A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations
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A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations M. Rasty · M. Hadizadeh
Received: 23 July 2008 / Accepted: 22 October 2008 / Published online: 30 October 2008 © Springer Science+Business Media B.V. 2008
Abstract This paper provides with a generalization of the work by Chelyshkov (Electron. Trans. Numer. Anal. 25(7): 17–26, 2006), who has introduced sequences of orthogonal polynomials over [0, 1] which can be expressed in terms of Jacobi polynomials. We develop a new approach of product integration algorithm based on these orthogonal polynomials including the numerical quadratures for solving the nonlinear weakly singular Volterra integral equations. The convergence analysis of the proposed scheme is derived and numerical results are given showing a marked improvement in comparison with recent numerical methods. Keywords Nonlinear weakly singular integral equation · Product integration · Orthogonal polynomials · Convergence analysis · Numerical treatments Mathematics Subject Classification (2000) Primary 65R20 · Secondary 45E10
1 Introduction Integral equations of Volterra type with weakly singular kernels arise in many modelling problems in mathematical physics and chemical reactions, such as stereology [16], heat conduction, crystal growth, electrochemistery, superfluidity [15], the radiation of heat from a semi infinite solid [11] and many other practical applications. We remark here that equations of this type have been the focus of many papers [5, 7, 10, 17, 22, 24, 25] in recent years. In this paper we consider the nonlinear Volterra integral equations of the second kind, x p(x, t)K(x, t, f (t))dt, x ∈ [0, T ] (1.1) f (x) = g(x) + 0
M. Rasty · M. Hadizadeh () Department of Mathematics, K.N. Toosi University of Technology, Tehran, Iran e-mail: [email protected] M. Rasty e-mail: [email protected]
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M. Rasty, M. Hadizadeh
where the kernel p(x, t) is weakly singular and the given functions g and K are assumed to be sufficiently smooth in order to guarantee the existence and uniqueness of a solution f (x) ∈ C[a, b] (see for instance, [1, 4]). Typical forms of p(x, t) are (i) p(x, t) = |x − t|−α , 0 < α < 1, (ii) p(x, t) = log|x − t|. It is well known that for Volterra equations with bounded kernels, the smoothness of the kernel and of the forcing function g(x) determines the smoothness of the solution on the closed interval [0, X], with X > 0. If we allow weakly singular kernels, then the resulting solutions are typically non smooth at the initial point of the interval of integration, where their derivatives become unbounded. Some results concerning the behavior of the exact solution of equations of type (1.1) are given in [4]. The numerical solvability of weakly singular integral equations and other related equations have been pursued by several authors. The existence and uniqueness results of solution (1.1) have been proved in [12] by Kershaw using the Banach’s fixed point theorem. Moreover, using Lagrange linear interpolation
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