Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method

Abstract

A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the second kind is given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation = , is bounded and its image space is the reproducing kernel space 1 2 [ , ]. It limits its application. Now, we modify the reproducing kernel method such that it can be more widely applicable. The n-term approximation solution obtained by the modified method is of high accuracy. The numerical example compared with other methods shows that the modified method is more efficient.

Analytic approximate solution for some integral equations by optimal homotopy analysis transform method

“…and then the mth-order deformation equation is given by The above result is in complete agreement with [33]. …”

The Optimal Homotopy Asymptotic Method with Application to Second Kind of Nonlinear Volterra Integral Equations

“…Recently, engineers and scientists known the applications of OHAM in linear and nonlinear problems [9] and [10], because this method continuously deforms complex problems into simple problems which can be solved very easily. This method gives a quick way to the convergence of approximate series and keep more proficiency and high potentiality in science and engineering for solving nonlinear problems.…”

On Reproducing Kernel and Applications

“…We suppose that this equation has a unique solution f : [0, 1] → R n (this section has glimpses from [1,11,13]). We assume (by simplicity) G(f (t)) = C(t)f (t), with C(t) = [c ij (t)] n×n (we may look it as an approximation to the nonlinear case G(u) = c 1 + c 2 u + r(u), with |r(u)|/|u| 0 if u 0).…”

“…Please see references [2,3,9,11,12,13] to more information on this subject. This paper is organized as follows.…”