Application of the New Iterative Method (NIM) to the Generalized Burgers–Huxley Equation

Abstract

In this paper, we propose the new iterative method (NIM) for solving the generalized Burgers–Huxley equation. NIM provides an approximate solution without the need for discretization and is based on a set of iterative equations. We compared the NIM with other established methods, such as Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), and the exact solution, and found that it is efficient and easy to use. NIM has the advantage of quick convergence, easy implementation, and handling of a wide range of initial conditions. The comparison of the present symmetrical results with the existing literature is satisfactory.

1. Introduction

Nonlinear partial differential equations (NPDEs) are essential in a range of fields, such as physics, chemistry, biology, mathematics, and engineering. These equations are crucial for describing phenomena like fluid dynamics, heat transfer, and others. However, solving nonlinear models for real-world problems is a challenging task, both theoretically and numerically. The complexity and nonlinearity of the models make it challenging to find accurate and reliable solutions. To make nonlinear models solvable, unnecessary assumptions are often required. These can include simplifying the equation, neglecting certain terms, or approximating the solution. While these assumptions can make the problem more manageable, they also introduce uncertainty to the solution. This can be problematic when applying the solution to real-world problems where accuracy and reliability are crucial. Various numerical methods are available to solve NPDEs, but they all have limitations. For instance, traditional numerical methods such as finite difference and finite element methods are based on domain discretization, which can lead to errors and instability. In conclusion, solving NPDEs is a challenging task that requires a combination of theoretical and numerical approaches. The development of new techniques and methods to solve NPDEs will remain an active area of research and development across various fields [1,2,3,4,5,6,7,8].

The Burgers–Huxley equation is a partial differential equation that combines two different mathematical models: the Burgers’ equation, which describes the dynamics of viscous fluids, and the FitzHugh–Nagumo model, which is used to study the behavior of excitable cells. This equation is often used to model complex phenomena in physics, biology, and engineering. One interesting aspect of the Burgers–Huxley equation is its symmetry. This equation is invariant under Galilean transformations, which means that its solutions are not affected by a change in the frame of reference. Additionally, the Burgers–Huxley equation is also symmetric under time reversal, which means that the equation remains the same when time is reversed. These symmetries have important implications for the behavior of solutions to the equation, and can help researchers better understand and predict the behavior of complex systems [9,10].

Solving the Burgers–Huxley equation can be challenging due to its complexity and nonlinear nature. One approach to solving this equation is through iterative methods, such as the finite difference method or the finite element method. These methods involve breaking the problem down into smaller pieces and solving each piece iteratively until a solution to the entire problem is found. The symmetries of the Burgers–Huxley equation can be useful in developing iterative methods for solving the equation. For example, the Galilean invariance symmetry can be used to simplify the problem by transforming it into a simpler reference frame. Similarly, the time reversal symmetry can be used to develop algorithms that exploit the reversible nature of the equation. Overall, understanding the symmetries of the Burgers–Huxley equation can help researchers develop more efficient and accurate iterative methods for solving this complex partial differential equation [11,12,13].

Discovering accurate solutions to nonlinear equations is an intriguing and significant task. Several techniques have been developed to tackle NPDEs, including the pseudospectral method [14], spectral collocation method [15], Adomian decomposition method (ADM) [16,17] and homotopy perturbation method [18,19,20].

Daftardar-Gejji and Jafari [21] have introduced a new mathematical technique called the new iterative method (NIM) for solving linear and nonlinear functional equations. This method has been proven effective in addressing a wide range of nonlinear equations, including integral equations, algebraic equations, and ordinary or partial differential equations of fractional and integer order. NIM is straightforward to understand and implement using computer software and has been found to produce superior results [22] when compared to well-known methods such as the Adomian Decomposition Method (ADM) [16], the Homotopy Perturbation Method (HPM) [23], and the Variational Iteration Method (VIM) [24]. Overall, NIM is a valuable tool for solving linear and nonlinear PDE and the generalized Burgers–Huxley equation.

This paper delves into the analysis of the generalized nonlinear Burgers–Huxley equation. This equation is known to model the complex interplay between various physical phenomena, such as reaction mechanisms, convection effects, and diffusion transports. Studying this equation is crucial for understanding the underlying dynamics of these systems and for developing accurate models to predict their behavior. The analysis presented in this paper aims to provide a deeper understanding of this equation and its solutions, and the results obtained will contribute to the existing body of knowledge in this field.

In this paper, we present an analysis of the generalized nonlinear Burgers–Huxley equation:

$$\begin{array}{c}\hfill u_t+\alpha u^{\delta }{u}_x-u_{xx}=\beta u(1-u^{\delta })(u^{\delta }-\gamma ),0\le x\le 1,t\ge 0.\end{array}$$

(1)

In Equation (1), the coefficients $\alpha$, $\beta \ge 0$ are real constants, $\delta$ is a positive integer, and $\gamma$ lies in the interval $(0,1)$, under the given initial condition,

$$\begin{array}{c}\hfill u(x,0)={\left[\frac{\gamma }{2}+\frac{\gamma }{2}tanh\left(\sigma \gamma x\right)\right]}^{1/\delta }.\end{array}$$

(2)

The exact solution to Equation (1) subject to the initial condition was obtained by Wang, Zhu, and Lu [25] through the use of nonlinear transformations. This solution is presented as follows:

$$\begin{array}{ccc}\hfill u(x,t)& =& {\left[\frac{\gamma }{2}+\frac{\gamma }{2}tanh\left[\sigma \gamma \left(x-\left\{\frac{\gamma \alpha }{1+\delta }-\frac{(1+\delta -\gamma )(\rho -\alpha )}{2(1+\delta )}\right\}t\right)\right]\right]}^{1/\delta },\hfill \end{array}$$

(3)

where $\sigma =\delta (\rho -\alpha )/4(1+\delta )$ and $\rho =\sqrt{{\alpha }^2+4\beta (1+\delta )}$.

This study presents a numerical solution of the generalized nonlinear Burgers–Huxley equation found by using the new iterative method (NIM). The results obtained by NIM are compared with the exact solution and other iterative methods such as VIM and ADM, and it was found that the results are consistent and have good agreement with those obtained utilizing ADM, VIM, and the exact solution.

2. The New Iterative Method (NIM)

In this section, the NIM numerical method will be outlined as follows [26,27,28,29]:

$$\begin{array}{c}\hfill u=f+L\left(u\right)+N\left(u\right),\end{array}$$

(4)

In the equation above, f is a known function, and L and N are linear and nonlinear operators, respectively.

The NIM solution for Equation (4) has the form

$$\begin{array}{c}\hfill u=\sum _{i=0}^{\infty }{u}_i.\end{array}$$

(5)

Since L is linear then

$$\begin{array}{c}\hfill L\left(\sum _{i=0}^{\infty }{u}_i\right)=\sum _{i=0}^{\infty }L\left(u_i\right).\end{array}$$

(6)

The nonlinear operator N in Equation (4) is decomposed as below

$$\begin{array}{ccc}\hfill N\left(\sum _{i=0}^{\infty }{u}_i\right)& =& N\left(u_0\right)+\sum _{i=1}^{\infty }\left\{N\left(\sum _{j=0}^i{u}_j\right)-N\left(\sum _{j=0}^{i-1}{u}_j\right)\right\}.\hfill \\ & =& \sum _{i=0}^{\infty }{A}_i,\hfill \end{array}$$

(7)

where

$$\begin{array}{ccc}\hfill A_0& =& N\left(u_0\right)\hfill \\ \hfill A_1& =& N(u_0+u_1)-N\left(u_0\right)\hfill \\ \hfill A_2& =& N(u_0+u_1+u_2)-N(u_0+u_1)\hfill \\ \hfill ⋮\\ \hfill A_i& =& \left\{N\left(\sum _{j=0}^i{u}_j\right)-N\left(\sum _{j=0}^{i-1}{u}_j\right)\right\},i\ge 1.\hfill \end{array}$$

Using Equations (5)–(7) in Equation (4), we get

$$\begin{array}{c}\hfill \sum _{i=0}^{\infty }{u}_i=f+\sum _{i=0}^{\infty }L\left(u_i\right)+\sum _{i=0}^{\infty }{A}_i.\end{array}$$

(8)

The solution of Equation (4) can be expressed as

$$\begin{array}{c}\hfill u=\sum _{i=0}^{\infty }{u}_i=u_0+u_1+u_2+\cdots +u_n+\cdots ,\end{array}$$

(9)

where

$$\begin{array}{ccc}\hfill u_0& =& f\hfill \\ \hfill u_1& =& L\left(u_0\right)+A_0\hfill \\ \hfill u_2& =& L\left(u_1\right)+A_1\hfill \\ \hfill ⋮\\ \hfill u_n& =& L\left(u_{n-1}\right)+A_{n-1}\hfill \\ \hfill ⋮\end{array}$$

(10)

Algorithm 1: The new iterative method (NIM)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}INPUT& :& ReadM\left(Numberofiterations\right);\hfill \\ \hfill & & ReadL\left(u\right);N\left(u\right);f\hfill \\ \hfill & & Step-1:u_{-1}=0,u_0=f\hfill \\ \hfill & & Step-2:For(n=0,n\le M,n++)\hfill \\ \hfill & & \{\hfill \\ \hfill & & Step-3:A_n=f\left(u_n\right)-f\left(u_{n-1}\right);\hfill \\ \hfill & & Step-4:u_{n+1}=f+L\left(u_n\right)+A_n;\hfill \\ \hfill & & Step-5:u=u_{n+1}\hfill \\ \hfill & & \}end\hfill \\ \hfill OUTPUT& :& u\hfill \end{array}$$

The Convergence of the NIM

Theorem 1.

For any n and for some real $L>0$ and $\left|\right|{\mathfrak{u}}_i\left|\right|\le M<\frac{1}{e},i=1,2,...,$ if N is $C^{(\infty )}$ in the neighborhood of ${\mathfrak{u}}_0$ and $\left|\right|N^{\left(n\right)}\left({\mathfrak{u}}_0\right)\left|\right|\le L$, then ${\sum }_{n=0}^{\infty }{H}_n$ is convergent absolutely and $\left|\right|H_n\left|\right|\le L{M}^n{e}^{n-1}(e-1),n=1,2,...$.

Proof. 

The full details of the proof can be found in [30]. □

Theorem 2.

The series ${\sum }_{n=0}^{\infty }{H}_n$ is convergent absolutely if N is $C^{(\infty )}$ and $\left|\right|N^{\left(n\right)}\left({\mathfrak{u}}_0\right)\left|\right|\le M\le e^{-1},\forall n$.

Proof. 

The full details of the proof can be found in [30]. □

3. Numerical Results and Discussion

This section will utilize NIM to uncover the solution of the generalized Burgers–Huxley equation.

To solve the generalized Burgers–Huxley Equation (1) with initial condition (2) we integrate Equation (1) and use Equation (2) to get:

$$u={\int }_0^t\left(\frac{{\partial }^{2}u}{\partial x\partial x}+\left(\beta u\right)\left(1-u^{\delta }\right)\left(u^{\delta }-\gamma \right)-\alpha u^{\delta }\frac{\partial u}{\partial x}\right)dt$$

(11)

By using Equation (10) we obtain:

$$\begin{array}{ccc}\hfill u_0& =& R^{1/\delta }\hfill \\ \hfill u_1& =& t\frac{{\gamma }^4\left(\frac{1}{\delta }-1\right)\delta M^2{R}^{\frac{1}{\delta }-2}{\sec \mathrm{h}}^4\left(K\right)}{4{(4\delta +4)}^2}\hfill \\ \hfill & -& t\frac{{\gamma }^3\delta M^{2}tanh\left(K\right)R^{\frac{1}{\delta }-1}{\sec \mathrm{h}}^2\left(K\right)}{{(4\delta +4)}^2}\hfill \\ \hfill & -& t\frac{\alpha {\gamma }^{2}M{\left(R^{1/\delta }\right)}^{\delta }{R}^{\frac{1}{\delta }-1}{\sec \mathrm{h}}^2\left(K\right)}{2(4\delta +4)}\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& +& t\beta \left({\left(R^{1/\delta }\right)}^{\delta }-\gamma \right)\left(1-{\left(R^{1/\delta }\right)}^{\delta }\right)R^{1/\delta },\hfill \end{array}$$

(13)

where $R=\frac{\gamma }{2}+\frac{1}{2}\gamma tanh\left(K\right)$, $M=\sqrt{{\alpha }^2+4\beta \delta +4\beta }-\alpha$, $K=\frac{\gamma \delta xM}{4\delta +4}$.

So,

$$\begin{array}{ccc}\hfill \sum _{i=0}^1{u}_i& =& u_0+u_1.\hfill \end{array}$$

(14)

The remaining components of the repetition formula can be obtained easily by using computer algebra software like Mathematica. We will show the accuracy and efficiency of the new iterative method (NIM) when applied to the generalized Burgers–Huxley equation (Equation (1)) in comparison to the exact solution, the Adomian decomposition method (referenced in [31]), and the variational iteration method (VIM) (referenced in [32]). To do this, we will use the same parameter values for the equation as were used in [31].

In

Table 1. The absolute errors for the case where $\gamma =0.001$, $\alpha =\beta =\delta =1$ using NIM, ADM and VIM are presented.

x	t	NIM	ADM, [31]	VIM [32]
0.1	0.05	1.87405 $\times {10}^{-8}$	1.87406 $\times {10}^{-8}$	1.87405 $\times {10}^{-8}$
	0.1	3.74811 $\times {10}^{-8}$	3.74812 $\times {10}^{-8}$	3.74813 $\times {10}^{-8}$
	1	3.74811 $\times {10}^{-7}$	3.74812 $\times {10}^{-7}$	3.74812 $\times {10}^{-7}$
0.5	0.05	1.87405 $\times {10}^{-8}$	1.87406 $\times {10}^{-8}$	1.87405 $\times {10}^{-8}$
	0.1	3.74811 $\times {10}^{-8}$	3.74812 $\times {10}^{-8}$	1.37481 $\times {10}^{-8}$
	1	3.74811 $\times {10}^{-7}$	3.74812 $\times {10}^{-7}$	3.74813 $\times {10}^{-7}$
0.9	0.05	1.87405 $\times {10}^{-8}$	1.87406 $\times {10}^{-8}$	1.87405 $\times {10}^{-8}$
	0.1	3.74811 $\times {10}^{-8}$	3.74812 $\times {10}^{-8}$	3.74813 $\times {10}^{-8}$
	1	3.74811 $\times {10}^{-7}$	3.74812 $\times {10}^{-7}$	3.74813 $\times {10}^{-7}$

, we compare the solution obtained using the NIM, VIM (referenced in [32]) and the solution obtained using the ADM (referenced in [31]) for the case where $\gamma =0.001$, $\alpha =\beta =\delta =1$. The comparison for the case where $\gamma =0.01$, $\alpha =\beta =1$, and $\delta =2$ is shown in

Table 2. Absolute errors when $\gamma =0.01$, $\alpha =\beta =1$ and $\delta =2$, using NIM, ADM and VIM.

x	t	NIM	ADM, [31]	VIM [32]
0.1	0.1	5.51552 $\times {10}^{-5}$	5.51554 $\times {10}^{-5}$	5.51580 $\times {10}^{-5}$
	0.2	1.10340 $\times {10}^{-4}$	1.10342 $\times {10}^{-4}$	1.10310 $\times {10}^{-4}$
	0.3	1.65525 $\times {10}^{-4}$	1.65529 $\times {10}^{-4}$	1.65457 $\times {10}^{-4}$
	0.4	2.2070 $\times {10}^{-4}$	2.20708 $\times {10}^{-4}$	2.20598 $\times {10}^{-4}$
	0.5	2.75945 $\times {10}^{-4}$	2.75950 $\times {10}^{-4}$	2.75734 $\times {10}^{-4}$
0.3	0.1	5.51380 $\times {10}^{-5}$	5.51381 $\times {10}^{-5}$	5.51340 $\times {10}^{-5}$
	0.2	1.10291 $\times {10}^{-4}$	1.10293 $\times {10}^{-4}$	1.10262 $\times {10}^{-4}$
	0.3	1.65455 $\times {10}^{-4}$	1.65458 $\times {10}^{-4}$	1.65385 $\times {10}^{-4}$
	0.4	2.20632 $\times {10}^{-4}$	2.20635 $\times {10}^{-4}$	2.20502 $\times {10}^{-4}$
	0.5	2.75830 $\times {10}^{-4}$	2.75832 $\times {10}^{-4}$	2.75614 $\times {10}^{-4}$
0.5	0.1	5.51131 $\times {10}^{-5}$	5.51134 $\times {10}^{-5}$	5.51099 $\times {10}^{-5}$
	0.2	1.10244 $\times {10}^{-4}$	1.10243 $\times {10}^{-4}$	1.10214 $\times {10}^{-4}$
	0.3	1.65400 $\times {10}^{-4}$	1.65402 $\times {10}^{-4}$	1.65313 $\times {10}^{-4}$
	0.4	2.20541 $\times {10}^{-4}$	2.20543 $\times {10}^{-4}$	2.20406 $\times {10}^{-4}$
	0.5	2.75714 $\times {10}^{-4}$	2.75716 $\times {10}^{-4}$	2.75493 $\times {10}^{-4}$

and

Table 3. Absolute errors when $\gamma =0.01$, $\alpha =\beta =1$ and $\delta =4$, using NIM, ADM, and VIM solutions.

x	t	NIM	ADM, [31]	VIM [32]
0.1	0.1	2.17787 $\times {10}^{-4}$	2.17787 $\times {10}^{-4}$	2.17687 $\times {10}^{-4}$
	0.2	4.35691 $\times {10}^{-4}$	4.35690 $\times {10}^{-4}$	4.35293 $\times {10}^{-4}$
	0.3	6.53717 $\times {10}^{-4}$	6.53711 $\times {10}^{-4}$	6.52817 $\times {10}^{-4}$
	0.4	8.71833 $\times {10}^{-4}$	8.71847 $\times {10}^{-4}$	8.70258 $\times {10}^{-4}$
	0.5	1.09050 $\times {10}^{-3}$	1.09010 $\times {10}^{-3}$	1.08762 $\times {10}^{-3}$
0.3	0.1	2.17548 $\times {10}^{-4}$	2.17552 $\times {10}^{-4}$	2.17453 $\times {10}^{-4}$
	0.2	4.35228 $\times {10}^{-4}$	4.35222 $\times {10}^{-4}$	4.34824 $\times {10}^{-4}$
	0.3	6.53010 $\times {10}^{-4}$	6.53008 $\times {10}^{-4}$	6.52113 $\times {10}^{-4}$
	0.4	8.70915 $\times {10}^{-4}$	8.70910 $\times {10}^{-4}$	8.69320 $\times {10}^{-4}$
	0.5	1.08891 $\times {10}^{-3}$	1.08893 $\times {10}^{-3}$	1.08644 $\times {10}^{-3}$
0.5	0.1	2.17320 $\times {10}^{-4}$	2.17318 $\times {10}^{-4}$	2.17218 $\times {10}^{-4}$
	0.2	4.34745 $\times {10}^{-4}$	4.34753 $\times {10}^{-4}$	4.34354 $\times {10}^{-4}$
	0.3	6.52311 $\times {10}^{-4}$	6.52304 $\times {10}^{-4}$	6.51408 $\times {10}^{-4}$
	0.4	8.69963 $\times {10}^{-4}$	8.69972 $\times {10}^{-4}$	8.68380 $\times {10}^{-4}$
	0.5	1.08755 $\times {10}^{-3}$	1.08776 $\times {10}^{-3}$	1.08527 $\times {10}^{-3}$

with the same value but with $\delta =4$. Based on these results, we can conclude that the NIM solution is of comparable accuracy to the VIM and ADM solutions. The NIM eliminates the need to compute the Adomian polynomials, which can sometimes be complicated.

4. Conclusions

This paper demonstrates the successful application of the new iterative method (NIM) in finding approximate solutions to the generalized Burgers–Huxley equation. The results obtained through NIM were compared with those obtained through the Variational Iteration Method (VIM), the Adomian Decomposition Method (ADM), and the exact solution, and it was found that NIM is both efficient and easy to use. The NIM could be used to solve other strong nonlinear partial differential equations.