Lie analysis and laws of conservation for the two-dimensional model of Newell–Whitehead–Segel regarding the Riemann operator fractional scheme in a time-independent variable

Abstract

The framework of fractional partial differential models is the first-rate hyperlink between mathematics and applied physics. This article project intends to utilize outcomes for the time-fractional Newell–Whitehead–Segel model in $2D$-space including conservation laws, Lie point symmetry analysis, and series solutions. We introduce a particular fractional model that is free of the type of approximate methods, whilst, environmental flow and magnetohydrodynamics processes are considered to be the main real-world phenomena treated with such a model. Herein, the method of power series is exercised to provide an analytical solution to the current governing model. The idea and undertaking of the method lie in the assumption that the solution is a power series of coefficients that are determined by a recurrence relation obtained by substituting the series solution in the considered model. Also, the Riemann approach is utilized as a total derivative. Simulation effects are systematically demonstrated through a chain of check cases. Strong proofs with some related plots are performed to confirm the accuracy and fitness of the model version and the presented approach. Moreover, the laws of conservation depend on the existence of a Lagrangian of the fractional Newell–Whitehead–Segel model utilized. Ultimately, a few related comments and future proposals are epitomized.

HIGHLIGHT

• The Lie point symmetries of the time-fractional Newell–Whitehead–Segel equation in two-dimensional space is utilized and derived.

• The technique of the power series is applied to conclude the explicit solutions for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space for the first time.

• The conservation laws for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space are built using a novel conservation theorem.

• Several graphical countenances were utilized to award a visual performance of the obtained solutions.

Keywords:

• Extended series solution
• fractional differential operator
• Newell–Whitehead–Segel model
• symmetry analysis

1. Preliminary

Fractional derivatives are more adequate in formulating models concerning various real word problems than the usual derivative of integer order. By any means, fractional models provide suitable descriptions for the hereditary properties and memory of several substances in comparison with models of integer order. Hence, the importance of treating problems with fractional orders has attracted the attention of researchers to investigate suitable techniques for providing solutions to such problems. FPDEs are considered to perform the main relationship between mathematical concepts and their basic applications in engineering, chemistry, physics, and other scientific areas. The most familiar applications of FPDEs appear in fluid dynamics, image processing, inverse problem, electric engineering, inviscid fluid, shallow-water waves, and fluid flow (Mainardi, 2010; Zaslavsky, 2005; Podlubny, 1999; Kilbas, Srivastava, & Trujillo, 2006). Many physical formations including time-FPDEs of nonlinear shapes are extremely hard to deal with and they do not have a uniform or fixed behaviour to control them; for this purpose, many researchers have developed methods and algorithms to deal with solutions to these fractional models numerically, analytically, and approximately forms as utilized next. Goswami, Singh, Kumar, Gupta, and Sushila (2019) have utilized an efficient scheme for solving FPDEs occurring in ion-acoustic waves. Kurt et al. (2019) have treated two approaches for handling FPDEs arising in long waves. Khater, Attia, and Lu (2019) have utilized three approximate schemes to handle with FPDEs arising in water surface versus. Wang et al. (2013a) have applied the kernel method for solving inverse heat fractional problems. Az-Zo’bi, Al-Maaitah, Tashtoush, and Osman, (2022) have adopted the cubic–quintic–septic NLSE approach and its optical approximation. Al-Deiakeh, Abu Arqub, Al-Smadi, and Momani (2021) have applied the power series expansion to describe the approximation of Fisher’s fractional model. Further results and concepts can be obtained from Osman et al. (2020); Nisar et al. (2021); Park et al. (2020); Yao et al. (2022); Haque, Akbar, and Osman (2022); Gao, Alotaibi, and Ismail (2022); Liu (2022); Chen and Ren (2022); Wen, Liu, Chen, Fakieh, and Shorman (2022); Zhirong and Alghazzawi (2022); Li (2022).

The FNWSM is an ultimate communal amplitude model that utilizes the event of fixed spatial stripe designs and the powerful mentality closest to the node of bifurcation in the Rayleigh–Benard transmission of twofold liquid blends. This model was studied by several researchers as follows: in the work of Mangoub and Sedeeg (2016), its solution and some related properties are described, in the work of Saravanan and Magesh (2013) comparisons between differential transform and Adomian decomposition methods are given, in the work of Macias-Diaz and Ruiz-Ramirez (2011), the symmetry-preserving solution is introduced, in the work of Kumar and Sharma (2016), a new iterative solution has been utilized, and in Prakash, Goyal, and Gupta (2019) the fractional variational scheme has described. A couple of modalities could be mentioned; the roll modality and the hexagonal pattern. For like, the slides modalities are established in the apparent cortex or human fingerprints. It is beneficial to focus on the fact that the shape hexagonal could be masterful in diffusion and chemical reaction (Newell & Whitehead, 1969). Whilst, the LPS strives a big deal in assorted fields of applied numerical sciences, especially in frameworks reconciliation, where vastly numerous symmetries happen. From that point, the LPS approach can hypothesize as a viable strategy for building scientific answers for FPDEs of a nonlinear kind. By and large, immense compositions are faithful in the investigation of LPS and its executions of models (Bluman & Kumei, 1989; Olver, 1993). Anyhow, detailed results on conservation laws for FPDEs can be gained from the work of Yang, Guo, and He (2019) and Gazizov, Ibragimov, and Lukashchuk (2015).

This article is devoted to theorizing the LPS and more results for the FNWSM in $2D$-space of the shape (Mangoub & Sedeeg, 2016; Saravanan & Magesh, 2013; Macias-Diaz & Ruiz-Ramirez, 2011; Kumar & Sharma, 2016; Prakash et al., 2019): (1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1)

Herein, $1\ge б>0$ and $D_t^{б}$ is given with the Riemann approach as (2) $$D_t^{б}\vartheta \left(x,t\right)=\{\begin{array}{l}\frac{1}{\Gamma \left(б\right)}{\partial }_t\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{б-1}\vartheta \left(x,s\right)ds,1>б>0, 0>s>t,\\ \vartheta \left(x,t\right),б=0,\\ {\partial }_t\vartheta \left(x,t\right),б=1.\end{array}$$(2)

Substantially, in (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) and (2)(2) $$D_t^{б}\vartheta \left(x,t\right)=\{\begin{array}{l}\frac{1}{\Gamma \left(б\right)}{\partial }_t\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{б-1}\vartheta \left(x,s\right)ds,1>б>0, 0>s>t,\\ \vartheta \left(x,t\right),б=0,\\ {\partial }_t\vartheta \left(x,t\right),б=1.\end{array}$$(2) , $a,b,c\in \mathbb{R}$ and $p\in \mathbb{N}.$ Whilst, $\vartheta$ symbolizes the function of the spatial independent variable $x$ and the temporal independent variable $t,$ $D_t^{б}\vartheta$ symbolizes the variations with $t$ at a fixed position, $D_x^2\vartheta$ symbolizes the variations with spatial independent variable $x$ at a specific $t,$ and $b\vartheta -c{\vartheta }^p$ symbolizes the effect of the source term. Indeed, it can be considered that the nonlinear temperature distribution in an infinitely long and thin rod or the fluid flow velocity in an infinitely long tube of small diameter.

After a suitable reduction of the FNWSM to FDE of nonlinear type, the power series method is employed to treat the considered problem. It is well-known, effective, and provides analytic solutions for different forms of differential equations. The power series expansion can be easily differentiated for any order.

The rest of the article’s contents are as follows.Section 2 introduces the LPS epitome: plan and properties. Section 3 presents the LPSs and decreasing trademark recipes with Erdélyi–Kober. Section 4 discusses the power series examination: express arrangement, intermingling, representations, and defences. Section 5 presents the connected protection laws: determination and conversation. Section 6 presents the end: feature and future.

2. Exemplification of the LPS

In this area, a few primary elaboration ideas and brief subtleties concerning fundamental concepts that concern the LPS investigation are implemented. Mainly, the infinitesimal invariance criterion, the invariant solution function, and the rule of Leibnitz are utilized to transform problem (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) into other acquired forms. Here, the uses are connected with the detailing of the FNWSM.

Allow us first to think about the FPDE of order $б\in \left[0,1\right]:$ (3) $$D_t^{б}\vartheta -F\left(t,x,\vartheta ,{\vartheta }_x,{\vartheta }_{xx},\dots \right)=0,$$(3) where $\vartheta$ is a dependent variable that depends on the independent variables $t$ and $x.$

Definition 1

(Wang et al., 2013b). Let $V=\xi \left(t,x,\vartheta \right)\frac{\partial }{\partial x}+\tau (t,x,\vartheta )\frac{\partial }{\partial t}+\mathrm{ŋ}(t,x,\vartheta )\frac{\partial }{\partial \vartheta }$ be a collection of vector fields and $\vartheta =Ө\left(x,t\right)$ is a function such that (4) $$VӨ=0\iff \left[\xi \left(t,x,\vartheta \right)\frac{\partial }{\partial x}+\tau \left(t,x,\vartheta \right)\frac{\partial }{\partial t}+ŋ\left(t,x,\vartheta \right)\frac{\partial }{\partial \vartheta }\right]Ө=0.$$(4)

Then, $\vartheta$ is indicated as an invariant.

Definition 2

(Wang, Liu, et al., 2013). For a function, $\vartheta =Ө(x,t)$ where

1. $\vartheta =Ө\left(x,t\right)$ is an invariant.

2. $\vartheta =Ө\left(x,t\right)$ agree well (4)(4) $$VӨ=0\iff \left[\xi \left(t,x,\vartheta \right)\frac{\partial }{\partial x}+\tau \left(t,x,\vartheta \right)\frac{\partial }{\partial t}+ŋ\left(t,x,\vartheta \right)\frac{\partial }{\partial \vartheta }\right]Ө=0.$$(4) .

A function $\vartheta$ is called an invariant solution of (3)(3) $$D_t^{б}\vartheta -F\left(t,x,\vartheta ,{\vartheta }_x,{\vartheta }_{xx},\dots \right)=0,$$(3) .

Commonly, for the Lie group coefficient parameter, one can generate $\left(\mathrm{\epsilon }\ll 1\right)$ Lie group formations as (5) $$\begin{array}{c}{t}^{\mathrm{*}}=t+\mathrm{\epsilon }.\tau \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ x^{\mathrm{*}}=x+\mathrm{\epsilon }.\xi \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {\vartheta }^{\mathrm{*}}=\vartheta +\mathrm{\epsilon }.\mathrm{ŋ}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {D_{t^{\mathrm{*}}}^{\mathrm{б}}\vartheta }^{\mathrm{*}}=D_t^{\mathrm{б}}\vartheta +\mathrm{\epsilon }.{\mathrm{ŋ}}_{\mathrm{б}}^0\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{\partial {\vartheta }^{\mathrm{*}}}{\partial x^{\mathrm{*}}}={\vartheta }_x+\mathrm{\epsilon }.{\mathrm{ŋ}}^x\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{{\partial }^2{\vartheta }^{\mathrm{*}}}{\partial {x^{\mathrm{*}}}^2}={\vartheta }_{xx}+\mathrm{\epsilon }.{\mathrm{ŋ}}^{xx}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right).\end{array}$$(5) wherein $\xi , ŋ,$ and $\tau$ are the formations for both independent/dependent variables. Now, consider the explicit formation of ${\mathrm{ŋ}}^x$ and ${\mathrm{ŋ}}^{xx}$ which are (6) $$\begin{array}{c}{\mathrm{ŋ}}^x=D_x\left(\mathrm{ŋ}\right)-{\vartheta }_x{D}_x\left(\xi \right)-{\vartheta }_t{D}_x\left(\tau \right),\\ {\mathrm{ŋ}}^{xx}=D_x\left({\mathrm{ŋ}}^x\right)-{\vartheta }_{xx}{D}_x\left(\xi \right)-{\vartheta }_{xt}{D}_x\left(\tau \right),\end{array}$$(6) where $D_x$ is given with (7) $$D_x=\frac{\partial }{\partial x}+{\vartheta }_x\frac{\partial }{\partial \vartheta }.$$(7)

The algebra of symmetries can be formulated accordingly to (8) $$V=\xi \left(t,x,\vartheta \right)\frac{\partial }{\partial x}+\tau (t,x,\vartheta )\frac{\partial }{\partial t}+\mathrm{ŋ}(t,x,\vartheta )\frac{\partial }{\partial \vartheta }.$$(8)

According to the infinitesimal invariance criterion, we have (9) $${Pr}^{\left(2\right)}V\left[\nabla \right]=0\mathrm{ }\mathrm{when} \nabla =0,$$(9) where ${\nabla =D}_t^{\mathrm{б}}\vartheta -F\left(t,x,\vartheta ,{\vartheta }_x,{\vartheta }_{xx},\dots \right).$

Thereafter, $\mathrm{ }\mathrm{when} t=0,$ the invariance status gives (10) $$\tau \left(t,x,\vartheta \right)=0.$$(10)

In (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) , let $D_t^{б}$ be the total FDO. Then, the $б\mathrm{th}$ extended infinitesimal associated with (9)(9) $${Pr}^{\left(2\right)}V\left[\nabla \right]=0\mathrm{ }\mathrm{when} \nabla =0,$$(9) can be symbolized with (11) $${\mathrm{ŋ}}_{б}^0=D_t^{б}\left(\mathrm{ŋ}\right)+\xi D_t^{б}\left({\vartheta }_x\right)-D_t^{б}\left({\xi \vartheta }_x\right)+D_t^{б}\left(D_x\left(\tau \right)\vartheta \right)-D_t^{б+1}\left(\tau \vartheta \right)+\tau D_t^{б+1}\left(\vartheta \right).$$(11) Thus, the general form of the Leibnitz rule gives (12) $$D_t^{б}\left(f\left(t\right)g\left(t\right)\right)=\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{б-\wp }f\left(t\right)D_t^{б}g\left(t\right),$$(12) where $\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)$ is written with (13) $$\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)=\frac{\mathrm{\Gamma }(\wp -б)}{\mathrm{\Gamma }(1-б)\mathrm{\Gamma }(\wp +1)}{(-1)}^{\wp -1}б.$$(13)

Still, the rule of Leibnitz and (11)(11) $${\mathrm{ŋ}}_{б}^0=D_t^{б}\left(\mathrm{ŋ}\right)+\xi D_t^{б}\left({\vartheta }_x\right)-D_t^{б}\left({\xi \vartheta }_x\right)+D_t^{б}\left(D_x\left(\tau \right)\vartheta \right)-D_t^{б+1}\left(\tau \vartheta \right)+\tau D_t^{б+1}\left(\vartheta \right).$$(11) can turn into (14) $${\mathrm{ŋ}}_{б}^0=D_t^{б}\left(\mathrm{ŋ}\right)-бD_t\left(\tau \right)D_t^{б}\vartheta -\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{\wp }\left(\xi \right)D_t^{б-\wp }{\vartheta }_x-\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D_t^{\wp }\left(\tau \right)D_t^{б-\wp }\vartheta .$$(14)

Subsequently, one can be defined (15) $$\frac{d^{\wp }\varphi \left(h\right(t\left)\right)}{d{t}^{\wp }\varphi }=\sum _{k=0}^{\wp }\sum _{r=0}^k\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)\frac{1}{k!}{\left[-h\left(t\right)\right]}^r\frac{d^{\wp }}{d{t}^{\wp }}{\left(h\left(t\right)\right)}^{k-r}\frac{d^{\wp }\varphi \left(h\right)}{d{h}^k}.$$(15)

Utilizing Leibnitz rules to $f\left(t\right)=1,$ one gets (16) $$D_t^{б}\left(\mathrm{ŋ}\right)=\frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}+{\mathrm{ŋ}}_{\vartheta }\frac{{\partial }^{б}\vartheta }{{\partial \mathrm{t}}^{б}}-\vartheta \frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}+\sum _{\wp =0}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}{D}_t^{б-\wp }\left(\xi \right)D_t^{б-\wp }\vartheta +ϻ,$$(16) where $ϻ$ is constructed as (17) $$ϻ=\frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}-{\mathrm{ŋ}}_{\vartheta }\frac{{\partial }^{б}\vartheta }{{\partial \mathrm{t}}^{б}}-\vartheta \frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}+\sum _{\wp =1}^{\infty }\sum _{\mathfrak{H}=2}^{\wp }\sum _{k=2}^{\mathfrak{H}}\sum _{r=0}^{k-1}\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\left(\genfrac{}{}{0pt}{}{\wp }{\mathfrak{H}}\right)\left(\genfrac{}{}{0pt}{}{k}{r}\right)\frac{1}{k!}\frac{\left({\mathrm{t}}^{\wp -б}\right)}{\left(\Gamma \left(\wp +1-б\right)\right)}(-\vartheta {)}^r\frac{{\partial }^{\mathfrak{H}}}{{\partial \mathrm{t}}^{\mathfrak{H}}}({\vartheta }^{k-r})\frac{{\partial }^{\mathfrak{\wp }-\mathfrak{H}+k}\mathrm{ŋ}}{{\partial \mathrm{x}}^{\mathfrak{\wp }-\mathfrak{H}}{\partial \vartheta }^k}.$$(17)

Consequently, the $б\mathrm{th}$ extended infinitesimal of (14)(14) $${\mathrm{ŋ}}_{б}^0=D_t^{б}\left(\mathrm{ŋ}\right)-бD_t\left(\tau \right)D_t^{б}\vartheta -\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{\wp }\left(\xi \right)D_t^{б-\wp }{\vartheta }_x-\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D_t^{\wp }\left(\tau \right)D_t^{б-\wp }\vartheta .$$(14) can be transformed into (18) $${\mathrm{ŋ}}_{б}^0=\frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}+\left({\mathrm{ŋ}}_{\vartheta }-бD_t\left(\tau \right)\right)\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}-u\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}+ϻ+\sum _{\wp =1}^{\infty }\left[\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\frac{{\partial }^{\wp }{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{\wp }}-{\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D}_t^{\wp +1}\left(\tau \right)\right]D_t^{б-\wp }\left(\vartheta \right)-\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{\wp }\left(\xi \right)D_t^{б-\wp }{\vartheta }_x.$$(18)

3. The LPSs and reduction

Hereby, we attempt to get the trademark recipes of vector fields referenced already, and afterward, we will involve it for getting the decreased conditions using three unique situations. Thus, we establish two theorems in which the FNWSM (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) is reduced to an FDE of nonlinear type with the Erdélyi–Kober FDO that achieves the desired goal of the current area.

With the invariant assumption of (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) under formation (5)(5) $$\begin{array}{c}{t}^{\mathrm{*}}=t+\mathrm{\epsilon }.\tau \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ x^{\mathrm{*}}=x+\mathrm{\epsilon }.\xi \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {\vartheta }^{\mathrm{*}}=\vartheta +\mathrm{\epsilon }.\mathrm{ŋ}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {D_{t^{\mathrm{*}}}^{\mathrm{б}}\vartheta }^{\mathrm{*}}=D_t^{\mathrm{б}}\vartheta +\mathrm{\epsilon }.{\mathrm{ŋ}}_{\mathrm{б}}^0\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{\partial {\vartheta }^{\mathrm{*}}}{\partial x^{\mathrm{*}}}={\vartheta }_x+\mathrm{\epsilon }.{\mathrm{ŋ}}^x\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{{\partial }^2{\vartheta }^{\mathrm{*}}}{\partial {x^{\mathrm{*}}}^2}={\vartheta }_{xx}+\mathrm{\epsilon }.{\mathrm{ŋ}}^{xx}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right).\end{array}$$(5) , we have (19) $${\mathfrak{D}}_t^{б}{\vartheta }^{*}-a{\vartheta }_{xx}^{*}-b{\vartheta }^{*}+c{\left({\vartheta }^{*}\right)}^p=0.$$(19)

The attached subordinate symmetry determining equation is obtained when the point formation of (5)(5) $$\begin{array}{c}{t}^{\mathrm{*}}=t+\mathrm{\epsilon }.\tau \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ x^{\mathrm{*}}=x+\mathrm{\epsilon }.\xi \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {\vartheta }^{\mathrm{*}}=\vartheta +\mathrm{\epsilon }.\mathrm{ŋ}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {D_{t^{\mathrm{*}}}^{\mathrm{б}}\vartheta }^{\mathrm{*}}=D_t^{\mathrm{б}}\vartheta +\mathrm{\epsilon }.{\mathrm{ŋ}}_{\mathrm{б}}^0\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{\partial {\vartheta }^{\mathrm{*}}}{\partial x^{\mathrm{*}}}={\vartheta }_x+\mathrm{\epsilon }.{\mathrm{ŋ}}^x\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{{\partial }^2{\vartheta }^{\mathrm{*}}}{\partial {x^{\mathrm{*}}}^2}={\vartheta }_{xx}+\mathrm{\epsilon }.{\mathrm{ŋ}}^{xx}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right).\end{array}$$(5) is utilized in (19)(19) $${\mathfrak{D}}_t^{б}{\vartheta }^{*}-a{\vartheta }_{xx}^{*}-b{\vartheta }^{*}+c{\left({\vartheta }^{*}\right)}^p=0.$$(19) : (20) $${\mathrm{ŋ}}_{б}^0-a {\mathrm{ŋ}}^{xx}-b\mathrm{ŋ}+c{\mathrm{ŋ}}^p=0.$$(20)

By substituting (5)(5) $$\begin{array}{c}{t}^{\mathrm{*}}=t+\mathrm{\epsilon }.\tau \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ x^{\mathrm{*}}=x+\mathrm{\epsilon }.\xi \left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {\vartheta }^{\mathrm{*}}=\vartheta +\mathrm{\epsilon }.\mathrm{ŋ}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ {D_{t^{\mathrm{*}}}^{\mathrm{б}}\vartheta }^{\mathrm{*}}=D_t^{\mathrm{б}}\vartheta +\mathrm{\epsilon }.{\mathrm{ŋ}}_{\mathrm{б}}^0\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{\partial {\vartheta }^{\mathrm{*}}}{\partial x^{\mathrm{*}}}={\vartheta }_x+\mathrm{\epsilon }.{\mathrm{ŋ}}^x\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right),\\ \frac{{\partial }^2{\vartheta }^{\mathrm{*}}}{\partial {x^{\mathrm{*}}}^2}={\vartheta }_{xx}+\mathrm{\epsilon }.{\mathrm{ŋ}}^{xx}\left(t,x,\vartheta )+o({\mathrm{\epsilon }}^2\right).\end{array}$$(5) and (18)(18) $${\mathrm{ŋ}}_{б}^0=\frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}+\left({\mathrm{ŋ}}_{\vartheta }-бD_t\left(\tau \right)\right)\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}-u\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}+ϻ+\sum _{\wp =1}^{\infty }\left[\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\frac{{\partial }^{\wp }{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{\wp }}-{\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D}_t^{\wp +1}\left(\tau \right)\right]D_t^{б-\wp }\left(\vartheta \right)-\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{\wp }\left(\xi \right)D_t^{б-\wp }{\vartheta }_x.$$(18) into (20)(20) $${\mathrm{ŋ}}_{б}^0-a {\mathrm{ŋ}}^{xx}-b\mathrm{ŋ}+c{\mathrm{ŋ}}^p=0.$$(20) , we acquire (21) $$0=\frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}+\left({\mathrm{ŋ}}_{\vartheta }-бD_t\left(\tau \right)\right)\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}-\vartheta \frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}+ϻ+\sum _{\wp =1}^{\infty }\left[\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\frac{{\partial }^{\wp }{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{\wp }}-{\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D}_t^{\wp +1}\left(\tau \right)\right]D_t^{б-\wp }\left(\vartheta \right)-\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{\wp }\left(\xi \right)D_t^{б-\wp }{\vartheta }_x-a[{\mathrm{ŋ}}_{xx}+\left(2{\mathrm{ŋ}}_{x\vartheta }-{\xi }_{xx}\right){\vartheta }_x-{\tau }_{xx}{\vartheta }_t-\left({\mathrm{ŋ}}_{\vartheta \vartheta }-2{\xi }_{x\vartheta }\right){(\vartheta }_x{)}^2-2{\tau }_{x\vartheta }{\vartheta }_x{\vartheta }_t-{\xi }_{\vartheta \vartheta }{(\vartheta }_x{)}^3-{\tau }_{\vartheta \vartheta }{(\vartheta }_x{)}^2{\vartheta }_t-\left({\mathrm{ŋ}}_{\vartheta }-2{\xi }_x\right){\vartheta }_{xx}-2{\tau }_x{\vartheta }_{xt}-3{\xi }_{\vartheta }{\vartheta }_{xx}{\vartheta }_x-{\tau }_{\vartheta }{\vartheta }_{xx}{\vartheta }_t-2{\tau }_{\vartheta }{\vartheta }_{xt}{\vartheta }_x]-b\mathrm{ŋ}+c{\mathrm{ŋ}}^p.$$(21)

In (21)(21) $$0=\frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}+\left({\mathrm{ŋ}}_{\vartheta }-бD_t\left(\tau \right)\right)\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}-\vartheta \frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}+ϻ+\sum _{\wp =1}^{\infty }\left[\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\frac{{\partial }^{\wp }{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{\wp }}-{\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D}_t^{\wp +1}\left(\tau \right)\right]D_t^{б-\wp }\left(\vartheta \right)-\sum _{\wp =1}^{\infty }\left(\genfrac{}{}{0pt}{}{б}{\wp }\right)D_t^{\wp }\left(\xi \right)D_t^{б-\wp }{\vartheta }_x-a[{\mathrm{ŋ}}_{xx}+\left(2{\mathrm{ŋ}}_{x\vartheta }-{\xi }_{xx}\right){\vartheta }_x-{\tau }_{xx}{\vartheta }_t-\left({\mathrm{ŋ}}_{\vartheta \vartheta }-2{\xi }_{x\vartheta }\right){(\vartheta }_x{)}^2-2{\tau }_{x\vartheta }{\vartheta }_x{\vartheta }_t-{\xi }_{\vartheta \vartheta }{(\vartheta }_x{)}^3-{\tau }_{\vartheta \vartheta }{(\vartheta }_x{)}^2{\vartheta }_t-\left({\mathrm{ŋ}}_{\vartheta }-2{\xi }_x\right){\vartheta }_{xx}-2{\tau }_x{\vartheta }_{xt}-3{\xi }_{\vartheta }{\vartheta }_{xx}{\vartheta }_x-{\tau }_{\vartheta }{\vartheta }_{xx}{\vartheta }_t-2{\tau }_{\vartheta }{\vartheta }_{xt}{\vartheta }_x]-b\mathrm{ŋ}+c{\mathrm{ŋ}}^p.$$(21) , making the powers of derivatives of $\vartheta$ to be $0,$ one has the subordinate system: (22) $$\begin{array}{c}{\tau }_x=0,\\ {\tau }_{\vartheta }=0,\\ {\xi }_{\vartheta }=0,\\ {\tau }_{\vartheta }=0,\\ {\xi }_{\vartheta \vartheta }=0,\\ {\tau }_{xx}=0,\\ {\tau }_{x\vartheta }=0,\\ \frac{{\partial }^{б}\mathrm{ŋ}}{{\partial \mathrm{t}}^{б}}-\frac{{\partial }^{б}{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{б}}-\lambda \mathrm{ŋ}\left(1-\mathrm{ŋ}\right)=0,\\ \left(\genfrac{}{}{0pt}{}{б}{\wp }\right)\frac{{\partial }^{\wp }{\mathrm{ŋ}}_{\vartheta }}{{\partial \mathrm{t}}^{\wp }}-{\left(\genfrac{}{}{0pt}{}{б}{\wp +1}\right)D}_t^{\wp +1}\left(\tau \right)=0,\wp =1,2,3,\dots ,\\ D_t^{\wp }\left(\xi \right)=0,\wp =1,2,3,\dots ,\\ 2{\mathrm{ŋ}}_{x\vartheta }-{\xi }_{xx}=0,\\ {\mathrm{ŋ}}_{\vartheta \vartheta }-2{\xi }_{x\vartheta }=0,\\ {\mathrm{ŋ}}_{\vartheta }-бD_t\left(\tau \right)=0,\\ {\mathrm{ŋ}}_{\vartheta }-2{\xi }_x=0.\end{array}$$(22)

With the use of an appropriate software approach or a convenient hand method, we gain the following infinitesimals: (23) $$\begin{array}{c}\xi =2бx{\mathrm{c}}_1+{\mathrm{c}}_2,\\ \tau =4t{\mathrm{c}}_1,\\ \mathrm{ŋ}=\left(3б-2\right)\vartheta {\mathrm{c}}_1+\vartheta {\mathrm{c}}_3,\end{array}$$(23) where $c_1,c_2,c_3\in \mathbb{R}.$

Anyhow, Lie algebra for infinitesimal symmetries for (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) could be (24) $$\begin{array}{c}{V}_1=\frac{\partial }{\partial x},\\ V_2=2бx\frac{\partial }{\partial x}+4t\frac{\partial }{\partial t}+\left(3б-2\right)\vartheta \frac{\partial }{\partial \vartheta },\\ V_3=\vartheta \frac{\partial }{\partial \vartheta }.\end{array}$$(24)

Now, to get the reduction equations, we employ (24)(24) $$\begin{array}{c}{V}_1=\frac{\partial }{\partial x},\\ V_2=2бx\frac{\partial }{\partial x}+4t\frac{\partial }{\partial t}+\left(3б-2\right)\vartheta \frac{\partial }{\partial \vartheta },\\ V_3=\vartheta \frac{\partial }{\partial \vartheta }.\end{array}$$(24) and consider the attached cases for the Lie algebra of infinitesimal symmetries that are previously mentioned:

• Case I: Related to the infinitesimal generator $V_1,$ we have the following characteristic formula: (25) $$\frac{\mathrm{d}x}{1}=\frac{\mathrm{d}t}{0}=\frac{\mathrm{d}\vartheta }{0}.$$(25)

The solution of (25)(25) $$\frac{\mathrm{d}x}{1}=\frac{\mathrm{d}t}{0}=\frac{\mathrm{d}\vartheta }{0}.$$(25) will lead to $\omega =t$ and $\vartheta =h\left(\omega \right)$ where $D_t^{б}h(\omega )=0.$ Thus, for an arbitrary parameter $k_1;$ ${\vartheta }_1=k_1{\mathrm{t}}^{б-1}$ is the group-invariant solution and clearly, it is congruent to $V_1.$

• Case II: Related to the infinitesimal generator $V_2,$ we have the following characteristic formula: (26) $$\frac{\mathrm{d}x}{2бx}=\frac{\mathrm{d}t}{4t}=\frac{\mathrm{d}\vartheta }{(3б-2)\vartheta }.$$(26)

To summarize and by solving (26)(26) $$\frac{\mathrm{d}x}{2бx}=\frac{\mathrm{d}t}{4t}=\frac{\mathrm{d}\vartheta }{(3б-2)\vartheta }.$$(26) , we get (27) $$\begin{array}{c}{\omega }_1=x{t}^{\frac{-б}{2}},\\ {\omega }_2=\vartheta t^{\frac{-(3б-2)}{4}}.\end{array}$$(27)

So, considering the symmetry $V_3$ and for an arbitrary $Q$ in $\omega ,$ the group invariant solution is (28) $$\begin{array}{c}\omega =x{t}^{\frac{-б}{2}},\\ \vartheta =t^{\frac{(3б-2)}{4}} Q\left(\omega \right).\end{array}$$(28)

Theorem 1.

Considering the formation of (29)(29) $$\left(P_{\frac{2}{б}}^{\frac{(2-б)}{4},б}Q\right)\left(\mathrm{z}\right)-\mathrm{a}{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right)=0,$$(29) , the FNWSM (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) is minimized for (29) $$\left(P_{\frac{2}{б}}^{\frac{(2-б)}{4},б}Q\right)\left(\mathrm{z}\right)-\mathrm{a}{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right)=0,$$(29) with ${(\mathrm{P}}_{\mathrm{\beta }}^{\mathrm{\tau },\mathrm{\alpha }})$ defined as (30) $$\left(P_{\beta }^{\zeta ,б}f\right)\left(\omega \right)=\prod _{j=0}^{\wp -1}\left(\zeta -j-\frac{1}{\beta }\omega \frac{d}{d\omega }\right)\left(K_{\beta }^{\zeta +б,\wp -б} f\right)\left(\omega \right),$$(30) (31) $$\wp =\{\begin{array}{c}\left[б\right]+1,б\ne \mathbb{N},\\ б,б=1,2,3,\dots .\end{array}$$(31)

In which $\omega ,\mathrm{\beta },б>0$ and (32) $$\left(K_{\beta }^{\zeta ,б}f\right)\left(\omega \right)=\{\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(б\right)}\underset{0}{\overset{\infty }{\int }}{\left(u-1\right)}^{б-1}{u}^{(\zeta +б)}f\left(\omega u^{\frac{1}{\beta }}\right)du,б>0,\\ f\left(\omega \right),б=0.\end{array}$$(32)

Proof.

Assume $\wp >б>\wp -1$ with $\wp =1,2,3,\dots$ and using (19)(19) $${\mathfrak{D}}_t^{б}{\vartheta }^{*}-a{\vartheta }_{xx}^{*}-b{\vartheta }^{*}+c{\left({\vartheta }^{*}\right)}^p=0.$$(19) , one gets (33) $$\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[ \frac{1}{\mathrm{\Gamma }(\wp -б)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\wp -б-1}{\mathrm{s}}^{\frac{(3б-2)}{4}}Q\left(x{\mathrm{s}}^{\frac{-б}{2}}\right)ds\right].$$(33)

Letting $r=\frac{t}{\mathrm{s}},$ we have $d\mathrm{s}=-\frac{t}{r^2} dr.$ Thus, (34)(34) $$\begin{array}{c}\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}{s^{\wp -б-1}\left(\frac{t}{s}-1\right)}^{\wp -б-1}{s}^{\frac{\left(3б-2\right)}{4}}Q\left(x\frac{t^{\frac{б}{2}}}{s^{\frac{б}{2}}}{t}^{\frac{-б}{2}}\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\frac{s^{\wp -б-1+\frac{\left(3б-2\right)}{4}}}{t^{\wp -б-1+\frac{\left(3б-2\right)}{4}}}{t}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(\frac{t}{s}-1\right)}^{\wp -б-1}Q\left(x{t}^{\frac{-б}{2}}{\left(\frac{t}{s}\right)}^{\frac{б}{2}}\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б-1+}\frac{\left(3б-2\right)}{4}}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\left({\left(\frac{s}{t}\right)}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(\frac{t}{s}-1\right)}^{\wp -б-1}Q\left(x{t}^{\frac{-б}{2}}{\left(\frac{t}{s}\right)}^{\frac{б}{2}}\right)\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б-1+}\frac{\left(3б-2\right)}{4}}{\Gamma \left(\wp -б\right)}\underset{+\infty }{\overset{1}{\int }}{\left( \frac{1}{r} \right)}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(r-1\right)}^{\wp -б-1}Q\left(\omega r^{\frac{б}{2}}\right)\left(-\frac{t}{r^2}\right)ds]\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б+\frac{\left(3б-2\right)}{4}}}{\Gamma \left(\wp -б\right)}\underset{1}{\overset{\infty }{\int }}{r}^{-\left(\wp -б+1+\frac{\left(3б-2\right)}{4}\right)}{\left(r-1\right)}^{\wp -б-1}Q\left(\omega r^{\frac{б}{2}}\right)dr]\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},n-б}Q\right)\left(\omega \right)\right].\end{array}$$(34) can be established as (34) $$\begin{array}{c}\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}{s^{\wp -б-1}\left(\frac{t}{s}-1\right)}^{\wp -б-1}{s}^{\frac{\left(3б-2\right)}{4}}Q\left(x\frac{t^{\frac{б}{2}}}{s^{\frac{б}{2}}}{t}^{\frac{-б}{2}}\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\frac{s^{\wp -б-1+\frac{\left(3б-2\right)}{4}}}{t^{\wp -б-1+\frac{\left(3б-2\right)}{4}}}{t}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(\frac{t}{s}-1\right)}^{\wp -б-1}Q\left(x{t}^{\frac{-б}{2}}{\left(\frac{t}{s}\right)}^{\frac{б}{2}}\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б-1+}\frac{\left(3б-2\right)}{4}}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\left({\left(\frac{s}{t}\right)}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(\frac{t}{s}-1\right)}^{\wp -б-1}Q\left(x{t}^{\frac{-б}{2}}{\left(\frac{t}{s}\right)}^{\frac{б}{2}}\right)\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б-1+}\frac{\left(3б-2\right)}{4}}{\Gamma \left(\wp -б\right)}\underset{+\infty }{\overset{1}{\int }}{\left( \frac{1}{r} \right)}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(r-1\right)}^{\wp -б-1}Q\left(\omega r^{\frac{б}{2}}\right)\left(-\frac{t}{r^2}\right)ds]\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б+\frac{\left(3б-2\right)}{4}}}{\Gamma \left(\wp -б\right)}\underset{1}{\overset{\infty }{\int }}{r}^{-\left(\wp -б+1+\frac{\left(3б-2\right)}{4}\right)}{\left(r-1\right)}^{\wp -б-1}Q\left(\omega r^{\frac{б}{2}}\right)dr]\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},n-б}Q\right)\left(\omega \right)\right].\end{array}$$(34)

Simplifying (34)(34) $$\begin{array}{c}\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}{s^{\wp -б-1}\left(\frac{t}{s}-1\right)}^{\wp -б-1}{s}^{\frac{\left(3б-2\right)}{4}}Q\left(x\frac{t^{\frac{б}{2}}}{s^{\frac{б}{2}}}{t}^{\frac{-б}{2}}\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\frac{s^{\wp -б-1+\frac{\left(3б-2\right)}{4}}}{t^{\wp -б-1+\frac{\left(3б-2\right)}{4}}}{t}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(\frac{t}{s}-1\right)}^{\wp -б-1}Q\left(x{t}^{\frac{-б}{2}}{\left(\frac{t}{s}\right)}^{\frac{б}{2}}\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б-1+}\frac{\left(3б-2\right)}{4}}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\left({\left(\frac{s}{t}\right)}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(\frac{t}{s}-1\right)}^{\wp -б-1}Q\left(x{t}^{\frac{-б}{2}}{\left(\frac{t}{s}\right)}^{\frac{б}{2}}\right)\right)ds\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б-1+}\frac{\left(3б-2\right)}{4}}{\Gamma \left(\wp -б\right)}\underset{+\infty }{\overset{1}{\int }}{\left( \frac{1}{r} \right)}^{\wp -б-1+\frac{\left(3б-2\right)}{4}}{\left(r-1\right)}^{\wp -б-1}Q\left(\omega r^{\frac{б}{2}}\right)\left(-\frac{t}{r^2}\right)ds]\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[\frac{t^{\wp -б+\frac{\left(3б-2\right)}{4}}}{\Gamma \left(\wp -б\right)}\underset{1}{\overset{\infty }{\int }}{r}^{-\left(\wp -б+1+\frac{\left(3б-2\right)}{4}\right)}{\left(r-1\right)}^{\wp -б-1}Q\left(\omega r^{\frac{б}{2}}\right)dr]\right]\\ =\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},n-б}Q\right)\left(\omega \right)\right].\end{array}$$(34) and utilizing $\omega ={tx}^{-\frac{1}{2}б},$ one has (35) $$\begin{array}{c}t\frac{\partial }{\partial t}\phi \left(\omega \right)=tx\left(-\frac{1}{2}б\right)t^{-\frac{1}{2}б-1} {\phi }^{\prime }\left(\omega \right)\\ =-\frac{1}{2}б\omega \frac{\partial }{\partial \mathrm{z}}\phi \left(\omega \right).\end{array}$$(35)

Anyhow, we have (36) $$\begin{array}{c}\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}Q\right)\left(\omega \right)\right]=\frac{{\partial }^{\wp -1}}{{\partial t}^{\wp -1}}\left[\frac{\partial }{\partial t}\left( t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{3б-2}{4},\wp -б}Q\right)\left(\omega \right)\right)\right]\\ =\frac{{\partial }^{\wp -1}}{{\partial t}^{\wp -1}}\left[t^{\wp -б-1+\frac{3б-2}{4}}\left(\wp -б+\frac{\left(3б-2\right)}{4}-\frac{б}{2}\omega \frac{\partial }{\partial \omega }\right)\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}Q\right)\left(\omega \right).\right]\end{array}$$(36)

Further results for $\wp -1$ times is (37) $$\begin{array}{c}\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[ t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left({(K}_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}f\right)\left(\omega \right)\right] =\cdots =t^{-б+\frac{\left(3б-2\right)}{4}}{\prod }_{k=1}^{\wp }\left[\left(1-б+\frac{\left(3б-2\right)}{4}+j-\frac{б}{2}\omega \frac{\partial }{\partial \omega }\right)\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}Q\right)\left(\omega \right)\right]\end{array}.$$(37) Furthermore, (37)(37) $$\begin{array}{c}\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[ t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left({(K}_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}f\right)\left(\omega \right)\right] =\cdots =t^{-б+\frac{\left(3б-2\right)}{4}}{\prod }_{k=1}^{\wp }\left[\left(1-б+\frac{\left(3б-2\right)}{4}+j-\frac{б}{2}\omega \frac{\partial }{\partial \omega }\right)\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}Q\right)\left(\omega \right)\right]\end{array}.$$(37) may be formed with (38) $$\begin{array}{c}\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=t^{-б+\frac{\left(3б-2\right)}{4}}\left(P_{\frac{2}{б}}^{1-б+\frac{\left(3б-2\right)}{4},б}Q\right)\left(\omega \right)\\ =t^{-\frac{\left(б+2\right)}{4}}\left(P_{\frac{2}{б}}^{1-б+\frac{\left(3б-2\right)}{4},б}Q\right)\left(\omega \right).\end{array}$$(38)

The outcome of (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) , considering (38)(38) $$\begin{array}{c}\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=t^{-б+\frac{\left(3б-2\right)}{4}}\left(P_{\frac{2}{б}}^{1-б+\frac{\left(3б-2\right)}{4},б}Q\right)\left(\omega \right)\\ =t^{-\frac{\left(б+2\right)}{4}}\left(P_{\frac{2}{б}}^{1-б+\frac{\left(3б-2\right)}{4},б}Q\right)\left(\omega \right).\end{array}$$(38) , is formulated with (39) $$\begin{array}{c}0=\left(P_{\frac{2}{б}}^{1+\frac{(3б-2)}{4}-б,б}Q\right)\left(\omega \right)-a{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right)\\ =\left(P_{\frac{2}{б}}^{\frac{(2-б)}{4},б}Q\right)\left(\omega \right)-a{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right).\end{array}$$(39)

As required.

• Case III: Related to the infinitesimal generator $V_1+V_2+V_3,$ we have the following characteristic formula: (40) $$\frac{\mathrm{d}x}{2бx+1}=\frac{\mathrm{d}t}{4t}=\frac{\mathrm{d}\vartheta }{(3б-1)\vartheta },$$(40)

this gives the attached similarity variables for an arbitrary function $Q:$ (41) $$\begin{array}{c}\omega =\left(\frac{2бx+1}{2б}\right)t^{\frac{-б}{2}},\\ \vartheta =t^{\frac{(3б-1)}{4}} Q\left(\omega \right).\end{array}$$(41)

The following theorem is utilized to produce a different reduction of FNWSM (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) to an FDE.

Theorem 2.

Formation of (41)(41) $$\begin{array}{c}\omega =\left(\frac{2бx+1}{2б}\right)t^{\frac{-б}{2}},\\ \vartheta =t^{\frac{(3б-1)}{4}} Q\left(\omega \right).\end{array}$$(41) minimizes FNWSM (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) for (42) $$\left(P_{\frac{2}{б}}^{\frac{(3-б)}{4},б}Q\right)\left(\mathrm{z}\right)-\mathrm{a}{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right)=0.$$(42)

Proof.

Similar to Theorem 1, we have (43) $$\frac{{\partial }^{б}\vartheta }{{\partial \mathrm{t}}^{б}}=\frac{{\partial }^{\wp }}{{\partial \mathrm{t}}^{\wp }}\left[ \frac{1}{\Gamma (\wp -б)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\wp -б-1}{\mathrm{s}}^{\frac{(3б-1)}{4}}Q\left(\frac{2бx+1}{2б}{\mathrm{s}}^{\frac{-б}{2}}\right)ds\right].$$(43)

Considering a similar change of variable in Theorem 2, one has (44) $$\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[t^{\wp -\frac{(б+1)}{4}}\left(K_{\frac{2}{б}}^{\frac{\left(3б+1\right)}{2},\wp -б}Q\right)\left(\omega \right)\right].$$(44) From (36)(36) $$\begin{array}{c}\frac{{\partial }^{\wp }}{{\partial t}^{\wp }}\left[t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}Q\right)\left(\omega \right)\right]=\frac{{\partial }^{\wp -1}}{{\partial t}^{\wp -1}}\left[\frac{\partial }{\partial t}\left( t^{\wp -б+\frac{\left(3б-2\right)}{4}}\left(K_{\frac{2}{б}}^{1+\frac{3б-2}{4},\wp -б}Q\right)\left(\omega \right)\right)\right]\\ =\frac{{\partial }^{\wp -1}}{{\partial t}^{\wp -1}}\left[t^{\wp -б-1+\frac{3б-2}{4}}\left(\wp -б+\frac{\left(3б-2\right)}{4}-\frac{б}{2}\omega \frac{\partial }{\partial \omega }\right)\left(K_{\frac{2}{б}}^{1+\frac{\left(3б-2\right)}{4},\wp -б}Q\right)\left(\omega \right).\right]\end{array}$$(36) , we obtained (45) $$\begin{array}{c}t\frac{\partial }{\partial t}\phi \left(\omega \right)=tx\left(-\frac{1}{2}б\right)t^{-\frac{1}{2}б-1} {\phi }^{\prime }\left(\omega \right)\\ =-\frac{1}{2}б\omega \frac{\partial }{\partial \mathrm{z}}\phi \left(\omega \right).\end{array}$$(45) With the same process of Theorem 2, we gain the following formulate: (46) $$\frac{{\partial }^{б}\vartheta }{{\partial t}^{б}}=t^{-\frac{(б+1)}{4}}\left(P_{\frac{2}{б}}^{\frac{3-б}{4},б}Q\right)\left(\omega \right).$$(46)

This completes the proof.

4. Implementation of the power series approach

This section is mainly devoted to employing the power series method (Abu Arqub, 2019) to solve FNWSM (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) of Case II in form of an explicit series solution. The presented method is utilized to solve several nonlinear differential equations. It is known to be a simple, accurate, and effective approach.

First, consider the substance expansion expression: (47) $$Q\left(\omega \right)=\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }.$$(47)

By differentiating both sides of (47)(47) $$Q\left(\omega \right)=\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }.$$(47) , one obtains (48) $$\begin{array}{c}{Q}^{\prime }\left(\omega \right)=\sum _{\wp =0}^{\infty }{\left(\wp +1\right)a}_{\wp +1}{\omega }^{\wp },\\ Q^{″}\left(\omega \right)=\sum _{\wp =0}^{\infty }{\left(\wp +1\right)\left(\wp +2\right)a}_{\wp +2}{\omega }^{\wp }.\end{array}$$(48)

Putting (47)(47) $$Q\left(\omega \right)=\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }.$$(47) and (48)(48) $$\begin{array}{c}{Q}^{\prime }\left(\omega \right)=\sum _{\wp =0}^{\infty }{\left(\wp +1\right)a}_{\wp +1}{\omega }^{\wp },\\ Q^{″}\left(\omega \right)=\sum _{\wp =0}^{\infty }{\left(\wp +1\right)\left(\wp +2\right)a}_{\wp +2}{\omega }^{\wp }.\end{array}$$(48) in (29)(29) $$\left(P_{\frac{2}{б}}^{\frac{(2-б)}{4},б}Q\right)\left(\mathrm{z}\right)-\mathrm{a}{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right)=0,$$(29) , one obtains (49) $$\sum _{\wp =0}^{\infty }\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }{\omega }^{\wp }-a\sum _{\wp =0}^{\infty }{\left(\wp +1\right)\left(\wp +2\right)a}_{\wp +2}{\omega }^{\wp }-b\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }+c{\left(\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }\right)}^{\wp }=0.$$(49)

Consequently, (50) $$\sum _{\wp =0}^{\infty }\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }{\omega }^{\wp }-a\sum _{\wp =0}^{\infty }{\left(\wp +1\right)\left(\wp +2\right)a}_{\wp +2}{\omega }^{\wp }-b\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }+c\sum _{\wp =0}^{\infty }\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}\omega }^{\wp }=0.$$(50)

Balancing coefficients in (50)(50) $$\sum _{\wp =0}^{\infty }\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }{\omega }^{\wp }-a\sum _{\wp =0}^{\infty }{\left(\wp +1\right)\left(\wp +2\right)a}_{\wp +2}{\omega }^{\wp }-b\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }+c\sum _{\wp =0}^{\infty }\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}\omega }^{\wp }=0.$$(50) at $\wp =0,$ one has (51) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right).$$(51) But, when $\wp \ge 1,$ we are bound with (52) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}).$$(52)

Thus, we can write the solution of (29)(29) $$\left(P_{\frac{2}{б}}^{\frac{(2-б)}{4},б}Q\right)\left(\mathrm{z}\right)-\mathrm{a}{Q}^{″}\left(\omega \right)-bQ\left(\omega \right)+c{Q}^p\left(\omega \right)=0,$$(29) explicitly as (53) $$f\left(\omega \right)=a_0+a_1\omega +\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right){\omega }^2+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}){\omega }^{\wp +2}.$$(53)

Therefore, the explicit power series is (54) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(54)

Wherein, the parameters $a_0$ and $a_1$ are arbitrary and $a_{\wp },$ $\wp =2,3,4,\dots$ are computed equational by relations (51)(51) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right).$$(51) and (52)(52) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}).$$(52) .

Now, it is essential to establish the convergence study of (47)(47) $$Q\left(\omega \right)=\sum _{\wp =0}^{\infty }{a}_{\wp }{\omega }^{\wp }.$$(47) to surround the analysis in the desired domain space. By taking the absolute value of (52)(52) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}).$$(52) and applying the triangular inequality, one obtains (55) $$\left|a_{\wp +2}\right|⩽\frac{\left|\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)\right|}{\left|a\right|\left|\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)\right|}\left|a_{\wp }\right|+\left|b\right|\left|a_{\wp }\right|+\left|c\right|\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}\left|a_{i_{p-1}}\right|\left|a_{i_{p-2}-i_{p-1}}\right|\left|a_{i_{p-3}-i_{p-2}}\right|\cdots \left|a_{i_1-i_2}{a}_{\wp -i_1}\right|.$$(55)

But, $\frac{|\mathrm{\Gamma }\left(\wp \right)|}{|\mathrm{\Gamma }\left(\mathfrak{H}\right)|}<1$ at each $\wp (\mathfrak{H}),$ so, (55)(55) $$\left|a_{\wp +2}\right|⩽\frac{\left|\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)\right|}{\left|a\right|\left|\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)\right|}\left|a_{\wp }\right|+\left|b\right|\left|a_{\wp }\right|+\left|c\right|\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}\left|a_{i_{p-1}}\right|\left|a_{i_{p-2}-i_{p-1}}\right|\left|a_{i_{p-3}-i_{p-2}}\right|\cdots \left|a_{i_1-i_2}{a}_{\wp -i_1}\right|.$$(55) turn into (56) $$\left|a_{\wp +2}\right|⩽Ƙ\left[\left|a_{\wp }\right|+\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}\left|a_{i_{p-1}}\right|\left|a_{i_{p-2}-i_{p-1}}\right|\left|a_{i_{p-3}-i_{p-2}}\right|\cdots \left|a_{i_1-i_2}\left|\right|a_{\wp -i_1}\right|\right],$$(56) where $\mathrm{Ƙ}=\max \hspace{0.17em}\left\{\frac{1}{a},b,c,2\right\}$ and $a\ne 0.$

Utilized, a new version series as (57) $$M\left(\omega \right)=\sum _{\wp =0}^{\infty }{\mathcal{s}}_{\wp }{\omega }^{\wp }.$$(57)

For $i=0,1,2,\dots ,$ let ${\mathcal{s}}_i=\left|a_i\right|,$ and so (58) $${\mathcal{s}}_{\wp +2}⩽Ƙ\left[{\mathcal{s}}_{\wp }+\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{\mathcal{s}}_{i_{p-1}}{\mathcal{s}}_{i_{p-2}-i_{p-1}}{\mathcal{s}}_{i_{p-3}-i_{p-2}}\cdots {\mathcal{s}}_{i_1-i_2}{\mathcal{s}}_{\wp -i_1}\right].$$(58)

So, it is easily shown that $\left|a_{\wp }\right|⩽{\mathcal{s}}_{\wp }$ with $\wp =0,1,2,\dots$ and $M\left(\omega \right)={\sum }_{\wp =0}^{\infty }{\mathcal{s}}_{\wp }{\omega }^{\wp }.$ Anyhow, one gets (59) $$M\left(\omega \right)={\mathcal{s}}_0+{\mathcal{s}}_1\omega +{\mathcal{s}}_2{\omega }^2+Ƙ\sum _{\wp =1}^{\infty }{\mathcal{s}}_{\wp }{\omega }^{\wp +2}+Ƙ\sum _{\wp =1}^{\infty }\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{\mathcal{s}}_{i_{p-1}}{\mathcal{s}}_{i_{p-2}-i_{p-1}}{\mathcal{s}}_{i_{p-3}-i_{p-2}}\cdots {\mathcal{s}}_{i_1-i_2}{\mathcal{s}}_{\wp -i_1}{\omega }^{\wp +2}.$$(59)

Now, the series $M\left(\omega \right)={\sum }_{\wp =0}^{\infty }{\mathcal{s}}_{\wp }{\omega }^{\wp }$ is of the positive radius of convergence. This can be proven by considering the subsequent implicit equation in $\omega$-variable as (60) $$R\left(\omega ,M\right)=M{-\mathcal{s}}_0-{\mathcal{s}}_1\omega -{\mathcal{s}}_2{\omega }^2-Ƙ\sum _{\wp =1}^{\infty }{\mathcal{s}}_{\wp }{\omega }^{\wp +2}-Ƙ\sum _{\wp =1}^{\infty }\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{\mathcal{s}}_{i_{p-1}}{\mathcal{s}}_{i_{p-2}-i_{p-1}}{\mathcal{s}}_{i_{p-3}-i_{p-2}}\cdots {\mathcal{s}}_{i_1-i_2}{\mathcal{s}}_{\wp -i_1}{\omega }^{\wp +2}=M{-\mathcal{s}}_0-{\mathcal{s}}_1\omega -{\mathcal{s}}_2{\omega }^2-Ƙ(M-{\mathcal{s}}_0){\omega }^2-Ƙ\left(M^p-{{\mathcal{s}}_0}^p\right){\omega }^2.$$(60)

But $R\left(\omega ,M\right)$ is analytic in $\left(0,{\mathcal{s}}_0\right)$ where $R\left(0,{\mathcal{s}}_0\right)=0$ and $\frac{\partial }{\partial M}R\left(0,{\mathcal{s}}_0\right)\ne 0.$ Applying the implicit function lemma, convergence is held.

Now, we will present the subordinate statuses:

• Status I: At $p=1$ results in (51)(51) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right).$$(51) , (52)(52) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}).$$(52) , and (54)(54) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(54) can be written, simultaneously, as (61) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-a_0(c-b)\right).$$(61) (62) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }+a_{\wp }(c-b)\right).$$(62) (63) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-a_0(c-b)\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }+(c-b)a_{\wp }\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(63)

• Status II: At $p=2$ results in (51)(51) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right).$$(51) , (52)(52) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}).$$(52) , and (54)(54) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(54) can be written, simultaneously, as (64) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^2\right).$$(64) (65) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }{a}_k{a}_{\wp -k}\right).$$(65) (66) $$\vartheta (x,t{)=a}_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^2)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }{a}_k{a}_{\wp -k}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(66)

• Status III: When $p=3$ in results in (51)(51) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right).$$(51) , (52)(52) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}).$$(52) , and (54)(54) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-b{a}_0+c{\left(a_0\right)}^p\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-b{a}_{\wp }+c\sum _{i_1=0}^{\wp }\sum _{i_2=0}^{i_1}\cdots \sum _{i_{p-1}=0}^{i_{p-2}}{a}_{i_{p-1}}{a}_{i_{p-2}-i_{p-1}}{a}_{i_{p-3}-i_{p-2}}\cdots a_{i_1-i_2}{a}_{\wp -i_1}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(54) can be written, simultaneously, as (67) $$a_2=\frac{1}{2a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^3\right).$$(67) (68) $$a_{\wp +2}=\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }\sum _{j=0}^k{a}_j{a}_{k-j}{a}_{\wp -k}\right).$$(68) (69) $$\vartheta (x,t{)=a}_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^3)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(2+\frac{\left(3б-2\right)}{4}-б-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }\sum _{j=0}^k{a}_j{a}_{k-j}{a}_{\wp -k}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(69)

Next, to sketch the $5\mathrm{t}\mathrm{h}$ series truncated in (63)(63) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-a_0(c-b)\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }+(c-b)a_{\wp }\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(63) , (66)(66) $$\vartheta (x,t{)=a}_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^2)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }{a}_k{a}_{\wp -k}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(66) , and (69)(69) $$\vartheta (x,t{)=a}_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^3)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(2+\frac{\left(3б-2\right)}{4}-б-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }\sum _{j=0}^k{a}_j{a}_{k-j}{a}_{\wp -k}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(69) , we use the MATHCAD 14 program for $a,$ $b,$ $c,$ $a_0,$ $a_1,$ and $б.$ Plot 1 from (a–d) relates to the quintuple $\left(a,b,c,a_0,a_1\right)=\left(1,1,0,1,1\right).$ Plot 2 from (a–d) relates to the quintuple $\left(a,b,c,a_0,a_1\right)=\left(1,2,3,1,1\right).$ Plot 3 from (a–d) relates to the quintuple $\left(a,b,c,a_0,a_1\right)=\left(0.5,2,1,1,1\right).$ Thither, the considered values of fractional order derivatives are taken $б=1,$ $б=0.975,$ $б=0.95,$ and $б=0.925$ for the triple figures.

Plot 1. View of the approximate result of $5\mathrm{t}\mathrm{h}$ truncated terms in (63)(63) $$\vartheta \left(x,t\right)=a_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}\left( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-a_0(c-b)\right)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }+(c-b)a_{\wp }\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(63) at $\left(a,b,c,a_0,a_1\right)=\left(1,1,0,1,1\right)$ within (a) $б=1,$ (b) $б=0.975,$ (c) $б=0.95,$ and (d) $б=0.925.$

Plot 2. View of the approximate result of $5\mathrm{t}\mathrm{h}$ truncated terms in (66)(66) $$\vartheta (x,t{)=a}_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^2)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(\frac{6-б}{4}-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }{a}_k{a}_{\wp -k}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(66) at $\left(a,b,c,a_0,a_1\right)=\left(1,2,3,1,1\right)$ within (a) $б=1,$ (b) $б=0.975,$ (c) $б=0.95,$ and (d) $б=0.925.$

Plot 3. View of the approximate result of $5\mathrm{t}\mathrm{h}$ truncated terms in (69)(69) $$\vartheta (x,t{)=a}_0{t}^{\frac{\left(3б-2\right)}{4}}+a_1{xt}^{\frac{\left(б-2\right)}{4}}+\frac{1}{2a}( \frac{\mathrm{\Gamma }\left(\frac{6-б}{4}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}\right)}{a}_0-{ba}_0+c{\left(a_0\right)}^3)x^2{t}^{\frac{-\left(б+2\right)}{4}}+\sum _{\wp =1}^{\infty }\frac{1}{\left(\wp +1\right)\left(\wp +2\right)a}{\left(\frac{\mathrm{\Gamma }\left(2+\frac{\left(3б-2\right)}{4}-б-\frac{\wp б}{2}\right)}{\mathrm{\Gamma }\left(\frac{3б+6}{4}-\frac{\wp б}{2}\right)}{a}_{\wp }-{ba}_{\wp }+c\sum _{k=0}^{\wp }\sum _{j=0}^k{a}_j{a}_{k-j}{a}_{\wp -k}\right)x}^{\wp +2}{t}^{\frac{-б\left(2\wp +1\right)-2}{4}}.$$(69) at $\left(a,b,c,a_0,a_1\right)=\left(0.5,2,1,1,1\right)$ within (a) $б=1,$ (b) $б=0.975,$ (c) $б=0.95,$ and (d) $б=0.925.$

The results shown above indicate the ability to a rough comparison of the disposition in Plots 1, 2, and 3 in all shapes. This guarantees the historical backdrop of syllabic fractional subsidiaries utilized when trade б and affirms their joined express series arrangements.

Physically, from the last plots, one notes the obtained series approximations of the function of the spatial independent variable $x$ and the temporal independent variable $t$ for different order fractional derivatives are elucidated accurately and it can be considered that the nonlinear temperature distribution in an infinitely long and thin rod or the fluid flow velocity in an infinitely long tube of small diameter. Indeed, the $3D$ different fractional order plots concerning the effect of the source term showed that the series approximation approaches have similar attitudes, as well as the fractional order goes towards from the integer one.

5. Laws of conservation

Conservation laws are essential to examine physical models. Therefore, various ideas are found to build conservation laws for suggested systems. Therefore, the laws of conservation depend on the existence of a Lagrangian of the FNWSM (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) are utilized in this part in which a full clarification should be possible in light of the LPSs.

Allow us, first and foremost, to think about a vector ${C=(C}^x,C^t$) that admitting (70) $$D_x\left(C^x\right)+D_t\left(C^t\right)=0.$$(70)

Herein, $C^x=C^x(x,t,\vartheta ,\dots )$ and $C^t=C^t(x,t,\vartheta ,\dots )$ indicated to be conserved of (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) . Concurring with (Abu Arqub, 2019), the formal Lagrangian of (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) is (71) $$\begin{array}{c}L=w\left(x,t\right)\left[D_t^{б}\vartheta -a{\vartheta }_{xx}-b\vartheta +c{\vartheta }^p\right]\end{array}.$$(71)

Furthermore, the well-known action integral is (72) $$\underset{0}{\overset{t}{\int }}\underset{\mathrm{\Omega }}{\overset{ }{\int }}L\left(x,t,\vartheta ,w,D_t^{б},{\vartheta }_{xx}\right)dxdt.$$(72)

Herein, the following relation gives the Euler–Lagrange operator (73) $$\frac{\mathrm{\delta }}{\mathrm{\delta }\vartheta }=\frac{\partial }{\partial \vartheta }+{\left(D_t^{б}\right)}^{*}\frac{\partial }{\partial D_t^{б}\vartheta }+D_{xx}\frac{\partial }{\partial {\vartheta }_{xx}}.$$(73)

Now, the adjoint equation utilizing the Euler–Lagrange is (74) $$\frac{\delta L}{\delta \vartheta }=0.$$(74)

The adjoint ${(D_t^{б})}^{*}$ is defined as (75) $$\begin{array}{c}{\left(D_t^{б}\right)}^{*}={\left(-1\right)}^{\wp }{I}_T^{\wp -б}\left(D_t^{\wp }\right)\\ ≣{ }_t^C{D}_T^{б},\end{array}$$(75) where $I_T^{\wp -б}$ is defined as (76) $$I_T^{\wp -б}f\left(x,t\right)=\frac{1}{\mathrm{\Gamma }(\wp -б)}\underset{t}{\overset{T}{\int }}{\left(\tau -t\right)}^{\wp -б-1}f\left(\tau ,x\right)d\tau .$$(76)

Next, based on the dependent variable $\vartheta =\vartheta (x,t),$ we gain the result: (77) $${{ R}^{*}+D}_x\left(\xi \right)I+D_t\left(\tau \right)I=W\frac{\mathrm{\delta }}{\mathrm{\delta }\vartheta }+D_x\left(C^x\right)+D_t\left(C^t\right),$$(77) where $R^{*}$ and $W$ are (78) $${ R}^{*}=\xi \frac{\partial }{\partial x}+\tau \frac{\partial }{\partial t}+\mathrm{ŋ}\frac{\partial }{\partial \vartheta }+{\mathrm{ŋ}}_{б}^0\frac{\partial }{\partial D_t^{б}\vartheta }+{\mathrm{ŋ}}^{xx}\frac{\partial }{\partial {\vartheta }_{xx}}.$$(78) (79) $$W=\mathrm{ŋ}{-\xi u}_x{-\tau u}_t.$$(79) Furthermore, $C^t$ is (80) $$C^t=\tau L+\sum _{k=0}^{\wp -1}{\left(-1\right)}^k{D}_t^{б-1-k}\left(W_{\mathfrak{H}}\right)D_t^k \frac{\partial \mathrm{ }}{{\partial }_0{D}_t^{б}\vartheta }-{\left(-1\right)}^{\wp }J\left(W_{\mathfrak{H}},D_t^{\wp } \frac{\partial \mathrm{ }L}{{\partial }_0{D}_t^{б}\vartheta }\right),$$(80) where $J$ is given by (81) $$J\left(f,g\right)=\frac{1}{\Gamma \left(\wp -б\right)}\underset{0}{\overset{t}{\int }}\underset{t}{\overset{T}{\int }}\frac{f\left(\tau ,x\right)g\left(\mu ,x\right)}{{ \left(\mu -\tau \right)}^{б+1-\wp }}\mathit{\text{dμdτ}}.$$(81)

For the other component, the following relation is proposed: (82) $$C^x=\xi L+W_{\mathfrak{H}}\left(-D_x\frac{\partial }{\partial {\vartheta }_{xx}}\right)+D_x{(W}_{\mathfrak{H}} \left)\right(\frac{\partial }{\partial {\vartheta }_{xx}}).$$(82)

It remains now to present the conservation laws of (1)(1) $$\left\{\begin{array}{l}{D}_t^{\mathrm{б}}\vartheta =a{D}_x^2\vartheta +b\vartheta -c{\vartheta }^p,\\ \vartheta =\vartheta \left(x,t\right)\in C\left(\mathbb{R}\times [0,\left.\mathrm{\infty }\right)\to \mathbb{R}\right).\end{array}\right.$$(1) employing (80)(80) $$C^t=\tau L+\sum _{k=0}^{\wp -1}{\left(-1\right)}^k{D}_t^{б-1-k}\left(W_{\mathfrak{H}}\right)D_t^k \frac{\partial \mathrm{ }}{{\partial }_0{D}_t^{б}\vartheta }-{\left(-1\right)}^{\wp }J\left(W_{\mathfrak{H}},D_t^{\wp } \frac{\partial \mathrm{ }L}{{\partial }_0{D}_t^{б}\vartheta }\right),$$(80) and (82)(82) $$C^x=\xi L+W_{\mathfrak{H}}\left(-D_x\frac{\partial }{\partial {\vartheta }_{xx}}\right)+D_x{(W}_{\mathfrak{H}} \left)\right(\frac{\partial }{\partial {\vartheta }_{xx}}).$$(82) .

• Status I: For $W_1=-{\vartheta }_x,$ one has (83) $$\begin{array}{c}{C}^t=\tau L{+\left(-1\right)}^0.D_t^{б}\left(W_1\right)D_t^0 \frac{\partial L}{{\partial }_0{D}_t^{б}\vartheta }-{\left(-1\right)}^{0}J\left(W_1,D_t^0 \frac{\partial L}{{\partial }_0{D}_t^{б}\vartheta }\right)\\ =D_t^{б-1}\left(-{\vartheta }_x\right)w-J\left(-{\vartheta }_x,w_t\right).\end{array}$$(83) (84) $$\begin{array}{c}{C}^x=\xi L+W_1\left(-D_x\frac{\partial L}{\partial {\vartheta }_{xx}}\right)+D_x\left(W_1\right)\left( \frac{\partial L}{\partial {\vartheta }_{xx}}\right)\\ =-{\vartheta }_x\left({aD}_x\left(w\right)\right)-a{wD}_x\left(-{\vartheta }_x\right)\\ =w\left(D_t^{б}\vartheta -b\vartheta +c{\vartheta }^p\right)-a{\vartheta }_x{w}_x.\end{array}$$(84)

• Status 2: For $W_2=(3б-2)\vartheta {-2бx\vartheta }_x{-4t\vartheta }_t,$ one has

(85) $$C^t=4tw\left[D_t^{б}\vartheta -a{\vartheta }_{xx}-b\vartheta +c{\vartheta }^p\right]-{ I}^{1-б}\left({2бx\vartheta }_x+{4t\vartheta }_t-\left(3б-2\right)\vartheta \right)+J\left(\left(3б-2\right)\vartheta {-2бx\vartheta }_x{-4t\vartheta }_t,w_t\right).$$(85) (86) $$\begin{array}{c}{C}^x=\xi L+W_2\left(-D_x\frac{\partial L}{\partial {\vartheta }_{xx}}\right)+D_x\left(W_2\right)\left(\frac{\partial L}{\partial {\vartheta }_{xx}}\right)\\ =\left(\left(3б-2\right)\vartheta {-2бx\vartheta }_x{-4t\vartheta }_t\right)\left(a{D}_x\left(w\right)\right)-aw\left(\left(б-2\right){\vartheta }_x{-2бx\vartheta }_{xx}{-4t\vartheta }_{xt}\right)\\ =2бxw\left(D_t^{б}\vartheta -b\vartheta +c{\vartheta }^p\right)-aw \left(\left(б-2\right){\vartheta }_x{-4t\vartheta }_{xt}\right){+aw}_x\left(\left(3б-2\right)\vartheta {-2бx\vartheta }_x{-4t\vartheta }_t\right).\end{array}$$(86)

6. Future, highlight, and conclusion

In the meantime, in the considered investigations, we have utilized the invariance properties of the overall FNWSM in the sense of LPS. Also, the LPS decreases the FNWSM depiction alongside all mathematical vector fields that are determined and acquired. The decrease in aspect is in the balance variable-based math because the FPDE is known to be invariant under time interpretation balance. Far beyond, the treated partial model has been changed over into an FDE, and afterward, the power series strategy has been applied to discover all unequivocal series logical arrangements. The laws of conservation depend on the existence of a Lagrangian of the fundamental FNWSM have been emblematically and hypothetically processed to ensure certain properties of the model. Also, a few graphical perspectives concerning different boundaries and requested fragmentary subsidiaries’ impacts have been shown. The acquired outcomes indicate that our proposed LPS and our expanded power series support a decent role in the numerical field and exceptionally proficient incredible assets for getting express series arrangements. For future research, the FNWSM concerning the time-conformable derivative will be studied.

Disclosure statement

The authors declare that they have no conflicts of interest.

Data availability statement

No datasets are associated with this manuscript. The datasets used for generating the plots and results during the current study can be directly obtained from the numerical simulation of the related mathematical equations in the manuscript.