Haar wavelet method for the numerical solution of Klein–Gordan equations

Abstract

Wavelets have become a powerful tool for having applications in almost all the areas of engineering and science such as numerical simulation of partial differential equations. In this paper, we present the Haar wavelet method (HWM) to solve the linear and nonlinear Klein–Gordon equations which occur in several applied physics fields such as, quantum field theory, fluid dynamics, etc. The fundamental idea of HWM is to convert the Klein–Gordon equations into a group of algebraic equations, which involve a finite number of variables. The examples are given to demonstrate the numerical results obtained by HWM, are compared with already existing numerical method i.e. finite difference method (FDM) and exact solution to confirm the good accuracy of the presented scheme.

Laguerre wavelets collocation method for the numerical solution of the Benjamina–Bona–Mohany equations

“…Now a days, much attention has been given to the literature of the stable methods for the numerical solution of Benjamina-Bona-Mohany (BBM) equations. In addition to that, Considerable efforts have been made by many mathematicians to obtain exact and approximate solutions of partial differential equations such as Benjamina-Bona-Mohany equations and a number of efficient, accurate and powerful methods have been developed by those mathematicians such as, Backlund transformation method [3], Lie group method [4], Adomian's decomposition method [5], Integral method [6], Hirota's bilinear method [7], homotopy analysis method [8], He's Homotopy perturbation method [9], Exp-Function method [10], Haar wavelet method [11] and Cardinal B-Spline wavelets method [12]. The aim of the present work is to develop Laguerre wavelets collocation method, mutually for solving partial differential equations with initial and boundary conditions of the BBM equations, which is simple, fast and guarantees the necessary accuracy for a relative small number of grid points.…”

A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

“…Namely, Bujurke et al [10][11][12] used the single term Haar wavelet series for the numerical solution of stiff systems from nonlinear dynamics, nonlinear oscillator equations and Sturm-Liouville problems. Shiralashetti et al [13][14][15][16]18] applied for the numerical solution of Klein?Gordan equations, multi-term fractional differential equations, singular initial value problems,nonlinear Fredholm integral equations, Riccati and Fractional Riccati Differential Equations. Shiralashetti et al [17] have introduced the adaptive gird Haar wavelet collocation method for the numerical solution of parabolic partial differential equations.…”

Solving Three Dimensional and Time Depending PDEs by Haar Wavelets Method

“…Ü. Lepik has applied this technique for solving different 1-D problems [14] [15] [16] and [17]. Also he used two dimensional Haar wavelets in solving PDFs which contain two variables [18] [19] [20]. The method is fast and with low error.…”