Two-Dimensional Time-Fractional Nonlinear Drift Reaction–Diffusion Equation Arising in Electrical Field

Abstract

Diffusion equations play a crucial role in various scientific and technological domains, including mathematical biology, physics, electrical engineering, and mathematics. This article presents a new formulation of the diffusion equation in the context of electrical engineering. Specifically, the behaviour of the physical quantity of charge carriers (such as concentration) is examined within semiconductor materials. The primary focus of this work is to solve the two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation by applying an appropriate numerical scheme. In recent years, researchers working on nonlinear diffusion equations have proposed several numerical methods, with the shifted airfoil collocation method being one such efficient technique for solving nonlinear partial differential equations. This collocation approach effectively reduces the considered two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation to a system of algebraic equations. The efficiency and effectiveness of the proposed method are validated through an error analysis, comparing the exact solution and the proposed numerical solution for a specific form of the considered mathematical model. The variations in the concentration of charge carriers, driven by the effects of drift and reaction terms, are displayed graphically as the system transitions from a fractional order to an integer order.

1. Introduction

Diffusion-based schemes play a pivotal role in the analysis of nano-structured photoelectrochemical cells, particularly in the context of converting solar energy into electricity. Diffusion-controlled models are prevalent in various fields, including electrochemistry, photoelectrochemistry, and solid-state electronics. These diffusion-based models provide valuable support for the technological development of different types of rechargeable batteries, polymers, and semiconductor junctions. The ability to accurately model and understand diffusion processes in these systems is crucial for advancing various energy storage and conversion technologies. By leveraging the insights gained from diffusion-controlled models, researchers and engineers can optimize the design and performance of these critical components, ultimately contributing to the progress and innovation in areas such as renewable energy, energy storage, and electronic devices.

The concentration of charge carries $u(x,t)$ in a one-dimensional convection–diffusion equation is defined in [1] as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,t)}{\partial t}}=D{\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial x^2}}-\nu {\displaystyle \frac{\partial u(x,t)}{\partial x}},\end{array}$$

(1)

where D represents diffusivity and $\mu$ represents mobility, which are assumed to be constants for the case of quasi-equilibrium transport and for the materials with a small disorder. The convection coefficient $\nu$ is defined by $\nu =\mu F_0$, where $F_0$ is a constant for the uniform electric field.

The relation between D (of the tracer) and $\mu$ in an active bath of passive Brownian particles in a non-equilibrium medium was first given by Albert Einstein as $D=\mu K_{B}T$. For the ideal Gas law, the pressure and density are propositional, and it is defined by $\varphi ={\rho }_0{K}_{B}T$. For active case, the active temperature $T_a$ gives the relationship as $T_a={\displaystyle \frac{\varphi }{{\rho }_0{K}_B}}$, where $\varphi$ defines the mechanical pressure exerted by the active bath on the tracer.

Diffusion is also found in a porous semiconductor medium that contains a network of pores. Here, charge carriers are basically electrons and holes those move through the material towards conducting electricity. Here, the carriers move from the higher concentration region to the region of lower concentration. Porosity in semiconductor means the presence of pores or voids within the material, which may influence the optical, thermal, and electro-mechanical behaviours.

The drift reaction diffusion equation, used to design and analyse the semiconductor devices viz., transistors, diodes, solar cell, photo detectors, etc., is represented by

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,t)}{\partial t}}={\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial x^2}}-\nu {\displaystyle \frac{\partial u(x,t)}{\partial x}}-R\left(u\right)+f(x,t).\end{array}$$

(2)

Actually, the semiconductor transport equation is analogous to the reaction advection diffusion equation, which consists of diffusion, drift, and reaction (source/sink) terms to describe the current flow, satisfying the following equation

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,t)}{\partial t}}=\nabla ·(\nu u(x,t)+D\nabla u(x,t))-R\left(u\right),\end{array}$$

(3)

which is ultimately reduced to Equation (2) by integrating an extra term $f(x,t)$ for making the system non homogeneous. It helps to include the effect of external factors on the concentration of the charge carriers $u(x,t)$. The physical interpretation of this term may be defined as the rate of the addition or removal of the quantity $u(x,t)$. Mathematical model (3) has applications in the contamination of the groundwater, the transport of drugs in biological tissues, and the movement of plasma and magnetic fields in astrophysical environments.

Initially, mathematical modeling was confined to linear systems in which analytical treatment was manageable. After the advent of powerful computers and effective computational techniques, it was possible to tackle nonlinear systems. It was very much needed as the nonlinearity is a salient feature of any dynamical system. For this reason, in the last few decades, researchers have been motivated to study nonlinear problems.

The fractional operator deals with systems with long-range time correlations, related to decay in the inverse power-law kernel. The evolution equations in fractional-order systems are obtained by replacing the first order derivative of an integer order system by a fractional order $\alpha$$(0<\alpha \le 1)$. Gorenflo and Mainardi [2] showed how such evolution equations obtained from a more general master equations governing the continuous time random walk (CTRW) by properly scaled passage to the limit of compressed waiting times. CTRW is basically a combination of random walks in time and space. The anomalous diffusion based on the concept of CTRW shows random waiting times in between successive jumps. For limiting the dynamic of CTRW, the fractional diffusion equations are very much relevant. Gorenflo et al. [3] demonstrated that the essential assumption for fractional diffusion equations is that the probabilities for waiting times and jump widths should be asymptotically like powers with negative exponents involving the fractional orders. Recently, fractional-order modelling has become an active area of research from the theoretical and applied perspectives. To date, a number of researchers from the different parts of the world have studied a wide range of problems in science and engineering to explore its potential [4,5,6,7]. The usage of fractional-order system is not applicable for non-Gaussian and non-Markovian systems. As per Einstein’s theory of Brownian motion, for a random moving particle, the mean square displacement is proportional to time. After the advancement of fractional calculus, the concept was changed and it was found that the mean square displacement for an anomalous diffusion equation grows slowly with time. In a simple linear fractional-order diffusion equation, the mean square displacement is <$X^2\left(t\right)>\sim$$t^{\alpha }$, where $0<\alpha <1$ is the anomalous exponent. For the drift reaction–diffusion Equation (2), the mean square displacement becomes <$X^2\left(t\right)>\sim$$t^{{\displaystyle \frac{3\alpha }{2}}}$, when $D=1$, $f(x,t)=0$ and $R\left(u\right)=u$. These evolution equations generate the fractional Brownian motion, which is a generalization of Brownian motion. If the integer order is replaced by a fractional-order time derivative, the fundamental concept of time is changed and, thus, the concept of the foundations of physics is changed. The anomalous charge carrier transport in disordered organic semiconductors has been modelled through the fractional drift-diffusion equation by Choo et al. [8]. The fractional Brownian motions were shown through reaction–diffusion equations with a polynomial drift term by Zamani [9]. Keeping with the importance of the aforementioned mathematical model, it is required to find an effective and efficient numerical scheme specifically for two-dimensional case.

The main focus of the present article is to numerically solve the time-fractional order two-dimensional non-linear drift reaction–diffusion equation. For this purpose, a hybrid spectral collocation technique is proposed using a shifted version of the airfoil polynomials of the second kind. Actually, the spectral collocation technique is a kind of spectral method that is usually applied to solve various PDEs. Abdelkawy et al. [10] solved the space-fractional two-dimensional reaction–diffusion nonlinear equation. Jafari et al. [11] studied the spectral method solve inverse reaction–diffusion-convection problem. In [12], analytical and numerical results for the diffusion-reaction equation when the reaction coefficient are found simultaneously on the space and time coordinates. Rashidinia et al. [13] have studied the solution of convection-diffusion model in groundwater pollution. Through a literature survey, it can be observed that the said procedures have been successfully employed to deal with different kinds of mathematical models computationally by applying different basis functions viz., Fibonacci, Genocci, Bernoulli, Jacobi, Legendre, Morgan Voyce, Hermite, Chebyshev, Bernstein, Muntz, Bessel, and Vieta–Lucas polynomials, etc. [14,15,16,17,18,19,20,21,22,23,24,25]. The shifted airfoil polynomials were used for their superior properties in terms of parameter orthogonality as well as the fact that their computation complexities were less compared to other polynomials. In the method, first, the approximate solution is taken as a linear combinations of basis functions in the form of airfoil polynomials. Then, it is substituted in the concerned model, and the concerned initial and boundary conditions to obtain a system of nonlinear algebraic equations after collocation, which can be solved by applying the Newton’s method.

In the present article, the authors have aimed to solve the following two-dimensional time-fractional nonlinear drift reaction–diffusion equation with the integration of an extra force term under the prescribed initial and boundary conditions, given as

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }u(x,y,t)}{\partial t^{\alpha }}}=& {\displaystyle \frac{{\partial }^{2}u(x,y,t)}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u(x,y,t)}{\partial y^2}}-{\nu }_1{\displaystyle \frac{\partial u(x,y,t)}{\partial x}}-{\nu }_2{\displaystyle \frac{\partial \mathrm{u}(x,y,t)}{\partial y}}\hfill \\ & -\eta u(x,y,t)(1-u(x,y,t\left)\right)+f(x,y,t),\hfill \end{array}$$

(4)

under the initial and boundary conditions, as follows

$$\begin{array}{c}\hfill \left.\begin{array}{cc}{\displaystyle u(x,y,0)=h_1(x,y),}\hfill & u(0,y,t)=h_2(y,t),\hfill \\ {\displaystyle \frac{\partial u}{\partial x}(X,y,t)=h_3(y,t),}\hfill & u(x,0,t)=h_4(x,t),\hfill \\ {\displaystyle \frac{\partial u}{\partial y}(x,Y,t)=h_5(x,t),}\hfill \end{array}\right\}\end{array}$$

(5)

where $0\le x\le X$, $0\le y\le Y$, $0\le t\le T$, $0<\alpha \le 1$, ${\nu }_1$, ${\nu }_2$, are drift coefficients, $\eta$ is the reaction coefficient, and $f(x,y,t)$ is the force term. $h_1(x,y),h_2(y,t),h_3(y,t),h_4(x,t),h_5(x,t)$ are sufficiently smooth functions. By applying the shifted airfoil collocation method, the nature of the concentration of charge carriers for different particular cases have been found when the system moves from fractional order to the integer order. The efficacy of the numerical scheme is validated through error analysis between the proposed scheme and the exact solutions of the particular cases of the concerned model.

The article contains 6 sections, out of which: Section 2 presents helpful definitions and properties. Section 3 offers a detailed explanation of the proposed numerical method, which involves solving two-dimensional time-fractional nonlinear drift reaction–diffusion equation by transforming those into a system of algebraic equations, which are tackled using Newton’s method. One test problem is outlined in Section 4 to verify the effectiveness of the proposed numerical approach. Section 5 focuses on solving the fractional order model using the proposed method, followed by an overall conclusion of the research in Section 6.

2. Preliminaries

2.1. Caputo Fractional Derivative

For the function $u(x,t)$, the Caputo derivative is defined as [26]

$$\begin{array}{c}\hfill {}_0{}^{c}D_t^{\alpha }u(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (m-\alpha )}{\int }_0^t\frac{1}{{(t-\tau )}^{(\alpha -m+1)}}\frac{{\partial }^{m}u(x,\tau )}{\partial {\tau }^m}d\tau ,}\hfill & t>0,0\le m-1\le \alpha <m\in \mathbb{N},\hfill \\ \\ {\displaystyle \frac{{\partial }^{m}u(x,t)}{\partial t^m},}\hfill & \alpha =m\in \mathbb{N}.\hfill \end{array}\right.\end{array}$$

(6)

The Caputo derivative also follows the power function property given by

$$\begin{array}{c}\hfill {}_0{}^{c}D_t^{\alpha }{t}^p=\left\{\begin{array}{cc}{\displaystyle \frac{\Gamma (p+1)}{\Gamma (p+1-\alpha )}{t}^{p-\alpha },}\hfill & p\in \mathbb{N},\hfill \\ 0,\hfill & p=0.\hfill \end{array}\right.\end{array}$$

(7)

2.2. Airfoil Polynomial

The recursive relations of the second kind airfoil polynomials on $\left[-1,1\right]$ are given as [27]

$$\begin{array}{c}\hfill A_{i+1}\left(x\right)=2x{A}_i\left(x\right)-A_{i-1}\left(x\right),i=0,1,2,\dots ;\end{array}$$

(8)

with initial conditions as $A_0\left(x\right)=1$, $A_1\left(x\right)=2x+1$.

The second kind airfoil polynomials $A_i\left(x\right)$ have the explicit form

$$\begin{array}{c}\hfill A_i\left(x\right)={\displaystyle \frac{1}{2^i}}{\Sigma }_{s=0}^i{(-1)}^{\left(s\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{2i+1}{2s+1}}\right){(1-x)}^s{(1+x)}^{i-s},i=0,1,2,\dots ;\end{array}$$

(9)

The second kind shifted airfoil polynomials $A_i^{*}\left(x\right)$ on the interval $\left[0,X\right]$ have the following explicit form as

$$\begin{array}{c}\hfill A_i^{*}\left(x\right)={\Sigma }_{s=0}^i{(-1)}^{\left(s\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{2i+1}{2s+1}}\right){\left(1-{\displaystyle \frac{x}{X}}\right)}^s{\left({\displaystyle \frac{x}{X}}\right)}^{i-s},i=0,1,2,\dots ;\end{array}$$

(10)

or

$$\begin{array}{c}\hfill A_i^{*}\left(x\right)={\Sigma }_{s=0}^i{\displaystyle \frac{{(-1)}^{(i-s)}{2}^{2s}\Gamma (i+s+1)}{\Gamma (i-s+1)\Gamma (2i+1)X^i}}{x}^i,i=0,1,2,\dots ;\end{array}$$

(11)

with initial conditions as $A_0\left(x\right)=1$, $A_1\left(x\right)={\displaystyle \frac{4x}{X}}-1$.

$A_i^{*}\left(x\right)$ satisfies the orthogonality w.r.t. weight function ${\displaystyle w_{*}\left(x\right)=\sqrt{\frac{X-x}{x}}}$ as

$$\begin{array}{c}\hfill {\int }_0^X{w}_{*}\left(x\right)A_i^{*}\left(x\right)A_j^{*}\left(x\right)=X{\displaystyle \frac{\pi }{2}}{\delta }_{\mathrm{i}\mathrm{j}}.\end{array}$$

(12)

The first and second order derivatives of Equation (11) are

$$\begin{array}{c}\hfill D_x{A}_i^{*}\left(x\right)={\Sigma }_{s=1}^i{\displaystyle \frac{{(-1)}^{(i-s)}{2}^{2s}\Gamma (i+s+1)s}{\Gamma (i-s+1)\Gamma (2s+1)X^s}}{x}^{(s-1)},i=1,2,3,\dots ;\end{array}$$

(13)

and

$$\begin{array}{c}\hfill D_x^2{A}_i^{*}\left(x\right)={\Sigma }_{s=2}^i{\displaystyle \frac{{(-1)}^{(i-s)}{2}^{2s}\Gamma (i+s+1)s(s-1)}{\Gamma (i-s+1)\Gamma (2s+1)X^s}}{x}^{(s-2)}.i=2,3,4,\dots ;\end{array}$$

(14)

The Caputo fractional derivative of second kind shifted airfoil polynomial of the order $\alpha$ can be obtained by using Equation (7) as

$$\begin{array}{c}\hfill {}_0{}^{c}D_x^{\alpha }{A}_i^{*}\left(x\right)={\Sigma }_{s=1}^i{\displaystyle \frac{{(-1)}^{(i-s)}{2}^{2s}\Gamma (i+s+1)\Gamma (s+1)}{\Gamma (i-s+1)\Gamma (2s+1)\Gamma (s+1-\alpha )X^s}}{x}^{(s-\alpha )}.i=1,2,3,\dots ;\end{array}$$

(15)

3. Description of the Proposed Method

In this section, the shifted airfoil collocation method is used to solve a two-dimensional time-fractional nonlinear drift reaction–diffusion Equation (4) under the prescribed initial and boundary conditions.

Let us assume the approximation of $u(x,y,t)$ as follows

$$\begin{array}{c}\hfill u(x,y,t)\simeq {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t\right)A_i^{*}\left(x\right)A_j^{*}\left(y\right).\end{array}$$

(16)

where $a_{i,j,k}$ is an unknown coefficient.

The derivatives of approximate solutions w.r.t. x and y are calculated as

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{2}u(x,y,t)}{\partial x^2}}\simeq {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t\right)D_x^2{A}_i^{*}\left(x\right)A_j^{*}\left(y\right),\end{array}$$

(17)

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{2}u(x,y,t)}{\partial y^2}}\simeq {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t\right)A_i^{*}\left(x\right)D_y^2{A}_j^{*}\left(y\right),\end{array}$$

(18)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,y,t)}{\partial x}}\simeq {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t\right)D_x{A}_i^{*}\left(x\right)A_j^{*}\left(y\right),\end{array}$$

(19)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,y,t)}{\partial y}}\simeq {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t\right)A_i^{*}\left(x\right)D_y{A}_j^{*}\left(y\right).\end{array}$$

(20)

The Caputo derivative of approximate solution w.r.t. t as

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{\alpha }u(x,y,t)}{\partial t^{\alpha }}}\simeq {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{}_0{}^{c}D_t^{\alpha }{A}_k^{*}\left(t\right)A_i^{*}\left(x\right)A_j^{*}\left(y\right).\end{array}$$

(21)

According to the proposed approach, the residuals of Equation (4) are assumed to be zeros at $(N-1)(M-1)K$ collocation points, given by $x_p={\displaystyle \frac{pX}{N}}$, $y_q={\displaystyle \frac{qY}{M}}$, $t_r={\displaystyle \frac{rT}{K}}$. Using Equations (16)–(21), Equation (4) is reformulated as

$$\begin{array}{cc}\hfill & {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}({}_0{}^{c}D_t^{\alpha }{A}_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)A_j^{*}\left(y_q\right)-A_k^{*}\left(t_r\right)D_x^2{A}_i^{*}\left(x_p\right)A_j^{*}\left(y_q\right)-A_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)\hfill \\ & \times D_y^2{A}_j^{*}\left(y_q\right)+{\nu }_1{A}_k^{*}\left(t_r\right)D_x{A}_i^{*}\left(x_p\right)A_j^{*}\left(y_q\right)+{\nu }_2{A}_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)D_y{A}_j^{*}\left(y_q\right))\hfill \\ & +\eta {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)A_j^{*}\left(y_q\right)-\eta {\left({\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)A_j^{*}\left(y_q\right)\right)}^2\hfill \\ & -f(x_p,y_q,t_r)=0,\hfill \end{array}$$

(22)

where $p=1,2,3,\dots ,(N-1)$, $q=1,2,3,\dots ,(M-1)$ and $r=1,2,3,\dots ,K$, which consists of $(N-1)(M-1)K$ equations having $(N+1)(M+1)(K+1)$ unknown coefficients. The aforementioned system required $\left(1+M+N+NM+2K(N+M)\right)$ additional equations to find its unique solution. These additional equations are derived from the initial and boundary conditions given by

$${\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(0\right)A_i^{*}\left(x_p\right)A_j^{*}\left(y_q\right)=h_1(x_p,y_q),$$

(23)

where $p=1,2,3,\dots ,N-1$, $q=1,2,3,\dots ,M-1$.

$$\begin{array}{cc}\hfill & {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t_r\right)A_i^{*}\left(0\right)A_j^{*}\left(y_q\right)=h_2(y_q,t_r),\hfill \\ & {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t_r\right)D_x{A}_i^{*}\left(X\right)A_j^{*}\left(y_q\right)=h_3(y_q,t_r),\hfill \end{array}$$

(24)

where $q=0,1,2,\dots ,M$, $r=0,1,2,\dots ,K$.

$$\begin{array}{cc}\hfill & {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)A_j^{*}\left(0\right)=h_4(x_p,t_r),\hfill \\ & {\Sigma }_{i=0}^N{\Sigma }_{j=0}^M{\Sigma }_{k=0}^K{a}_{i,j,k}{A}_k^{*}\left(t_r\right)A_i^{*}\left(x_p\right)D_y{A}_j^{*}\left(Y\right)=h_5(x_p,t_r),\hfill \end{array}$$

(25)

where $p=1,2,3,\dots ,N-1,$ $r=0,1,2,\dots ,K$.

The above system provides $(N+1)(M+1)(K+1)$ equations with $(N+1)(M+1)(K+1)$ unknowns. By solving these algebraic equations, we can determine the unknowns and, thus, compute the numerical solution of Equation (4).

4. Numerical Analysis

By applying the aforementioned numerical technique for solving the problem with an accurate solution, the reliability of the technique is demonstrated in this section. Let us assume $u_E(x,y,t)$ and $u_{N,M}(x,y,t)$ denote the exact and numerical solutions, respectively. Then Maximum Absolute Error (MAE) at $t=T$ is determined as

$$\begin{array}{c}\hfill MA{E}_u={\displaystyle \underset{1\le i\le N,1\le j\le M}{max}}\left\{|u_E(x_i,y_j,T)-u_{N,M}(x_i,y_j,T)|\right\}.\end{array}$$

Example 1.

Here, we take the Equation (4) for $(x,y)\in \left[0,1\right]\times \left[0,1\right]$ and $t\in \left[0,1\right]$.

The force term is given by

$$\begin{array}{cc}\hfill f(x,y,t)=& y\eta -y^2\eta +{\nu }_2+e^{-t}ycos\left(x\right)-2e^{-t}{y}^2\eta cos\left(x\right)+e^{-t}{\nu }_{2}cos\left(x\right)\hfill \\ & -e^{-2t}{y}^2\eta co{s}^2\left(x\right)-e^{-t}y{\nu }_{1}sin\left(x\right).\hfill \end{array}$$

The exact solution can be used to figure out the initial and boundary conditions as

$${\displaystyle u(x,y,t)=(1+e^{-t}cos\left(x\right))y.}$$

Now, the proposed approach is applied to compute the numerical solution for the example. The example is solved using various values of N, M, and K with $\alpha =1$, ${\nu }_1={\nu }_2=0.5$ and $\eta =1$. Figure 1 illustrates the exact and numerical solution for (N, M, and K) = (5, 5, 5).

Table 1. MAE for $\alpha =1$.

($\mathit{N},\hspace{0.17em}\mathit{M},\hspace{0.17em}\mathit{K}$)	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{u}}$
(3, 3, 3)	$2.05592\times {10}^{-3}$
(4, 4, 4)	$1.36522\times {10}^{-4}$
(5, 5, 4)	$2.08003\times {10}^{-5}$
(5, 5, 5)	$1.59699\times {10}^{-5}$

presents the MAE, and it is observed that when N, M, and K increase, errors decrease. Figure 2 displays the variations of the absolute error of $u(x,y,t)$ vs. x and y at $t=1.$

5. Application

This section focuses on solving the mathematical model of a two-dimensional time-fractional nonlinear drift reaction–diffusion equation with $f(x,y,t)=0$, described in the form of the nonlinear Equation (4) along with the following initial and boundary conditions, which have been solved using the proposed efficient numerical scheme for different particular cases when $(x,y)\in \left[0,1\right]\times \left[0,1\right]$, $t\in \left[0,1\right]$.

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle u(x,y,0)=\frac{\Gamma (\alpha +1)}{\Gamma (2\alpha +1)}sin\left(x\right)\sqrt{y},}\hfill \\ {\displaystyle u(0,y,t)=0,}\hfill \\ {\displaystyle \frac{\partial u}{\partial x}(1,y,t)=\frac{0.540302\Gamma (\alpha +1)}{\Gamma (2\alpha +1)}{e}^{-t}\sqrt{y},}\hfill \\ {\displaystyle u(x,0,t)=0}\hfill \\ {\displaystyle \frac{\partial u}{\partial y}(x,1,t)=\frac{\Gamma (\alpha +1)}{2\Gamma (2\alpha +1)}{e}^{-t}sin\left(x\right),}\hfill \end{array}\right\}\end{array}$$

(26)

The proposed numerical scheme has solved the model (4) with initial and boundary conditions (26) for different specific cases. Each figure is given for $(N,M,K)=(3,3,3)$. Figure 3, Figure 4 and Figure 5, show the nature of the solution profiles for various values of the time-fractional derivative $\alpha =0.6,0.8,0.9,1$, with fixed-drift coefficients ${\nu }_1$ = ${\nu }_2=0.5$, and reaction coefficient $\eta =1$. It can be observed from Figure 3, Figure 4 and Figure 5 that the charge carrier concentration $u(x,o.5,t)$ vs. x, $u(o.5,y,t)$ vs. y and $u(x,y,t)$ vs. x and y at $t=1$ are decreased as the fractional-order system approaches the integer-order system. Figure 6, Figure 7 and Figure 8 show the nature of solution profiles for various values of the time-fractional derivative $\alpha =0.6,0.8,0.9,1$, with fixed-drift coefficients ${\nu }_1$ = ${\nu }_2=0$, and $\eta =1$. The figures follow a similar fashion as shown in previous figures, with a difference being that, in absence of an advection term, the charge carrier concentration increases. Figure 9, Figure 10 and Figure 11, show the nature of the solution profile for $\eta =-1,0,1$ at fixed $\alpha =0.95$ and ${\nu }_1$ = ${\nu }_2=0.5$. As expected, the charge carrier concentrations at $t=1$ are observed to be lower for the case when the sink term is considered $\eta =1$, compared to the case when the source term is considered, i.e., $\eta =-1$ or the case of the conservative system, i.e., $\eta =0$.

6. Conclusions

In this scientific contribution, the authors have developed a numerical algorithm, namely the shifted airfoil collocation numerical scheme, to solve the two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation. The authors have demonstrated the efficiency and effectiveness of the proposed scheme through the numerical solution of an example problem. The shifted airfoil collocation scheme is a flexible and computationally efficient approach that can be adapted to solve various types of partial differential equations (PDEs). Compared with other numerical schemes, this method is simpler to implement while achieving a high accuracy even with a relatively small number of basis functions. A notable feature of this work is the showcasing of the effects of drift and reaction terms on the solution profiles for different particular cases. Additionally, the authors have provided a graphical presentation of the variations in the concentration of charge carriers as the system transitions from a fractional-order to an integer-order model.

The authors believe that in the near future, their proposed scheme will be able to handle a wide range of nonlinear PDEs commonly encountered in the electrical engineering field.