Keywords:-
Article Content:-
Abstract
Elliptic partial differential equations arise in the mathematical modelling of many physical phenomena arising in science and engineering. In this paper, we use Haar wavelet method for the numerical solution of Laplace and Poisson equation. The basic idea of Haar wavelet collocation method is to convert the partial differential equation into a system of algebraic equations that involves a finite number of variables. The numerical results are compared with the exact solution to prove the accuracy of the Haar wavelet method.
References:-
References
D.J Jones, J.C South Jr., E.B Klunker, On the numerical solution of elliptic partial differential equations by the method of lines, J. Comput. Phys. 9(3) (1972) 496-527.
D.J Evans, S.O Okolie, The numerical solution of an elliptic PDE with periodic boundary conditions in a rectangular region by the spectral resolution method, J. Comput. Appl. Math. 8(4) (1982) 237-271.
J.G. Lewis, R.G. Rehm, The numerical solution of a nonseparable elliptic partial differential equation by preconditioned conjugate gradients, J. Res. Nat. Bur. Stand. 88(5) (1980) 367-390.
M. Dehghan, M. Shirzadi, Numerical solution of stochastic elliptic partial differential equations using the meshless method of radial basis functions, Eng. Anal. Bound. Elem. 50 (2015) 21-303.
P. Hashemzadeh, A.S. Fokas, S.A. Smitheman, A numerical technique for linear elliptic partial differential equations in polygonal domains, Proc. R. Soc. 471 (2175) (2015).
R. Rangogni, Numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method, Appl. Math. Model. 10(4) (1986) 266-270.
M. Tatari, M. Dehghan, Numerical solution of Laplace equation in a disk using the Adomian decomposition method, Phys. Scripta 72(5) (2005) 345.
M. Sohail, S.T.M. Din, Reduced differential transform method for Laplace equations, Int. J. Mod. Theor. Phys. 1(1) (2012) 6-12.
A. Aminataei, M.M. Mazarei, Numerical solution of Poisson’s equation using radial basis function networks on the polar coordinate, Comp. Math. Appl. 56(11) (2008) 2887-2895.
H. Bennour, M.S. Said, Numerical solution of Poisson equation with Dirichlet boundary conditions, Int. J. Open Problems Compt. Math. 5(4) (2012) 171-195.
R.S. Sumana, L.N. Achala, A short report on different wavelets and their structures, Int. J. Res. Engg. Sci. 4(2) (2016) 31-35.
C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.-Control Theory Appl. 144(1) (1997) 87-94.
U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68 (2005) 127-143.
U. Lepik, Applications of the Haar wavelet transform to solving integral and differential equations, Proc. Estonian Acad. Sci. Phys. Math. 56(1) (2007) 28-46.
U. Lepik, Numerical solution of evolution equations by Haar wavelet method, Appl. Math. Comput. 185 (2007) 695-704.
U. Lepik, Haar wavelet method for solving stiff differential equations, Math. Model. Anal. 14 (2007) 467-481.
N.M. Bujurke, C.S. Salimath, S.C. Shiralashetti, Computation of eigenvalues and solutions of regularSturm-Liouville problems using Haar wavelets, J. Comput. Appl. Math. 219 (2008) 90-101.
G. Hariharan, Haar wavelet method for solving the Klein-Gordon and the Sine-Gordon equations, Int. J. Nonlin. Sci. 11(2) (2011) 180-189.