Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations Using Haar Wavelet Collocation Method

Abstract

Abstract:In this paper, we present a numerical solution of nonlinear Volterra-Fredholm integral equations using Haar wavelet collocation method. Properties of Haar wavelet and its operational matrices are utilized to convert the integral equation into a system of algebraic equations, solving these equations using MATLAB to compute the Haar coefficients. The numerical results are compared with exact and existing method through error analysis, which shows the efficiency of the technique.

Haar wavelets method for solving class of coupled systems of linear fractional Fredholm integro-differential equations

“…Therefore, the Haar wavelets are an effective tool for obtaining approximate solutions for a lot of the applied problems such as integral and differential equations as a result of their simplicity for instance, (see e.g. [10] , [11] , [12] , [13] , [14] ). Thus, it is deduced that Haar wavelets approach has gained much attention in the recent years in obtaining approximate solutions for integro-differential equations, and that is because its simplicity and efficiency.…”

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

“…Collocation methods are very popular. They are based on splines [21,29,92,93], general approximate functions [22,82], Chebyshev polynomials [16], shifted Chebyshev polynomials [81], Bernstein polynomials [34,63], Chelyshkov Polynomials [36], Lagrange polynomials [42], Taylor polynomials [47], Fibonacci polynomials [57], Bell polynomials [65], first Boubaker polynomials [60], Müntz-Legendre polynomials [70], generalized Lucas polynomials [87], Jacobi polynomials [89], block-pulse functions [27], hybrid of block-pulse functions and Lagrange polynomials [26], hybrid block-pulse function and Taylor polynomials [41] (see also [73]), block-pulse functions and Bernoulli polynomials [45], hybrid block-pulse functions and Bernstein polynomials [62], Haar wavelets [4,66], rationalized Haar functions [19], Legendre wavelets [13,97], triangular functions [23,100], fuzzy transforms [32], Sinc function [37,40], radial basis functions [35], pseudospectral integration matrices [54] and shifted piecewise cosine basis functions [75]. Galerkin methods are also popular and are commonly used in conjunction with general approximate functions [22,82], Legendre polynomials…”

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.…”