On The Haar Wavelet-Finite Difference Method For Solving The Generalized Burgers-Fisher Equation

In this paper, the numerical solution of the generalized Burgers-Fisher equation with the initial boundary value conditions based on the Haar wavelet-finite difference method is studied. The generalized Burgers-Fisher equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport. It has the extremely vital significance in the modern physics. The Haar wavelet family on the closed interval is given. The Haar wavelet matrix of n-tuple integration is established. Haar wavelet-finite difference method adopted the matrix of n-tuple integration. The matrix is to convert the integral operations into the matrix operations. According to this method the spatial operators are approximated by the Haar wavelet family and the time derivation operators by the first-order precision backward difference quotient. The Haar wavelet-finite difference method is proposed to solve the generalized Burgers-Fisher equation with the general initial boundary value conditions. The paper gives the algorithm, program diagram and MATLAB code.This paper analyzes the stability of Haar wavelet-finite difference method, and numerical proves that the algorithm is conditional stability. The algorithm property of the positivity of the numerical solutions and their boundedness is testing. The results show that the algorithm can maintain property of the positivity of the numerical solutions and their boundedness.This paper fully combines the features of Haar wavelet multi-resolution analysis flexible, efficiency and the advantage of finite difference easy to realize. This method improves the computation speed and accuracy. It is due to the sparsity of the Haar wavelet matrix of n-tuple integration. Numerical results, obtained by computer simulation, are compared with the finite difference method and the Adomian decomposition method. The result showed that the method for solving problems of small time is more than Adomian decomposition method. The algorithm precision is higher than the finite difference method. The algorithm is feasible, effective and high accuracy. In order to study the numerical solution for the generalized Burgers-Fisher equation with initial boundary value problem providing new ideas and new methods.