Numerical solution of elliptic partial differential equations by Haar wavelet operational matrix method / Nor Artisham Che Ghani

Nor Artisham, Che Ghani (2012) Numerical solution of elliptic partial differential equations by Haar wavelet operational matrix method / Nor Artisham Che Ghani. Masters thesis, University of Malaya.

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Abstract

The purpose of this study is to establish a simple numerical method based on the Haar wavelet operational matrix of integration for solving two dimensional elliptic partial differential equations of the form, Ñ2u(x, y) + ku(x, y) = f (x, y) with the Dirichlet boundary conditions. To achieve the target, the Haar wavelet series were studied, which came from the expansion for any two dimensional functions g(x, y) defined on L2 ([0,1)´ [0,1)), i.e. g(x, y) =Σc h (x)h ( y) ij i j or compactly written as HT (x)CH( y) , where C is the coefficient matrix and H(x) or H( y) is a Haar function vector. Wu (2009) had previously used this expansion to solve first order partial differential equations. In this work, we extend this method to the solution of second order partial differential equations. The main idea behind the Haar operational matrix for solving the second order partial differential equations is the determination of the coefficient matrix, C. If the function f (x, y) is known, then C can be easily computed as H × F × HT , where F is the discrete form for f (x, y) . However, if the function u(x, y) appears as the dependent variable in the elliptic equation, the highest partial derivatives are first expanded as Haar wavelet series, i.e. u HT (x)CH( y) xx = and u HT (x)DH( y) yy = , and the coefficient matrices C and D usually can be solved by using Lyapunov or Sylvester type equation. Then, the solution u(x, y) can easily be obtained through Haar operational matrix. The key to this is the identification for the form of coefficient matrix when the function is separable. Three types of elliptic equations solved by the new method are demonstrated and the results are then compared with exact solution given. For the beginning, the computation was carried out for lower resolution. As expected, the more accurate results can be obtained by increasing the resolution and the convergence are faster at collocation points. This research is preliminary work on two dimensional space elliptic equation via Haar wavelet operational matrix method. We hope to extend this method for solving diffusion equation, k u t u = Ñ2 ¶ ¶ and wave equation, c u t u 2 2 2 2 = Ñ ¶ ¶ in a plane.

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