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

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Introduction
Te Burgers-Huxley equation is given as Its initial and boundary conditions are v(x, 0) � f(x), v(a, t) � g 0 (t), v(b, t) � g 1 (t). ( Here a ≤ x ≤ b, t ≥ 0, and nonlinear reaction term is f(v) � (1 − v)(v − c).α and β ≥ 0 are advection and reaction coefcients, respectively.c is a parameter such that c ∈ (a, b).When α � 0, equation ( 1) is reduced to the Huxley equation which describes nerve pulse propagation in nerve fbres and wall motion in liquid crystals [1].When β � 0, it is reduced to the well-known Burgers equation which describes the far feld in wave propagation in nonlinear dissipative systems.When α � 0 and β � 1, it becomes the FitzHugh-Nagumo equation which is the reaction difusion equation used in circuit theory and biology [1].When α � 0 and β � 0, the equation turns into an important equation called Burgers-Huxley equation which describes many physical phenomena encountered in models where reaction, convection, and difusion take place.
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 diferent methods which are based on interpolation scaling functions as well as mixed collocation fnite diference schemes for the numerical solution of the nonlinear generalized Burgers-Huxley equation [18].Recently, in 2023, Chin [19] used the coupling of the nonstandard fnite 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 felds 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 diferential quadrature method and collocation method to deal with the diferential equations.Te cubic B-spline collocation method was developed by Goh et al. [20] to numerically solve onedimensional heat and advection difusion equations.Dag and Saka [21] used this method for equal width equation.Diferent 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 diferential 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 difusion 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][30][31][32][33][34] to fnd solution of some important problems occurring in the feld 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.Te present method does not involve any integrals to get the fnal set of equations, and thus computational eforts have been reduced to a great extent.Fourth-order approximation for single as well as double derivatives is employed.Tis is done by using different end conditions and taking one extra term in the Taylor series expansion which has resulted in accurate and efcient numerical solutions.Te aim of this work is to study the numerical solutions of the Burgers-Huxley equation for diferent 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.
Te 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.Te proposed method is quite simple and produces highly efcient results and hence reduces complexity and computational cost.

Mathematical Formulation
Let us consider an equal partition of the domain Ω by the knots x j , j � 0, 1, 2, . . .., N, such that h � x j − x j− 1 is the length of each interval.Te third-degree B-spline termed as cubic B-spline is given as where [F − 1 (x), F 0 (x), F 1 (x), . . . . . ., F N (x), F N+1 (x)] are basis functions over the interval.Exact solution v(x, t) is approximated by K(x, t) in the cubic B-spline collocation method as where a j (t)'s are unknown quantities which we have to fnd.K(x, t) is considered to satisfy the following interpolatory and boundary conditions: (4) x j , t  , j � 0, N. (5)

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If K(x, t) is a unique cubic spline interpolant which satisfes above boundary conditions and v(x, t) is a smooth function, then from [38], we have Using Taylor's expansion and fnite diferences, the approximate values of K(x, t) and its frst-order derivatives at the knots are defned as follows.
For j � 0, For j � N,

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 4

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For j � 0, We may write it as We may write it as

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We may write it as Finally, the following system of linear equations is obtained: where Te resulting system of equation ( 28) is semi-implicit.We can clearly observe that there are N + 1 equations in N + 3 unknowns.a − 1 and a N+1 are eliminated with the help of Dirichlet or Neumann's boundary conditions.After elimination, we get N + 1 equations in N + 1 unknowns.Bspline approximation of initial condition is used to get the initial vector [a (0)  0 , a (0) 1 , a (0) 2 , . . . . . ., a (0) N ] T .Now, the system of equations can be solved at any desired time level with the help of initial vector.

Stability Analysis
In equation ( 21), we assume Ten Assume , where h is step length, D is amplitude, and m is mode number, we have where For stability of the present method, we should have We need to show For minimum value of Hence, the proposed method is unconditionally stable.

Numerical Experiments
Te exact solution of equation ( 1) was derived by Wang et al.
[1] using nonlinear transformations and is given by where Tis exact solution satisfes the following initial and boundary conditions: Error norms are given by where u(x, t) and v(x, t) are the exact and numerical solutions, respectively.Formula for the rate of convergence is given by where E(N 1 ) and E(N 2 ) are L ∞ errors at grid sizes N 1 and N 2 , respectively.Accuracy and efciency of the proposed method have been verifed 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 fnd numerical solution, the space and time step lengths are taken as h � 0.1 and Δt � 0.1 or Δt � 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 efcient.Solutions have been calculated for large values of t and it can be seen that it is taking very small CPU-time.Tere 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 α � 0, β � 1, equation ( 1) becomes the FitzHugh-Nagumo equation and can be written as For c � 0.001, comparison of absolute errors of the present method with Bratsos [37] and Mohammadi [17] at diferent grid points at times T � 0.05, 0.10, 1 is shown in Table 1.Similarly, Table 2 compares L ∞ error of the present method with three diferent methods of Dehghan [18] at times T � 0.3, 0.6, 1.For c � 0.0001, comparison of L ∞ error of the present method with three diferent methods of Dehghan [18] is established in Table 3.For large T, absolute errors of the present method are shown in Table 4. Tis shows that the method is giving good results for large times as well.Te 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 fgures 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 efcient and requires minimal computational eforts.Example 2. We set α � 2 and β � 1. Table 5 compares L ∞ errors of the present scheme with three diferent methods of Dehghan [18] at times T � 0.3, 0.6, 1 for c � 0.001 and c � 0.0001.We get similar nature of results as that of Example 1.
Example 3. In this example, we take negative value of convection coefcient, i.e., α � − 0.1.β and c 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.and Δt are taken as 5 and 0.003, respectively.Te L ∞ 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 β � 10 to β � 100 with an increment of 10.For this fgure, c � 0.00001 and Δt � 0.01.It can inferred that as β or c decreases, error decreases.Tus, the smaller the difusion coefcient, the better the accuracy.
Example 5. We set α � c � 0.1 and β � 0.001.Table 8 shows comparison of absolute errors of the present method with diferent methods given in the literature such as ADM [2][3][4], VIM [6,7], DTM [39], and LDM [39] at times T � 0.05, 0.1, 1. Te results of the present method show better accuracy  Journal of Mathematics and require less computational eforts.Figure 4 depicts absolute errors for T � 0.05 to T � 1 with Δt � 0.05.So, it can be easily seen from this fgure that error decreases with decrease in time.
Example 7. In this example, α, β, and c 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 efective for large times.Table 11 compares L ∞ error of the present method with diferent 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 L ∞ error comparison with HBCSM [41] for diferent n.

Conclusion
In this work, we have proposed a fourth-order cubic B-spline collocation method to solve the second-order nonlinear Burgers-Huxley equation.Te various numerical experiments show that this method can produce accurate as well as efcient solutions.MATLAB programming is done for calculations and plotting.Te 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.Te method satisfes the physical behavior of the nonlinear Burgers-Huxley equation.(3) Stability of the present method has been verifed and found to be unconditionally stable.(4) In diferent settings of the parameters, this method can successfully provide highly efcient solutions.(5) Te 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 diferential equations.
(7) Te 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.

Figure 2 :
Figure 2: Evolution of numerical solution with space and time variables for T �1.

Table 8 :
Comparison of absolute error at grid points with diferent schemes taking α � c � 0.1 and β � 0.001.

Table 13 :
Comparison of absolute error at grid points with diferent schemes taking α � β � 1 and c � 0.001.