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- W2110059847 abstract "The purpose of this study is to establish a simple numerical method based on theHaar wavelet operational matrix of integration for solving two dimensional elliptic partialdifferential equations of the form, N2u(x, y) + ku(x, y) = f (x, y) with the Dirichletboundary conditions. To achieve the target, the Haar wavelet series were studied, whichcame from the expansion for any two dimensional functions g(x, y) defined onL2 ([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 thiswork, 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 orderpartial differential equations is the determination of the coefficient matrix, C. If thefunction f (x, y) is known, then C can be easily computed as H × F × HT , where F is thediscrete form for f (x, y) . However, if the function u(x, y) appears as the dependentvariable in the elliptic equation, the highest partial derivatives are first expanded as Haarwavelet series, i.e. u HT (x)CH( y)xx = and u HT (x)DH( y)yy = , and the coefficientmatrices 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 keyto 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 andthe 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 obtainedby increasing the resolution and the convergence are faster at collocation points.This research is preliminary work on two dimensional space elliptic equation viaHaar wavelet operational matrix method. We hope to extend this method for solving diffusion equation, k utu = N2¶¶ and wave equation, c utu 2 222= N¶¶ in a plane." @default.
- W2110059847 created "2016-06-24" @default.
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- W2110059847 date "2012-01-01" @default.
- W2110059847 modified "2023-09-27" @default.
- W2110059847 title "Numerical solution of elliptic partial differential equations by Haar wavelet operational matrix method / Nor Artisham Che Ghani" @default.
- W2110059847 hasPublicationYear "2012" @default.
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