Wavelet Numerical Method For Solving 2D Poisson Equations And Biharmonic Equations

Haar wavelets are composed of a series of piecewise constant functions,and they are the simplest orthogonal wavelets with a compact support.Due to the simplicity of expression and the computability of wavelet functions,Haar wavelets turned out to be an effective mathematical tool for solving differential equations.In the paper,Haar wavelet collocation method is used to discrete the original differential equations and solve the boundary value problem of 1D and 2D Poisson equations and biharmonic equations.The feasibility and efficiency of the method is demonstrated by some numerical examples.Due to the Haar wavelets are piecewise continuous functions and not derivative,The highest derivative is expanded into Haar wavelet series and the expression of the unknown function is obtained through integrations.Using boundary conditions to get add additional equations.The crucial issue is through converting differential equations to algebraic equations to obtain numerical solutions.The main content of this article is divided into three parts:Firstly,solving 2D Poisson equations in a rectangular domain by using Haar wavelets method,then using Haar wavelets method to solve 2D biharmonic equations over the rectangle.As to several equations with different boundary conditions,we achieved very good results.Finally,Haar wavelets method is used to solve 2D Poisson equations in an irregular domain with jump conditions.An irregular domain can be treated by embedding the region into a rectangular domain.Using Haar wavelet collocation method to solve the equations on formal area and obtain the numerical solution of high accuracy.In this paper,the error analysis of Haar wavelet numerical approximation method for solving differential equations is proposed,which provides a theoretical basis for numerical calculation.