Numerical Solution of Burgers–Huxley Equation Using a Higher Order Collocation Method

Singh, Aditi; Dahiya, Sumita; Emadifar, Homan; Khademi, Masoumeh

doi:https://doi.org/10.1155/2024/2439343

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Abstract Introduction Conclusion Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 2439343 | https://doi.org/10.1155/2024/2439343

Numerical Solution of Burgers–Huxley Equation Using a Higher Order Collocation Method

Aditi Singh,¹Sumita Dahiya,²Homan Emadifar,^3,4,5and Masoumeh Khademi⁵

Academic Editor: M. M. Bhatti

Received20 Aug 2023

Revised28 Dec 2023

Accepted07 Feb 2024

Published29 Feb 2024

Abstract

In this paper, the collocation method with cubic B-spline as basis function has been successfully applied to numerically solve the Burgers–Huxley equation. This equation illustrates a model for describing the interaction between reaction mechanisms, convection effects, and diffusion transport. Quasi-linearization has been employed to deal with the nonlinearity of equations. The Crank–Nicolson implicit scheme is used for discretization of the equation and the resulting system turned out to be semi-implicit. The stability of the method is discussed using Fourier series analysis (von Neumann method), and it has been concluded that the method is unconditionally stable. Various numerical experiments have been performed to demonstrate the authenticity of the scheme. We have found that the computed numerical solutions are in good agreement with the exact solutions and are competent with those available in the literature. Accuracy and minimal computational efforts are the key features of the proposed method.

1. Introduction

The Burgers–Huxley equation is given as

Its initial and boundary conditions are

Here a x b, t 0, and nonlinear reaction term is . and are advection and reaction coefficients, respectively. is a parameter such that . When = 0, equation (1) is reduced to the Huxley equation which describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals [1]. When = 0, it is reduced to the well-known Burgers equation which describes the far field in wave propagation in nonlinear dissipative systems. When = 0 and = 1, it becomes the FitzHugh–Nagumo equation which is the reaction diffusion equation used in circuit theory and biology [1]. When = 0 and = 0, the equation turns into an important equation called Burgers–Huxley equation which describes many physical phenomena encountered in models where reaction, convection, and diffusion take place.

There are many computational methods for the solution of Burgers–Huxley equation, such as the Adomian decomposition method [2–4], homotopy analysis method [5], and variational iteration method [6, 7]. In 2006, Javidi [8] employed modified pseudospectral method to numerically obtain the solution of generalized Burgers–Huxley equation. Javidi and Golbabai [9] employed Chebyshev polynomials-based new domain decomposition algorithm, Ismail et al. [2] applied the Adomian decomposition method (ADM), Kaushik [10] used grid equidistribution, Babolian et al. [5] the applied homotopy analysis method (HAM), and Khattak [11] used the computational meshless method to get analytic or numerical solutions of the generalized Burgers–Huxley equation. Tomasiello [12] used the IDQ method, Zhang et al. [13] used local discontinuous Galerkin methods, and Duan et al. [14] developed the lattice Boltzmann model to obtain the solution of the Burgers–Huxley equation.

Recently, Mittal and Tripathi developed a collocation method using cubic B-splines as basis functions to numerically solve generalized Burgers–Fisher and generalized Burgers–Huxley equations [15]. Celik [16] employed the Haar wavelet method, Reza Mohammadi [17] used B-spline collocation algorithm, and Dehghan et al. made use of different methods which are based on interpolation scaling functions as well as mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation [18]. Recently, in 2023, Chin [19] used the coupling of the nonstandard finite difference approach in the time variables with the Galerkin method and the compactness methods in the space to obtain solution of the Burgers–Huxley equation.

Nowadays, B-spline functions are becoming popular and are being used as a powerful tool in various fields such as approximation theory, image processing, atomic and molecular physics, numerical simulations, and computer-aided designs. Basis spline functions have been incorporated in various numerical methods such as differential quadrature method and collocation method to deal with the differential equations. The cubic B-spline collocation method was developed by Goh et al. [20] to numerically solve one-dimensional heat and advection diffusion equations. Dag and Saka [21] used this method for equal width equation. Different variants of this method have been used by Kadalbajoo and Arora [22], Zahra [23], Dag [24], and Khater et al. [25] to solve various other important equations. Mittal and Dahiya [26] used quintic B-splines in the differential quadrature method to solve a class of Fisher–Kolmogorov equations. Fourth-order collocation methods have been developed by Mittal and Rohila [27] to numerically study the reaction diffusion Fisher’s equation. In 2020, Singh et al. [28] employed the fourth-order collocation method to get numerical solutions of the Burgers–Fisher equation. Roul et al. used B-spline collocation methods in various studies [29–34] to find solution of some important problems occurring in the field of science and engineering. Recently, Kumar et al. [35] used a spline approximation technique to solve the boundary value inverse problem associated with the generalized Burgers–Fisher and generalized Burgers–Huxley equations.

A novel method called the fourth-order cubic B-spline collocation method is adopted in the proposed work to solve the Burgers–Huxley equation. The present method does not involve any integrals to get the final set of equations, and thus computational efforts have been reduced to a great extent. Fourth-order approximation for single as well as double derivatives is employed. This is done by using different end conditions and taking one extra term in the Taylor series expansion which has resulted in accurate and efficient numerical solutions. The aim of this work is to study the numerical solutions of the Burgers–Huxley equation for different parametric values using collocation method with cubic B-splines as basis functions. A preprint of this work has been previously published by Singh et al. [36] in 2023.

The present scheme gives the approximate solution at any point of the solution domain. Accuracy obtained in this work is satisfactory and comparable with those present in the previous literature. The proposed method is quite simple and produces highly efficient results and hence reduces complexity and computational cost.

2. Mathematical Formulation

Let us consider an equal partition of the domain by the knots , , such that = is the length of each interval. The third-degree B-spline termed as cubic B-spline is given aswhere are basis functions over the interval.

Exact solution is approximated by in the cubic B-spline collocation method aswhere ’s are unknown quantities which we have to find. K(x, t) is considered to satisfy the following interpolatory and boundary conditions:

If K(x, t) is a unique cubic spline interpolant which satisfies above boundary conditions and is a smooth function, then from [38], we have

Using Taylor’s expansion and finite differences, the approximate values of K(x, t) and its first-order derivatives at the knots are defined as follows.

For j = 0,

For ,

For j = N,

Using equations (8)–(10) in (6) and (7), we get

For j = 0,

For ,

For j = N,

Using equations (3) and (4), equations (12)–(14) can be simplified as follows.

For j = 0,

For ,

For j = N,

3. Implementation of the Method

We now use the Crank–Nicolson scheme to discretize Burgers–Huxley equation (1) and then we get

Quasi-linearization formula is

By applying quasi-linearization in equation (18), we get

Now, terms of (n)th and (n + 1)th time levels are separated to get the equation of the form

For j = 0,

We may write it as

For ,

We may write it as

For j = N,

We may write it as

Finally, the following system of linear equations is obtained:where

The resulting system of equation (28) is semi-implicit. We can clearly observe that there are equations in unknowns. and are eliminated with the help of Dirichlet or Neumann’s boundary conditions. After elimination, we get equations in unknowns. B-spline approximation of initial condition is used to get the initial vector . Now, the system of equations can be solved at any desired time level with the help of initial vector.

4. Stability Analysis

In equation (21), we assume

Then

Assume = L.

Substituting = D , where h is step length, D is amplitude, and m is mode number, we haveorwhere

For stability of the present method, we should have

We need to show

For minimum value of , cos (h) = 1. Thus, on putting values of A, B, C from equation (35) in , we get which is obviously positive.

Hence, the proposed method is unconditionally stable.

5. Numerical Experiments

The exact solution of equation (1) was derived by Wang et al. [1] using nonlinear transformations and is given bywhere

This exact solution satisfies the following initial and boundary conditions:

Error norms are given bywhere u(x, t) and (x, t) are the exact and numerical solutions, respectively.

Formula for the rate of convergence is given bywhere and are errors at grid sizes and , respectively.

Accuracy and efficiency of the proposed method have been verified by comparing obtained results with the above exact solution and traditional methods available in the literature [2–4, 6, 7, 9, 13, 15–18, 39, 40].

In order to find numerical solution, the space and time step lengths are taken as h = 0.1 and t = 0.1 or t = 0.01 for all examples unless otherwise stated. CPU-time in seconds is calculated for all the examples and shown in their respective tables. It is found that accuracy of the present method is satisfactory and comparable to or rather higher than those available in the literature. As we can see in all the examples, the present method is taking very less computational time and thus it is highly efficient. Solutions have been calculated for large values of t and it can be seen that it is taking very small CPU-time. There is one more advantage of the present method, that is, it requires less number of grid points resulting in low memory storage.

Example 1. When = 0, = 1, equation (1) becomes the FitzHugh–Nagumo equation and can be written asFor = 0.001, comparison of absolute errors of the present method with Bratsos [37] and Mohammadi [17] at different grid points at times T = 0.05, 0.10, 1 is shown in Table 1. Similarly, Table 2 compares error of the present method with three different methods of Dehghan [18] at times T = 0.3, 0.6, 1. For = 0.0001, comparison of error of the present method with three different methods of Dehghan [18] is established in Table 3. For large T, absolute errors of the present method are shown in Table 4. This shows that the method is giving good results for large times as well. The approximate solutions of the present method are shown graphically at times T = 0.3, 0.6, 1 in Figure 1. 3D form of the approximate solution for T = 1 is shown in Figure 2. In the relevant nonlinear dissipative systems, the solutions obtained here describe the special coherent structures.
It can be concluded from these figures and tables that the results are found to be quite competent with the literature. It can be clearly seen that it requires very less CPU-time; hence, the proposed method is efficient and requires minimal computational efforts.

Example 2. We set = 2 and = 1. Table 5 compares errors of the present scheme with three different methods of Dehghan [18] at times T = 0.3, 0.6, 1 for = 0.001 and = 0.0001. We get similar nature of results as that of Example 1.

Example 3. In this example, we take negative value of convection coefficient, i.e., = −0.1. and are taken as 0.1 and 0.001, respectively. Comparison of absolute errors of the present method is done with Celik [16] at time T = 0.9 in Table 6. Results are comparable and show good accuracy.

Example 4. In this example, we consider different values of and as = 1, 10, 100 and = 0.001, 0.0001, 0.00001. Here, and t are taken as 5 and 0.003, respectively. The errors of the obtained results are presented for T = 0.3 and 0.9 in Table 7 and compared with those of CM [15]. Figure 3 represents the pictorial view of absolute errors at T = 1 for = 10 to = 100 with an increment of 10. For this figure, = 0.00001 and t = 0.01. It can inferred that as or decreases, error decreases. Thus, the smaller the diffusion coefficient, the better the accuracy.

Example 5. We set = = 0.1 and = 0.001. Table 8 shows comparison of absolute errors of the present method with different methods given in the literature such as ADM [2–4], VIM [6, 7], DTM [39], and LDM [39] at times T = 0.05, 0.1, 1. The results of the present method show better accuracy and require less computational efforts. Figure 4 depicts absolute errors for T = 0.05 to T = 1 with t = 0.05. So, it can be easily seen from this figure that error decreases with decrease in time.

Example 6. We consider = 0.01, = = 0.0001. Table 9 compares absolute errors of the present method with different methods such as ADM [2–4], VIM [6, 7], DTM [39], and LDM [39] at times T = 0.05, 0.1, 1.

Example 7. In this example, , , and are taken as 0.1, 0.001 and 0.0001, respectively. Absolute errors of the present scheme for large T are mentioned in Table 10. For T = 50, absolute errors of the present method can be seen in Figure 5. After getting the results in tabular as well as graphical form, it can inferred that the method is effective for large times. Table 11 compares error of the present method with different schemes like Javidi [9], Zhang [13], and CM [15] at T = 0.2 and T = 1. Also, Table 12 validates the accuracy of the method by making error comparison with HBCSM [41] for different n.

Example 8. We consider Burgers–Huxley equation (1) with = = 1, = 0.001. Absolute errors of the present method are compared with methods given in ADM [2–4], VIM [6, 7], DTM [39], and Bratsos [37] at different grids for times T = 0.05, 0.1, 1. The results are shown in Table 13. errors are calculated at times T = 0.2, 1 and compared with different methods such as Javidi [9], Zhang [13], and CM [15] as reported in Table 14.

6. Conclusion

In this work, we have proposed a fourth-order cubic B-spline collocation method to solve the second-order nonlinear Burgers–Huxley equation. The various numerical experiments show that this method can produce accurate as well as efficient solutions. MATLAB programming is done for calculations and plotting. The main inferences are as follows:(1)A technique based on fourth-order approximation of the solution has been used.(2)From the numerical section, it is evident that the results are in full agreement with the exact solution and are quite competent with those available in the literature. The method satisfies the physical behavior of the nonlinear Burgers–Huxley equation.(3)Stability of the present method has been verified and found to be unconditionally stable.(4)In different settings of the parameters, this method can successfully provide highly efficient solutions.(5)The method is reliable, easy to implement, and economical.(6)Results illustrate that the present scheme is a valuable tool for studying various nonlinear problems. It can be extended to higher dimensional partial differential equations.(7)The advantage of the present method over other methods is that the present method is convenient for solving boundary value problems with numerical ease, high accuracy, and minimal time consumption.

Data Availability

No data were used to support this study.

Disclosure

An earlier version of this manuscript is also available on ResearchGate and can be accessed through https://www.researchgate.net/publication/369069921_Numerical_solution_of_Burgers'-Huxley_equation_using_a_higher_order_collocation_method.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are very thankful to Prof. S. P. Singh for his unconditional guidance and support.

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