This paper presents a new application of the homotopy analysis method (HAM) for solving evolution equations described in terms of nonlinear partial differential equations (PDEs). The new approach, termed bivariate spectral homotopy analysis method (BISHAM), is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.
The study of nonlinear evolution partial differential equations (PDEs) is a vast area of research with well-developed and documented theories and applications in almost all areas of science and engineering. The PDEs are used to describe many complex nonlinear settings in applications such as vibration and wave propagation, fluid mechanics, plasma physics, quantum mechanics, nonlinear optics, solid state physics, chemical kinematics, physical chemistry, population dynamics, and many other areas of mathematical modelling. The development of both analytical and numerical methods for solving complicated highly nonlinear PDEs continues to be a fertile area of research geared towards enriching and deepening our understanding of these intriguing nonlinear problems.
The homotopy analysis method (HAM) has been widely discussed in the literature for solving both nonlinear ordinary and partial differential equations. A comprehensive exposition of the underlying concepts and applications of the HAM can be found in recently published books [
The main objective of this work is to introduce a new variant of the spectral homotopy analysis method for solving nonlinear partial differential equations. The proposed method is developed by defining a rule of solution expression based on bivariate Lagrange interpolation. The homotopy analysis method algorithm is then applied to decompose the governing nonlinear PDEs into a sequence of linear PDEs. The resulting linear sequence of PDEs contains variable coefficients and is impossible to solve exactly. Consequently, the Chebyshev spectral collocation method is applied independently in the space and time independent variables. In view of the application of the combination of bivariate interpolation and spectral collocation differentiation, the new method is called bivariate interpolated spectral homotopy analysis method (BI-SHAM). The study presents a general BI-SHAM algorithm that can be used to solve second order nonlinear evolution equations. The applicability, accuracy, and reliability of the proposed BI-SHAM is confirmed by solving the Fisher, Burger-Fisher, Burger-Huxley, and Fitzhurg-Nagumo equations. The BI-SHAM results are compared against known exact solutions that have been reported in the scientific literature.
The remainder of the paper is organized as follows. In Section
In this section we introduce the
To derive the HAM equations corresponding to the nonlinear equation (
If the initial condition for (
Equation (
The algorithm of the HAM begins with the construction of the homotopy for a given linear operator
Thus, substituting (
To demonstrate the applicability of the proposed Bi-SHAM algorithm as an appropriate tool for solving nonlinear partial differential equations, we apply the proposed algorithm to well-known nonlinear PDEs of the form (
We consider Fisher’s equation as follows:
We consider the generalized Burgers-Fisher’s equation [
Consider the Fitzhurg-Nagumo equation as follows:
Consider the Burgers-Huxley’s equation as follows:
The exact solution subject to the initial condition,
In this example the linear
In this section we present the numerical solutions of the implementation of the BI-SHAM algorithm on the nonlinear evolution equations as described in the previous section. The number of collocation points in the space
Using finite terms of the SHAM series we define
Assuming that
The residual error is used in establishing the suitable convergence controlling parameter
In Figures
Residual
Residual
Residual
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In Tables
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CPU Time (sec) | 0.023 | 0.026 | 0.039 | 0.051 |
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CPU Time (sec) | 0.012 | 0.024 | 0.029 | 0.048 |
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CPU Time (sec) | 0.012 | 0.024 | 0.029 | 0.048 |
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CPU Time (sec) | 0.041 | 0.065 | 0.099 | 0.182 |
This paper has presented a new variant of the spectral homotopy analysis method for solving general nonlinear evolution partial differential equations. The new method, called bivariate spectral homotopy analysis method (BISHAM), was developed from a combination of the homotopy analysis method algorithm with bivariate Lagrange interpolation and spectral collocation differentiation. The main goal of the current study was to assess the accuracy, applicability, and effectiveness of the proposed method in solving nonlinear partial differential equations. Numerical simulations were conducted on the Fisher equation, Burger-Fisher equation, Fitzhurg-Nagumo, and Burger-Huxley equations. This study has shown that the BISHAM gives very accurate results in a computationally efficient manner. Further evidence from this study is that the BISHAM gives solutions that are uniformly accurate and valid in large intervals of the governing space and time domains. The apparent success of the method can be attributed to the use of the nontrivial linear operators and the spectral collocation method for differentiating. This work contributes to the existing body of literature on homotopy analysis method based tools for solving complicated nonlinear partial differential equations. Further work needs to be done to establish whether the BISHAM can be equally successful in solving higher order nonlinear partial differential equations and coupled systems of two of more equations.
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is based on the research supported in part by the National Research Foundation of South Africa (Grant no. 85596).