\left(0\right)}{\mathrm{d\eta }}=0,\theta \left(1\right)=1.\end{array}$(8) Very interesting point is calculating skin friction is useful in estimating not only total frictional drag exerted on a fluid but also convectional heat transfer rate in fluids. Nusselt number which indicates the magnitude of convectional heat transfer and skin friction coefficient which is a dimensionless frictional stress. The thermal Nusselt number $N{u}_T$ and skin friction $C_f$, can be defined as (Mustafa et al. [58]) (9a) $\begin{array}{rl}& N{u}_T=\frac{-l{\kappa }_T{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}_{y=h\left(t\right)}}{{\kappa }_T{T}_H},\text{and}{C}_f=\frac{\mu {\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}_{y=h\left(t\right)}}{\rho v_w^2}\end{array}$(9a) (9b) $\begin{array}{rl}& \sqrt{1-\mathrm{\alpha t}}N{u}_T=-{\theta }^{\mathrm{\prime }}\left(1\right),\text{and}{l}^2/x^2(1-\mathrm{\alpha t})R{e}_x{C}_f=f^{\mathrm{\prime \prime }}\left(1\right)\end{array}$(9b) where $R{e}_x=\frac{\alpha x{(1-\mathrm{\alpha t})}^{1.5}}{2\nu }$.

3. The fundamental notion of the Bernoulli wavelet technique

The definition of the Bernoulli wavelets is [49–51] $\begin{array}{r}{\xi }_{n.m}\left(x\right)=\{\begin{array}{ll}{2}^{\frac{k-1}{2}}\widetilde{b_m}\left(2^{k-1}x-\hat{n}\right),& {\displaystyle \frac{-\hat{n}}{2^{k-1}}}\le x<{\displaystyle \frac{\hat{n}+1}{2^{k-1}}}\\ 0,& \mathrm{Otherwise}\end{array}\end{array}$with $\begin{array}{r}\widetilde{b_m}\left(x\right)=\{\begin{array}{ll}1,& m=0\\ {\displaystyle \frac{1}{\sqrt{\frac{{\left(-1\right)}^{m-1}{\left(m!\right)}^2}{\left(2m\right)!}{a}_{2m}}}}{b}_m\left(x\right),& m>0\end{array}\end{array}$ where $m=0,1,2,\dots ,M-1,n=1,2,\dots ,2^{k-1}$.

The coefficient $\frac{1}{\sqrt{\frac{{\left(-1\right)}^{m-1}{\left(m!\right)}^2}{\left(2m\right)!}{a}_{2m}}}$ is for normality, the dilation parameter is $f=2^{-(k-1)}$ and the translation parameter $g=\hat{n}{2}^{-(k-1)}$. Here, $b_m\left(x\right)$ are the familiar Bernoulli polynomials of order $m$ which can be defined by, $b_m\left(x\right)=\sum _{i=0}^m\left(\begin{array}{c}m\\ i\end{array}\right)a_{m-i}{x}^i$, where $a_i,i=0,1,\dots ,m$ are Bernoulli numbers. These numbers are a sequence of signed rational numbers that arise in the series expansion of trigonometric functions and can be defined by the identity,$\frac{x}{e^x-1}=\sum _{i=0}^{\mathrm{\infty }}{a}_i\frac{x^i}{i!}.$

The first few Bernoulli numbers are $\begin{array}{rl}& a_0=1,a_1=\frac{-1}{2},a_2=\frac{1}{6},a_4=\frac{-1}{30},a_6=\frac{1}{42},\\ & a_8=\frac{-1}{30},a_{10}=\frac{5}{66},a_{12}=-\frac{691}{2730},a_{14}=\frac{7}{6},\\ & a_{16}=-\frac{3617}{510},a_{18}=\frac{43867}{798},\dots \end{array}$ with $a_{2i+1}=0,i=1,2,3,\dots$.

The first few Bernoulli polynomials are given by $\begin{array}{rl}& b_0=1,b_1=-\frac{1}{2}+x,b_2=\frac{1}{6}-x+x^2,\\ & b_3=\frac{x}{2}-\frac{3x}{2}+x^3,\end{array}$ $\begin{array}{rl}& b_4=-\frac{1}{30}+x^2-2x^3+x^4,\\ & b_5=-\frac{x}{6}+\frac{5x^3}{3}-\frac{5x^4}{2}+x^5,\end{array}$ $\begin{array}{rl}& b_6=\frac{1}{42}-\frac{x^2}{2}+\frac{5x^4}{2}-3x^5+x^6,\\ & b_7=\frac{x}{6}-\frac{7x^3}{6}+\frac{7x^5}{2}-\frac{7x^6}{2}+x^7,\end{array}$ $\begin{array}{rl}& b_8=-\frac{1}{30}+\frac{2x^2}{3}-\frac{7x^4}{3}+\frac{14x^6}{3}-4x^7+x^8,\\ & b_9=-\frac{3x}{10}+2x^3-\frac{21x^5}{5}+6x^7-\frac{9x^8}{2}+x^9,\end{array}$ $\begin{array}{rl}& b_{10}=\frac{5}{66}-\frac{3x^2}{2}+5x^4-7x^6+\frac{15x^8}{2}-5x^9+x^{10},\dots .\end{array}$

Bernoulli wavelet integral matrixes

The following are some of the Bernoulli wavelet basis at k =1 $\begin{array}{r}{\xi }_{1,0}=1,\end{array}$ $\begin{array}{r}{\xi }_{1,1}=\sqrt{3}\left(-1+2x\right),\end{array}$ $\begin{array}{r}{\xi }_{1,2}=\sqrt{5}\left(1+6\left(-1+x\right)x\right),\end{array}$ $\begin{array}{r}{\xi }_{1,3}=\sqrt{210}\left(-1+x\right)x\left(-1+2x\right),\end{array}$ $\begin{array}{r}{\xi }_{1,4}=10\sqrt{21}\left(-\frac{1}{30}+{\left(-1+x\right)}^2{x}^2\right),\end{array}$ $\begin{array}{r}{\xi }_{1,5}=\sqrt{\frac{462}{5}}x\left(-1+x^2\left(10+3x\left(-5+2x\right)\right)\right),\end{array}$ $\begin{array}{rl}{\xi }_{1,6}& =\sqrt{\frac{1430}{691}}(1+21{\left(-1+x\right)}^2\\ & \times x^2\left(-1+2\left(-1+x\right)x\right)),\end{array}$ $\begin{array}{r}{\xi }_{1,7}=2\sqrt{\frac{143}{7}}\left(x-7x^3+21x^5-21x^6+6x^7\right),\end{array}$ $\begin{array}{rl}{\xi }_{1,8}& =\sqrt{\frac{7293}{3617}}(-1+10(2x^2-7x^4\\ & +14x^6-12x^7+3x^8\left)\right),\end{array}$ $\begin{array}{rl}{\xi }_{1,9}& =\sqrt{\frac{1939938}{219335}}x(-3+20x^2-42x^4\\ & +5x^6\left(12+x\left(-9+2x\right)\right)),\end{array}$ $\begin{array}{rl}{\xi }_{1,10}& =22\sqrt{\frac{125970}{174611}}(\frac{5}{66}-\frac{3x^2}{2}+5x^4-7x^6\\ & +\frac{15x^8}{2}-5x^9+x^{10}),\end{array}$ $\begin{array}{rl}{\xi }_{1,11}& =2\sqrt{\frac{676039}{854513}}z(5-33x^2+66x^4-66x^6\\ & +55x^8-33x^9+6x^{10}),\end{array}$ $\begin{array}{rl}{\xi }_{1,12}& =182\sqrt{\frac{222870}{236364091}}(-\frac{691}{2730}+5x^2-\frac{33z^4}{2}\\ & +22x^6-\frac{33z^8}{2}+11x^{10}-6x^{11}+x^{12}),\end{array}$where, $\begin{array}{rl}{\xi }_{10}\left(x\right)& =\left[{\xi }_{1,0}\right(x),{\xi }_{1,1}(x),{\xi }_{1,2}(x),{\xi }_{1,3}(x),{\xi }_{1,4}(x),{\xi }_{1,5}(x),\\ & {\xi }_{1,6}\left(x\right),{\xi }_{1,7}\left(x\right),{\xi }_{1,8}\left(x\right),{\xi }_{1,9}\left(x\right){]}^T.\end{array}$Integrate the above first ten basis in respect of $x$ range from $0$ to $x$, then exhibit in the form of a linear combination of Bernoulli wavelet basis as, $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,0}\left(\text{x}\right)\text{dx}& =[\frac{1}{2}\frac{1}{2\sqrt{3}}00000\\ & 000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,1}\left(\text{x}\right)\text{dx}& =[-\frac{1}{2\sqrt{3}}0\frac{1}{2\sqrt{15}}0000\\ & 000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,2}\left(\text{x}\right)\text{dx}& =[000\frac{1}{\sqrt{42}}000\\ & 000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,3}\left(\text{x}\right)\text{dx}& =[\frac{\sqrt{7}}{2\sqrt{30}}000\frac{1}{2\sqrt{10}}0\\ & 0000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,4}\left(\text{x}\right)\text{dx}& =[00000\frac{\sqrt{5}}{3\sqrt{22}}\\ & 0000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,5}\left(\text{x}\right)\text{dx}& =[-\sqrt{\frac{11}{210}}00000\\ & \frac{\sqrt{691}}{10\sqrt{273}}000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\xi }_{1,6}\left(\text{x}\right)\text{dx}& =[\phantom{\frac{1}{2\sqrt{3}}}0000000\\ & \sqrt{\frac{35}{1382}}00\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\xi }_{1,7}\left(\text{x}\right)\text{dx}=[\frac{\sqrt{143}}{20\sqrt{7}}0000000\\ & \frac{\sqrt{3617}}{20\sqrt{357}}0\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\xi }_{1,8}\left(\text{x}\right)\text{dx}=[\phantom{\frac{1}{2\sqrt{3}}}000000000\\ & \frac{\sqrt{219335}}{3\sqrt{962122}}\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\xi }_{1,9}\left(\text{x}\right)\text{dx}=[-\sqrt{\frac{146965}{2895222}}00000\\ & 0000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right)[phone]\frac{\sqrt{\mathrm{}}}{\mathrm{}\sqrt{\mathrm{}}}{\xi }_{1,10}\left(\text{x}\right).\end{array}$

Hence, $\begin{array}{r}{\int }_0^x\xi \left(x\right)\mathrm{dx}={\xi }_{10}\left(x\right)+{\overline{{\xi }^{\mathrm{\prime }}}}_{10}\left(x\right),\end{array}$where $\begin{array}{rl}{B}_{10\times 10}^{\mathrm{\prime }}& =[\begin{array}{ccccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2\sqrt{3}}}& 0& 0& 0\\ -{\displaystyle \frac{1}{2\sqrt{3}}}& 0& {\displaystyle \frac{1}{2\sqrt{15}}}& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{\sqrt{42}}}& 0\\ {\displaystyle \frac{\sqrt{7}}{2\sqrt{30}}}& 0& 0& 0& {\displaystyle \frac{1}{2\sqrt{10}}}\\ 0& 0& 0& 0& 0\\ -\sqrt{{\displaystyle \frac{11}{210}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ {\displaystyle \frac{\sqrt{143}}{20\sqrt{7}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\sqrt{{\displaystyle \frac{146965}{2895222}}}& 0& 0& 0& 0\end{array}\\ & \begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ {\displaystyle \frac{\sqrt{5}}{3\sqrt{22}}}& 0& 0& 0& 0\\ 0& {\displaystyle \frac{\sqrt{691}}{10\sqrt{273}}}& 0& 0& 0\\ 0& 0& \sqrt{{\displaystyle \frac{35}{1382}}}& 0& 0\\ 0& 0& 0& {\displaystyle \frac{\sqrt{3617}}{20\sqrt{357}}}& 0\\ 0& 0& 0& 0& {\displaystyle \frac{\sqrt{219335}}{3\sqrt{962122}}}\\ 0& 0& 0& 0& 0\end{array}],\\ {\overline{{\xi }^{{}^{\mathrm{\prime }}}}}_{10}\left(x\right)& =\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {\displaystyle \frac{\sqrt{1222277}}{10\sqrt{482537}}}{\xi }_{1,10}\left(x\right)\end{array}\right].\end{array}$

Next, twice integration of above ten basis is given below $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,0}\left(\text{x}\right)\text{dxdx}& =[\frac{1}{6}\frac{1}{4\sqrt{3}}\frac{1}{12\sqrt{5}}00\\ & 00000\phantom{\frac{1}{2\sqrt{3}}}\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,1}\left(\text{x}\right)\text{dxdx}& =[-\frac{1}{4\sqrt{3}}-\frac{1}{12}0\frac{1}{6\sqrt{70}}\\ & 000000\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,2}\left(\text{x}\right)\text{dxdx}& =[\frac{1}{12\sqrt{5}}000\frac{1}{4\sqrt{105}}\\ & 00000\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,3}\left(\text{x}\right)\text{dxdx}& =[\frac{\sqrt{7}}{4\sqrt{30}}\frac{\sqrt{7}}{12\sqrt{10}}000\\ & \frac{1}{12\sqrt{11}}0000\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,4}\left(\text{x}\right)\text{dxdx}& =[-\frac{1}{6\sqrt{21}}00000\\ & \frac{\sqrt{691}}{6\sqrt{30030}}000\phantom{\frac{1}{2\sqrt{3}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,5}\left(\text{x}\right)\text{dxdx}& =[-\frac{\sqrt{11}}{2\sqrt{210}}-\frac{\sqrt{11}}{6\sqrt{70}}00\\ & 000\frac{1}{2\sqrt{390}}\\ & 00\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,6}\left(\text{x}\right)\text{dxdx}=[\frac{\sqrt{143}}{4\sqrt{6910}}000\\ & 0000\frac{\sqrt{3617}}{4\sqrt{352410}}0\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,7}\left(\text{x}\right)\text{dxdx}=[\frac{\sqrt{143}}{40\sqrt{7}}\frac{\sqrt{143}}{40\sqrt{21}}00\\ & 00000\frac{\sqrt{43867}}{84\sqrt{9690}}\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,8}\left(\text{x}\right)\text{dxdx}& =[-\frac{5\sqrt{221}}{6\sqrt{119361}}00\\ & 0000000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right)\\ & [phone]\frac{\sqrt{\mathrm{}}}{\mathrm{}\sqrt{\mathrm{}}}{\xi }_{1,10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,9}\left(\text{x}\right)\text{dxdx}=[-\frac{\sqrt{146965}}{2\sqrt{2895222}}-\frac{\sqrt{146965}}{6\sqrt{965074}}\\ & 00000000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right)\\ & [phone]\frac{\sqrt{\mathrm{}}}{\mathrm{}\sqrt{\mathrm{}}}{\xi }_{1,11}\left(\text{x}\right).\end{array}$

Hence, $\begin{array}{r}{\int }_0^x{\int }_0^x\xi \left(x\right)\mathrm{dxdx}=..{\xi }_{10}\left(x\right)+{\overline{{\xi }^{\mathrm{\prime \prime }}}}_{10}\left(x\right),\end{array}$Where $\begin{array}{rl}{B}_{10\times 10}^{\mathrm{\prime \prime \prime }}& =[\begin{array}{ccc}{\displaystyle \frac{1}{6}}& {\displaystyle \frac{1}{4\sqrt{3}}}& {\displaystyle \frac{1}{12\sqrt{5}}}\\ -{\displaystyle \frac{1}{4\sqrt{3}}}& -{\displaystyle \frac{1}{12}}& 0\\ {\displaystyle \frac{1}{12\sqrt{5}}}& 0& 0\\ {\displaystyle \frac{\sqrt{7}}{4\sqrt{30}}}& {\displaystyle \frac{\sqrt{7}}{12\sqrt{10}}}& 0\\ -{\displaystyle \frac{1}{6\sqrt{21}}}& 0& 0\\ -{\displaystyle \frac{\sqrt{11}}{2\sqrt{210}}}& -{\displaystyle \frac{\sqrt{11}}{6\sqrt{70}}}& 0\\ {\displaystyle \frac{\sqrt{143}}{4\sqrt{6910}}}& 0& 0\\ {\displaystyle \frac{\sqrt{143}}{40\sqrt{7}}}& {\displaystyle \frac{\sqrt{143}}{40\sqrt{21}}}& 0\\ -{\displaystyle \frac{5\sqrt{221}}{6\sqrt{119361}}}& 0& 0\\ -{\displaystyle \frac{\sqrt{146965}}{2\sqrt{2895222}}}& -{\displaystyle \frac{\sqrt{146965}}{6\sqrt{965074}}}& 0\end{array}\\ & \begin{array}{cccc}0& 0& 0& 0\\ {\displaystyle \frac{1}{6\sqrt{70}}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{4\sqrt{105}}}& 0& 0\\ 0& 0& {\displaystyle \frac{1}{12\sqrt{11}}}& 0\\ 0& 0& 0& {\displaystyle \frac{\sqrt{691}}{6\sqrt{30030}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\\ & \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{1}{2\sqrt{390}}}& 0& 0\\ 0& {\displaystyle \frac{\sqrt{3617}}{4\sqrt{352410}}}& 0\\ 0& 0& {\displaystyle \frac{\sqrt{43867}}{84\sqrt{9690}}}\\ 0& 0& 0\\ 0& 0& 0\end{array}],\end{array}$ $\begin{array}{r}{\overline{{\xi }^{\mathrm{\prime \prime }}}}_{10}\left(\text{x}\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {\displaystyle \frac{\sqrt{174611}}{6\sqrt{7559530}}}{\xi }_{1,10}\left(\text{x}\right)\\ {\displaystyle \frac{\sqrt{77683}}{2\sqrt{30268230}}}{\xi }_{1,11}\left(\text{x}\right)\end{array}\right].\end{array}$

Once again, thrice the integration of the aforementioned ten basis is shown below $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,0}\left(\text{x}\right)\text{dxdxdx}=[\frac{1}{24}\frac{1}{12\sqrt{3}}\frac{1}{24\sqrt{5}}\\ & \frac{1}{12\sqrt{210}}000000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,1}\left(\text{x}\right)\text{dxdxdx}=[\frac{-\sqrt{3}}{40}\frac{-1}{24}\frac{-1}{24\sqrt{15}}\\ & 0\frac{1}{120\sqrt{7}}00000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,2}\left(\text{x}\right)\text{dxdxdx}=[\frac{1}{24\sqrt{5}}\frac{1}{24\sqrt{15}}00\\ & 0\frac{1}{12\sqrt{462}}0000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,3}\left(\text{x}\right)\text{dxdxdx}=[\frac{1}{2\sqrt{210}}\frac{\sqrt{7}}{24\sqrt{10}}\\ & \frac{\sqrt{7}}{120\sqrt{6}}00\frac{\sqrt{691}}{120\sqrt{3003}}\\ & 0000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,4}\left(\text{x}\right)\text{dxdxdx}=[\frac{-1}{12\sqrt{21}}\frac{-1}{36\sqrt{7}}00\\ & 000\frac{1}{12\sqrt{429}}00\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,5}\left(\text{x}\right)\text{dxdxdx}=[\frac{-17\sqrt{11}}{120\sqrt{210}}\frac{-\sqrt{11}}{12\sqrt{70}}\\ & \frac{-\sqrt{11}}{60\sqrt{42}}00000\frac{\sqrt{3617}}{120\sqrt{15470}}0\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,6}\left(\text{x}\right)\text{dxdxdx}=[\frac{\sqrt{143}}{8\sqrt{6910}}\\ & \frac{\sqrt{143}}{8\sqrt{20730}}0000000\\ & \frac{\sqrt{43867}}{24\sqrt{4687053}}\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,7}\left(\text{x}\right)\text{dxdxdx}=[\frac{\sqrt{91}}{90\sqrt{11}}\frac{\sqrt{143}}{80\sqrt{21}}\\ & \frac{\sqrt{143}}{240\sqrt{35}}0000000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right)\\ & [phone]\frac{\sqrt{\mathrm{}}}{\mathrm{}\sqrt{\mathrm{}}}{\xi }_{1,10}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,8}\left(\text{x}\right)\text{dxdxdx}=[\frac{-5\sqrt{221}}{12\sqrt{119361}}\frac{-5\sqrt{221}}{36\sqrt{39787}}\\ & 00000000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right)\\ & [phone]\frac{\sqrt{\mathrm{}}}{\mathrm{}\sqrt{\mathrm{}}}{\xi }_{1,11}\left(\text{x}\right),\end{array}$ $\begin{array}{rl}& {\int }_0^{\text{x}}{\int }_0^{\text{x}}{\int }_0^{\text{x}}{\xi }_{1,9}\left(\text{x}\right)\text{dxdxdx}=[\frac{-3859\sqrt{323}}{60\sqrt{1317326010}}\\ & \frac{-\sqrt{146965}}{12\sqrt{965074}}\frac{-\sqrt{29393}}{12\sqrt{2895222}}0000\\ & 000\phantom{\frac{\sqrt{11}}{6\sqrt{70}}}]{\xi }_{10}\left(\text{x}\right)+\frac{\sqrt{236364091}}{120\sqrt{1009949941}}{\xi }_{1,12}\left(\text{x}\right).\end{array}$

Hence, $\begin{array}{r}{\int }_0^x{\int }_0^x{\int }_0^x\xi \left(x\right)\mathrm{dxdxdx}={\xi }_{10}\left(x\right)+\overline{{\xi }^{\mathrm{\prime \prime }}{{}^{\mathrm{\prime }}}_{10}}\left(x\right),\end{array}$where $\begin{array}{rl}{B}_{10\times 10}^{\mathrm{\prime \prime \prime }}& =[\begin{array}{cc}{\displaystyle \frac{1}{24}}& {\displaystyle \frac{1}{12\sqrt{3}}}\\ {\displaystyle \frac{-\sqrt{3}}{40}}& {\displaystyle \frac{-1}{24}}\\ {\displaystyle \frac{1}{24\sqrt{5}}}& {\displaystyle \frac{1}{24\sqrt{15}}}\\ {\displaystyle \frac{1}{2\sqrt{210}}}& {\displaystyle \frac{\sqrt{7}}{24\sqrt{10}}}\\ {\displaystyle \frac{-1}{12\sqrt{21}}}& {\displaystyle \frac{-1}{36\sqrt{7}}}\\ {\displaystyle \frac{-\sqrt{11}}{120\sqrt{210}}}& {\displaystyle \frac{-\sqrt{11}}{12\sqrt{70}}}\\ {\displaystyle \frac{\sqrt{143}}{8\sqrt{6910}}}& {\displaystyle \frac{\sqrt{143}}{8\sqrt{20730}}}\\ {\displaystyle \frac{\sqrt{91}}{90\sqrt{11}}}& {\displaystyle \frac{\sqrt{143}}{80\sqrt{21}}}\\ {\displaystyle \frac{-5\sqrt{221}}{12\sqrt{119361}}}& {\displaystyle \frac{-5\sqrt{221}}{36\sqrt{39787}}}\\ {\displaystyle \frac{-3859\sqrt{323}}{60\sqrt{1317326010}}}& {\displaystyle \frac{-\sqrt{146965}}{12\sqrt{965074}}}\end{array}\\ & \begin{array}{ccc}{\displaystyle \frac{1}{24\sqrt{5}}}& {\displaystyle \frac{1}{12\sqrt{210}}}& 0\\ {\displaystyle \frac{-1}{24\sqrt{15}}}& 0& {\displaystyle \frac{1}{120\sqrt{7}}}\\ 0& 0& 0\\ {\displaystyle \frac{\sqrt{7}}{120\sqrt{6}}}& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{-\sqrt{11}}{60\sqrt{42}}}& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{\sqrt{143}}{240\sqrt{35}}}& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{-\sqrt{29393}}{12\sqrt{2895222}}}& 0& 0\end{array}\\ & \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{1}{12\sqrt{462}}}& 0& 0\\ 0& {\displaystyle \frac{\sqrt{691}}{120\sqrt{3003}}}& 0\\ 0& 0& {\displaystyle \frac{1}{12\sqrt{429}}}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\\ & \begin{array}{c}\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ {\displaystyle \frac{\sqrt{3617}}{120\sqrt{15470}}}& 0\\ 0& {\displaystyle \frac{\sqrt{43867}}{24\sqrt{4687053}}}\\ 0& 0\\ 0& 0\\ 0& 0\end{array}],\end{array}$ $\begin{array}{r}\overline{{\xi }^{\mathrm{\prime \prime }}{{}^{\mathrm{\prime }}}_{10}}\left(x\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {\displaystyle \frac{\sqrt{174611}}{120\sqrt{746130}}}{\xi }_{1,10}\left(x\right)\\ {\displaystyle \frac{\sqrt{77683}}{12\sqrt{33193209}}}{\xi }_{1,11}\left(x\right)\\ {\displaystyle \frac{\sqrt{236364091}}{120\sqrt{1009949941}}}{\xi }_{1,12}\left(x\right)\end{array}\right].\end{array}$

Similarly, for our convenience, we may design matrices of various sizes.

4. Bernoulli wavelet numerical method

(10) $\begin{array}{r}\frac{d^{4}f\left(x\right)}{d{x}^4}=G^T\xi \left(x\right)\end{array}$(10) Integrate Equation (10) with respect to $x$ and limit from 0 to $x$, (11) $\begin{array}{r}\frac{d^{3}f\left(x\right)}{d{x}^3}-\frac{d^{3}f\left(0\right)}{d{x}^3}=G^T\left[\xi \left(x\right)B+\overline{\xi }\left(x\right)\right]\end{array}$(11)

Integrate (11) with respect to $x$ & limit from 0 to $x$, (12) $\begin{array}{r}\frac{d^{2}f\left(x\right)}{d{x}^2}-\frac{d^{2}f\left(0\right)}{d{x}^2}=x\frac{d^{3}f\left(0\right)}{d{x}^3}+G^T\left[\xi \left(x\right)B^{\mathrm{\prime }}+\overline{{\xi }^{\mathrm{\prime }}}\left(x\right)\right]\end{array}$(12) Integrate (12) with respect to $x$ & limit from 0 to $x$, (13) $\begin{array}{rl}\frac{\mathrm{df}\left(x\right)}{\mathrm{dx}}-\frac{\mathrm{df}\left(0\right)}{\mathrm{dx}}& =x\frac{d^{2}f\left(0\right)}{d{x}^2}+\frac{x^2}{2!}\frac{d^{3}f\left(0\right)}{d{x}^3}\\ & +G^T\left[\xi \left(x\right)B^{\mathrm{\prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime }}}\left(x\right)\right]\end{array}$(13) Integrate (13) with respect to x & limit from 0 to x, (14) $\begin{array}{rl}f\left(x\right)-f\left(0\right)& =x\frac{\mathrm{df}\left(0\right)}{\mathrm{dx}}+\frac{x^2}{2!}\frac{d^{2}f\left(0\right)}{d{x}^2}+\frac{x^3}{3!}\frac{d^{3}f\left(0\right)}{d{x}^3}\\ & +G^T\left[\xi \left(x\right)B^{\mathrm{\prime \prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime \prime }}}\left(x\right)\right]\end{array}$(14) Substitute $f\left(0\right)=0\text{and}\frac{d^{2}f\left(0\right)}{d{x}^2}=0$ in Equation (14) we have, (15) $\begin{array}{r}f\left(x\right)=\frac{\mathrm{df}\left(0\right)}{\mathrm{dx}}x+\frac{d^{3}f\left(0\right)}{d{x}^3}\frac{x^3}{3!}+G^T\left[\xi \left(x\right)B^{\mathrm{\prime \prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime \prime }}}\left(x\right)\right]\end{array}$(15)

Substitute $\frac{\mathrm{df}\left(1\right)}{\mathrm{dx}}=0\text{and}f\left(1\right)=1$ in Equations (13) and (15) and solve those equations to find the values of $\frac{d^{3}f\left(0\right)}{d{x}^3}$ and $\frac{\mathrm{df}\left(0\right)}{\mathrm{dx}}$ we have, (16) $\begin{array}{rl}\frac{d^{3}f\left(0\right)}{d{x}^3}& =-3+3G^T[\xi \left(x\right)B^{\mathrm{\prime \prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime \prime }}}\left(x\right){]}_{x=1}\\ & -3G^T[\xi B^{\mathrm{\prime \prime }}\left(x\right)+\overline{{\xi }^{\mathrm{\prime \prime }}}\left(x\right){]}_{x=1}\end{array}$(16) (17) $\begin{array}{rl}\frac{\mathrm{df}\left(0\right)}{\mathrm{dx}}& =-\frac{1}{2}\frac{d^{3}f\left(0\right)}{d{x}^3}-G^T[\xi \left(x\right)B^{\mathrm{\prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime }}}\left(x\right){]}_{x=1}\end{array}$(17) Substitute (16) and (17) in (15) we have, (18) $\begin{array}{rl}f\left(x\right)& =\left(-G^T{\left[\xi \left(x\right)B^{\mathrm{\prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime }}}\left(x\right)\right]}_{x=1}\right)x\\ & +\left(\begin{array}{c}-3+3G^T[\xi \left(x\right)B^{\mathrm{\prime \prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime \prime }}}\left(x\right){]}_{x=1}\\ -3G^T[\xi B^{\mathrm{\prime \prime }}\left(x\right)+\overline{{\xi }^{\mathrm{\prime \prime }}}\left(x\right){]}_{x=1}\end{array}\right)\left(\frac{x^3}{3!}-\frac{1}{2}\right)\\ & +G^T\left[\xi \left(x\right)B^{\mathrm{\prime \prime \prime }}+\overline{{\xi }^{\mathrm{\prime \prime \prime }}}\left(x\right)\right]\end{array}$(18) Again, $\frac{d^2\theta \left(x\right)}{d{x}^2}=A^T\xi \left(x\right)$. (19) Integrate Equation (19) with respect to $x$ & limit from 0 to $x$, $\frac{\mathrm{d\theta }\left(x\right)}{\mathrm{dx}}=\frac{\mathrm{d\theta }\left(0\right)}{\mathrm{dx}}+A^T\left[\xi \left(x\right)B+\overline{\xi }\left(x\right)\right]$. (20) Integrate Equation (20) with respect to $x$ & limit from 0 to $x$, (21) $\begin{array}{r}\theta \left(x\right)=\theta \left(0\right)+x\frac{\mathrm{d\theta }\left(0\right)}{\mathrm{dx}}+A^T\left[\xi \left(x\right)B^{\mathrm{\prime }}+\overline{{\xi }^{\mathrm{\prime }}}\left(x\right)\right]\end{array}$(21) using $\theta \left(1\right)=1$ we have, (22) $\begin{array}{r}\theta \left(0\right)=1-A^T\left[\xi \left(x\right)B^{\mathrm{\prime }}+\overline{{\xi }^{\mathrm{\prime }}}\left(x\right)\right]{|}_{x=1}\end{array}$(22) Substitute (22) in (21), we have (23) $\begin{array}{rl}\theta \left(x\right)& =1-A^T\left[\xi \left(x\right)B^{\mathrm{\prime }}+\overline{{\xi }^{\mathrm{\prime }}}\left(x\right)\right]{|}_{x=1}\\ & +A^T\left[\xi \left(x\right)B^{\mathrm{\prime }}+\overline{{\xi }^{\mathrm{\prime }}}\left(x\right)\right]\end{array}$(23) In order to collocate these equations, substitute $\frac{d^{4}f}{d{x}^4},\frac{d^{3}f}{d{x}^3},\frac{d^{2}f}{d{x}^2},\frac{\mathrm{df}}{\mathrm{dx}},f,\frac{d^2\theta }{d{x}^2},\frac{\mathrm{d\theta }}{\mathrm{dx}},\theta$ and utilize collocation steps as $x_i=\frac{2i-1}{N}$ Where $i=1,2,3,\cdot \cdot \cdot ,N$. As a result, a collection of algebraic equations are formed. We may obtain the Bernoulli wavelet's unknown coefficients by solving this with the proper solver, which results in numerical Bernoulli wavelet's solutions for the equations (6)(6) $\begin{array}{rl}& \left(1+\frac{1}{\text{β}}\right)\frac{d^{4}f}{d{\eta }^4}-S\left(\eta \frac{d^{3}f}{d{\eta }^3}-f\frac{d^{3}f}{d{\eta }^3}+\frac{\mathrm{df}}{\mathrm{d\eta }}\frac{d^{2}f}{d{\eta }^2}+3\frac{d^{2}f}{d{\eta }^2}\right)\\ & =\left(\mathrm{Mp}+\mathrm{Mg}\right)\frac{d^{2}f}{d{\eta }^2}\end{array}$(6) and (7)(7) $\begin{array}{rl}& \frac{d^2\theta }{d{\eta }^2}+PrS\left(f\frac{\mathrm{d\theta }}{\mathrm{d\eta }}-\eta \frac{\mathrm{d\theta }}{\mathrm{d\eta }}\right)\\ & +Pr\mathrm{Ec}\left({\left(\frac{d^{2}f}{d{\eta }^2}\right)}^2+4{\delta }^2{\left(\frac{\mathrm{df}}{\mathrm{d\eta }}\right)}^2\right)\\ & +\mathrm{Hs}\theta =0.\end{array}$(7) subjected to the conditions Equation (8).

5. Results and discussion

Based on the numerical analysis of previous section, the Bernoulli wavelet numerical technique is developed to study the problem of an unsteady incompressible Casson fluid flow through a porous medium with an internal heat source under the influence of a magnetic field between two parallel plates. We have presented the velocity, temperature profile, skin friction and heat transfer for conjugate parameter for porous medium parameter, MHD parameter, squeeze number Prandtl number, Eckert number and an internal heat source through graphs. The Bernoulli wavelet numerical results of the skin friction coefficient, and Nusselt for different values of $S$ are compared with different numerical methods (Mustafa et al. [58]) and good agreement is noticed as shown in Table 1.

Table 1. Validation of BWM with Mustafa et al. [58]

	$-f^{\mathrm{\prime \prime }}\left(1\right)$		$-{\theta }^{\mathrm{\prime }}\left(1\right)$
$S$	BWM	NUM [58]	BWM	NUM [58]
−1	2.170090	2.170090	3.319899	3.319899
−0.5	2.614038	2.614038	3.129491	3.129491
0.01	3.007134	3.007134	3.047092	3.047092
0.5	3.336449	3.336449	3.026324	3.026324
2	4.167389	4.167389	3.118551	3.118551

Figures 2–7 shows effects of $S$ on velocity and temperature for fixed values of $\mathrm{Mg}=\mathrm{Mp}=1=\text{β}$. For $S>0$ and $S<0$ are used to describe how surfaces move as they get closer to one another and as they move away. For $S>0$ case, the velocity decreases with increasing the value of $S$(Figure 3) but opposite trend could be seen when $S<0$ (Figure 2). Thus, squeezing of surfaces causes the fluid to be forced out of the channel, which slows down the velocity field. When the surfaces continue to separate, the fluid flow an increase in velocity into the channel. In Figure 4 it is clearly shows that an increase the positive values of S, the velocity goes on decreases. On the other hand, there is an increase in the velocity profile with decrease in the negative values of $S$, as we observed in the Figure 5. Figures 6 and 7, show the impact of $S$ on the temperature field. The temperature falls with increase in positive values of $S$ (Figure 6), but temperature rises when increase in negative values of $S$ (Figure 7). The kinetic energy in fluid particles slows down as the surfaces become closer, which lowers the flow's temperature. It is quite obvious that the temperature is relatively high when the plates are moving towards each other. The temperature close to the upper border was unaffected by plate movement at $\eta$ =  1. Figure 8 illustrates the impact of the Casson parameter $\left(\text{β}\right)$, on $f^{\mathrm{\prime }}\left(\eta \right)$. As β increases with $S=-2,\mathrm{Mg}=\mathrm{Mp}=1$ the velocity grows closer to the lower plates and stops near the upper plate. Figure 9 demonstrates the impact of β with $S=3,\mathrm{Mg}=\mathrm{Mp}=1$, indicating that the velocity component declines with increasing β (Figure 9).

Figure 2. Influence of S on $f\left(\eta \right)$.

Figure 3. Influence of S on $f\left(\eta \right)$.

Figure 4. Influence of S on $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 5. Influence of S on $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 6. Influence of S on $\theta \left(\eta \right)$.

Figure 7. Influence of S on $\theta \left(\eta \right)$.

Figure 8. Influence of $\text{β}$ on $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 9. Influence of $\text{β}$ on $f^{\mathrm{\prime }}\left(\eta \right)$.

The variation of $\mathrm{Mg}$ on the velocity and temperature field is shown in Figures 10–12, for fixed values of $S=1,\mathrm{Sc}=\mathrm{Pr}=1,\text{β}=1$. Figure 10 shows that as $\mathrm{Mg}$ increases, velocity decreases. Physically, the magnetic field creates the Lorentz force in the electrically conducted fluid and the channel's resistance to flow is heightened by the existence of this energy. Figure 11 demonstrates that the axial velocity decreases for $\eta$≤ 0.5 and increases for $\eta$> 0.5 as $\mathrm{Mg}$ increases. Figure 12 illustrates how Mg effects on temperature and it is observed that a increase in $\mathrm{Mg}$ causes a temperature decrease. Figure 13 illustrates how $\mathrm{Mp}$ has an impact on axial velocity. As $\mathrm{Mp}$ increases for $\eta$≤ 0.5, the velocity increases; however, for $\eta$> 0.5, it is decreases. The fluid flow through porous media near the centre of the channel increases as the permeability of the porous medium increases.

Figure 10. Influence of Mg on $f\left(\eta \right)$.

Figure 11. Influence of Mg on $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 12. Influence of Mg on $\theta \left(\eta \right)$.

Figure 13. Influence of Mp on $f^{\mathrm{\prime }}\left(\eta \right)$.

The thermal boundary layer thickness is found to decrease upon increasing $\mathrm{Pr}$ and $\mathrm{Ec}$. It is apparent that an increase in the values of $\mathrm{Pr}$ largely decreases the thermal diffusivity which therefore decays the thermal boundary layer thickness. The effect of Pr on the temperature field is shown in Figure 14, for fixed values of $\text{β}=1,S=\mathrm{Ec}=1,\delta =0.1$. As $\mathrm{Pr}$ increases, the flow's temperature increases. The Prandtl number measures the relationship between thermal and momentum diffusivity. A drop in the thermal diffusion of viscous fluid caused by increased $\mathrm{Pr}$ increases the flow temperature.

Figure 14. Influence of Pr on $\theta \left(\eta \right)$.

Figure 15. Influence of Ec on $\theta \left(\eta \right)$.

Figure 16. Influence of $\text{β}$ on $\theta \left(\eta \right)$.

Figure 15 shows the effects of $\mathrm{Ec}$ on the temperature field for fixed values of $\text{β}=1,S=\mathrm{Pr}=1,\delta =0.1$. It has been discovered that improvements in $\mathrm{Ec}$ are associated with increasing temperature. The viscous dissipation effect is the name given to the heat generation brought on by fluid particle friction in a high-viscosity flow. Increased kinetic energy results from high $\mathrm{Ec}$ values. As a result, it increases the friction between fluid particles, which elevates the temperature field. Figure 16 shows how the temperature profile behaves as $\text{β}$ increases for fixed values of $\mathrm{Ec}=\mathrm{Pr}=1,\delta =0.1S=1$. From Figure 15 clears that the temperature profile is increasing with increasing $\text{β}$.

6. Conclusion

In this paper, the Bernoulli wavelet technique was employed to obtain a numerical solution for heat transfer on MHD Casson fluid flow through porous media with an internal heat source under the control of a magnetic field. The study examined the effects of various known physical parameters, including $S,\text{β},\mathrm{Mp},\mathrm{Mg},\mathrm{Ec},\mathrm{Hs}$ and $\mathrm{Pr}$ on velocity, and temperature profiles. The following conclusions can be drawn from the present study:

From Table 1 it is obvious that the computed Bernoulli wavelet numerical solution is very close to up to a numerical solution.

Bernoulli wavelet numerical method is more capable than any numerical method to solve this type of modelling problems.

The velocity and temperature profiles are increases with increasing Caason fluid parameter ($\text{β}$).

The velocity near the upper plate region increases as the surfaces get closer together ($S>0$), and it drops as they get further apart ($S<0$).

As $S$ and $\mathrm{Mg}$ increases, the wall shear stress increases while decreasing for increasing $\mathrm{Mp}$.

The increase in $\mathrm{Mg}$ and $\mathrm{Mp}$ causes a reduction in velocity close to the upper plate region.

Raising $\mathrm{Pr}$ and $\mathrm{Ec}$ results in an increase in temperature and heat transfer rate.

The internal heat source increases, the amount of heat energy being added to the fluid increases. This can cause the temperature of the fluid increases.

Acknowledgments

The authors thank the reviewers for their constructive comments and useful suggestions which helped in improving the paper considerably.

Disclosure statement

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

Data availability

The data that support the findings of this study are available within the article.