Efficiency and Accuracy of Numerical Solution of Laguerre's Differential Equation Using Haar Wavelet

Abstract

From past literature, it is well known that Haar wavelet is a powerful mathematical tool for solving various type of differential equations and the solution obtained by Haar wavelet are more accurate than that obtained by other methods. Our aim in the present paper is to illustrate the slow computational convergence of Laguerre's differential equation using Haar wavelet, noting that Laguerre's differential equation has polynomial solutions.

Solution of Laguerre’s Differential Equations via Modified Adomian Decomposition Method

“…In particular, Laguerre's differential equation is a type of differential equation that is found in a variety of engineering problems [1], and in quantum mechanics, because it is one of several equations that appear in the quantum mechanical description of the hydrogen atom [2].…”

“…Further, Laguerre's differential equation was solved in [1] by using the Haar wavelet method; while [4] solved the same model by using G-transform, a generalized Laplace-typed transform method. Also, the differential transformation method [5] was applied to solve Laguerre's differential equation.…”

Solving Some Oscillatory Problems using Adomian Decomposition Method and Haar Wavelet Method

“…Haar wavelet is the simplest orthonormal wavelet with compact support and has been utilized for solving linear as well as non-linear differential and integral equations. Haar wavelet method has been used for solving ordinary and partial differential equations [11][12][13][14][15][16][17][18]. Since solutions of ordinary and partial differential equations which are not enough smooth, when approximated by cubic, quadratic and linear polynomials results in poor convergence or no convergence in results and in such cases, an approximation of zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are more suitable and successful.…”

Numerical Solution of Volterra Integral Equations by Using Hybrid Block-Pulse Functions and Bernstein Polynomials

“…Bernstein polynomials (BP) have been recently used for the solution of some nonlinear integro-differential equations, both BVP, by Yuzbasi [1] and Islam & Hossain [2]. Also these have been used to solve some classes of mathematical equations by [3,4,5,6,7,8]. These were further used to solve Falkner-Skan equation by Tavassoli Kajani et al [9].…”