Solving Partial Differential Equations Based On Haar Wavelet

Wavelet analysis is an emerging discipline, is widely used in mathematics, medicine and military. Partial differential equations(PDE) are commonly used to describe many physical phenomena of the nature, since most of these equations cannotfind the analytic solution. Hence, the study of the numerical approximationbecomes particularly important, but the traditional numerical methods solving partial differential equations have some limitations. Therefore, lots of scholars have developed some new methods, such as,using wavelet to solve partial differential equations. The main contents of this thesis are as follows: Firstly, we introduce thedevelopment of waveletbrieflyand the background and significance of the topic based on Haar wavelet to solve the partial differential equation.Then we introduce the related theory of wavelet, such as wavelet function, continuous wavelet transform, discrete wavelet transform, multi-resolution analysis and so on.Afterwards,wepresent theiteration method solving linear equation system. Secondly,the Haar wavelets and their integral operator matrices are studied, which lays a theoretical foundation for the following research. Finally, we study the technique solving three kind of different partial differential equation by using the Haar wavelet, which are the one-dimensional heat conduction equation of parabolic type, 1D hyperbolic second-order telegraph equation, Laplacian equation in the elliptic.Theirbasic idea is:using Haar wavelet to approximate the high-order mixed partial derivative of the unknown function, obtaining the discrete form of original equation by means of definite integral, getting their coefficients of approximation applying MATLAB, obtaining the numerical solutions of the unknown function.Based on the examples, the results of numerical experiments illustrate that using the Haar wavelets to solve elliptic equation, hyperbolic equations and parabolic equations are all feasible, and the higher the order number, the higher the accuracy of the approximation.