Abstract
An effective technique based on fractional calculus in the sense of Riemann–Liouville has been developed for solving weakly singular Volterra integral equations of the first and second kinds. For this purpose, orthogonal Chebyshev polynomials are applied. Properties and some operational matrices of these polynomials are first presented and then the unknown functions of the integral equations are represented by these polynomials in the matrix form. These matrices are then used to reduce the singular integral equations to some linear algebraic system. For solving the obtained system, Galerkin method is utilized via Chebyshev polynomials as weighting functions. The method is computationally attractive, and the validity and accuracy of the presented method are demonstrated through illustrative examples. As shown in the numerical results, operational matrices, even for first kind integral equations, have relatively low condition numbers, and thus, the corresponding matrices are well posed. In addition, it is noteworthy that when the solution of equation is in power series form, the method evaluates the exact solution.
Highlights
The aim of this study is to present a high-order computational method for solving special cases of singular Volterra integral equations of the first and second kinds, namelyAbel’s integral equations, defined byZx f ðxÞ 1⁄4 jx À tjÀayðtÞdt; ð1Þ andZx yðxÞ 1⁄4 f ðxÞ þ jx À tjÀayðtÞdt; ð2Þ0\a\1; 0 x T; where f ðxÞ 2 C1⁄20; T is the known function and y(x) is the unknown function that to be determined, and T is a positive constant.Abel’s equation is one of the integral equations derived directly from a concrete problem of mechanics or physics
An effective technique based on fractional calculus in the sense of Riemann–Liouville has been developed for solving weakly singular Volterra integral equations of the first and second kinds
For solving the obtained system, Galerkin method is utilized via Chebyshev polynomials as weighting functions
Summary
The aim of this study is to present a high-order computational method for solving special cases of singular Volterra integral equations of the first and second kinds, namely. The construction of high-order methods for the equations is, not an easy task because of the singularity in the weakly singular kernel In this case, the solution y is generally not differentiable at the endpoints of the interval [3], and due to this, to the best of the authors’ knowledge, the best convergence rate ever achieved remains only at polynomial order. We use the Chebyshev polynomials operational matrices via Galerkin method for solving weakly singular integral equations. Our method consists of reducing the given weakly singular integral equation to a set of algebraic system by expanding the unknown function by Chebyshev polynomials of the first kind. X 2 1⁄2a; b; a [ 0: Definition 2 For f 2 C1⁄2a; b, the left and right Riemann–
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