Qualitative Analysis of The Burgers-Huxley Equation

There are many well-known techniques for obtaining exact solutions of differential equations, but some of them only work for a very limited class of problems and are merely special cases of a few power symmetry methods. These approaches can be applied to nonlinear differential of unfamiliar type; they do not rely on special “tricks. Instead, a given differential equation can be made to reveal its symmetries, which are then used to construct exact solutions. In this thesis, we briefly present the theory of the Lie symmetry method for finding exact solutions of nonlinear differential equations, then apply it to the study of the generalized Burgers-Huxley equation. Through analyzing the linearized symmetry condition and the associated determining system, we find two nontrivial infinitesimal generators, and obtain exact solutions by solving the reduced differential equation under certain parametric conditions. An approximate solution of the generalized Burgers-Huxley equation is established by means of the Adomian decomposition method

Application of Differential Transform Method to the Generalized Burgers–Huxley Equation

In this paper, the differential transform method (DTM) will be applied to the generalized Burgers-Huxley equation, and some special cases of the equation, say, Huxley equation and Fitzhugh-Nagoma equation. The DTM produces an approximate solution for the equation, with few and easy computations. Numerical comparison between differential transform method, Adomian decomposition method and Variational iteration method for Burgers-Huxley, Huxley equation and Fitzhugh-Nagoma equation reveal that differential transform method is simple, accurate and efficient

A finite volume-complete flux scheme for the singularly perturbed generalized Burgers-Huxley equation

In this paper the finite volume-complete flux scheme is proposed to numerically solve the generalized Burgers-Huxley equation. The scheme is applied in an iterative manner. Numerical computations are performed for traveling wave-type problems as a validation of the method. Convection-dominated problems are used to assess the method on boundary layers. The results are in good agreement with reference results

Finite difference method for numerical solution of a generalized burgers-huxley equation

There are many applications of the generalized Burgers-Huxley equation which is a form of nonlinear Partial Differential Equation such as in the work of physicist which can effectively models the interaction between reaction mechanisms, convection effects and diffusion transports. This study investigates on the implementation of numerical method for solving the generalized Burgers-Huxley equation. The method is known as the Finite Difference Method which can be employed using several approaches and this work focuses on the Explicit Method, the Modified Local Crank-Nicolson (MLCN) Method and Nonstandard Finite Difference Schemes (NFDS). In order to use the NFDS, due to a lack of boundary condition provided in the problem, this research used the Forward Time Central Space (FTCS) Method to approximate the first step in time. Thomas Algorithm was applied for the methods that lead to a system of linear equation. Computer codes are provided for these methods using the MATLAB software. The results obtained are compared among the three methods with the exact solution for determining their accuracy. Results shows that NFDS has the lowest relative error and one of the best way among these three methods in order to solve the generalized Burgers-Huxley equations

Haar wavelet method for solving generalized Burgers–Huxley equation

In this paper, an efficient numerical method for the solution of nonlinear partial differential equations based on the Haar wavelets approach is proposed, and tested in the case of generalized Burgers–Huxley equation. Approximate solutions of the generalized Burgers–Huxley equation are compared with exact solutions. The proposed scheme can be used in a wide class of nonlinear reaction–diffusion equations. These calculations demonstrate that the accuracy of the Haar wavelet solutions is quite high even in the case of a small number of grid points. The present method is a very reliable, simple, small computation costs, flexible, and convenient alternative method. © 201

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature

Reduced Differential Transform Method for (2+1) Dimensional type of the Zakharov-Kuznetsov ZK(n,n) Equations

In this paper, reduced differential transform method (RDTM) is employed to approximate the solutions of (2+1) dimensional type of the Zakharov-Kuznetsov partial differential equations. We apply these method to two examples. Thus, we have obtained numerical solution partial differential equations of Zakharov-Kuznetsov. These examples are prepared to show the efficiency and simplicity of the method

A multi-domain implementation of the pseudo-spectral method and compact finite difference schemes for solving time-dependent differential equations

Abstract : In this dissertation, we introduce new numerical methods for solving time-dependant differential equations. These methods involve dividing the domain of the problem into multiple sub domains. The nonlinearity of the differential equations is dealt with by using a Gauss-Seidel like relaxation or quasilinearisation technique. To solve the linearized iteration schemes obtained we use either higher order compact finite difference schemes or spectral collocation methods and we call the resulting methods the multi-domain compact finite difference relaxation method (MD-CFDRM), multi-domain compact finite difference quasilinearisation method (MD-CFDQLM) and multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) respectively. We test the applicability of these methods in a wide variety of differential equations. The accuracy is compared against other methods as well as other results from literature. The MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. Chaotic and hyperchaotic systems are characterized by high sensitivity to small perturbation on initial data and rapidly changing solutions. Such rapid variations in the solution pose tremendous problems to a number of numerical approximations. We modify the CFDs to be able to deal with such systems of equations. We also used the MD-CFDQLM to solve the nonlinear evolution partial differential equations, namely, the Fisher’s equation, Burgers- Fisher equation, Burgers-Huxley equation and the coupled Burgers’ equations over a large time domain. The main advantage of this approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method for large time domain. We also studied the generalized Kuramoto-Sivashinsky (GKS) equations. The KS equations exhibit chaotic behaviour under certain conditions. We used the multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) to approximate the numerical solutions for the generalized KS equations.M.Sc. (Pure and Applied Mathematics

Numerical resolution of Emden's equation using Adomian polynomials

Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.This work has been supported by the Ministerio de Ciencia e Innovación, project TIN2009-10581