Numerical Solution of Nonlinear Partial Differential Equation by Legendre Multiwavelet Method
Magdy Ahmed Mohamed · International Journal of Scientific Research · 2012
In this work, the Legendre multiwavelet basis with considering the standard Galerkin method has been applied to give the approximate solution of nonlinear partial differential equations (NPDE's). The properties of the Legendre multiwavelet presented. These properties together with the standard Galerkin method are then utilized to reduce nonlinear partial differential equations to the solution of an algebraic system. Numerical results and comparison with exact solution given to demonstrate the applicability and efficiency of the method. Abstract In this work, the Legendre multiwavelet basis with considering the standard Galerkin method has been applied to give the approximate solution of nonlinear partial differential equations (NPDE's). The properties of the Legendre multiwavelet presented. These properties together with the standard Galerkin method are then utilized to reduce nonlinear partial differential equations to the solution of an algebraic system. Numerical results and comparison with exact solution given to demonstrate the applicability and efficiency of the method. 1. Introduction In 1807, Joseph Fourier developed a method for representing a signal with a series of coefficients based on an analysis function. He laid the mathematical basis from which the wavelet theory is developed. The first to mention wavelets was Alfred Haar in 1909 in his PhD thesis. In the 1930's, Paul Levy found the scale-varying Haar basis function superior to Fourier basis functions. Jean Morlet and Alex Grossman again derive the transformation method of decomposing a signal into wavelet coefficients and reconstructing the original signal in 1981. In 1986, Stephane Mallat and Yves Meyer developed a multiresolution analysis using wavelets. They mentioned the scaling function of wavelets for the first time, it allowed researchers and mathematicians to construct their own family of wavelets using the derived criteria. Around 1998, Ingrid Daubechies used the theory of multiresolution wavelet analysis to construct her own family of wavelets. Her set of wavelet orthonormal basis functions have become the cornerstone of wavelet applications today. Wavelet analysis can be performed in several ways, a continuous wavelet transform, a discretized continuous wavelet transform and a true discrete wavelet transform. The application of wavelet analysis becomes more widely spread as the analysis technique becomes more generally known. The fields of application vary from science, engineering, medicine to finance. Types of wavelets are Haar Wavelets (orthogonal in L