Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

Abstract

a b s t r a c tAn effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.

On integral equations with Weakly Singular kernel by using Taylor series and Legendre polynomials

“…In [9] Babolian et al solved (1) with k(x, y) = , they removed singularity with Taylor expansion of ϕ(y) at point x, and then used Legendre functions as basis and computed all de nite integrals involved without numerical quadratures. In [10] Lakestani et al utilized Legendre multiwavelets as basis to reduce the solution of Fredholm integro-di erential equation to the solution of sparse linear system of algebraic equations. In [11] Jiang et al gave representation of the exact solution of third or rst kind weakly singular Fredholm integral equations by a series in the reproducing kernel space.…”

“…Table 4 shows the absolute errors which computed by using proposed method. Also, Comparison is made with the result presented in [10,18]. There the maximum error is de ned as MaxError = max|ϕ(x) − ϕ N (x)|, to calculate the errors in Table 4.…”

Comparative experiment on the numerical solutions of Hammerstein integral equation arising from chemical phenomenon

“…Saadatmandi et al have solved class of fractional convectiondiffusion equation with variable coefficients by Sinc-Legendre collocation method [18]. Haar wavelet method has been applied to solve fractional differential equation by Saha Ray et al [19,20].Legendre multiwavelets have been used for solving weakly singular Fredholm integrodifferential equations [21]. Some iterative techniques and quadrature formulae [7,8] have been applied to solve Hammerstein integral equations by Saha Ray et al Also Hammerstein integral equation has been solved by B-spline wavelets [10].…”

Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis

“…A numerical solution of weakly singular integral equation was also discussed in [2,11]. Also see [1,7,13,21,22].…”