The Numerical Solution of Weakly Singular Volterra Integral Equations by Collocation on Graded Meshes

Abstract

Abstract. Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.

Solving integral equations with logarithmic kernels by Chebyshev polynomials

“…Note in passing that the particular choice: C 0 = 0 and C ν = 1 in (2) implies that the approximate solution will be continuous on the entire interval [0, T] [1,2]. For t ∈ [t i , t i+1 ] and i = 0(1)N − 1, consider the approximation…”

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“…For the research for IEs and IDEs we refer to Refs. [2,[6][7][8][15][16][17][18]20,22,23,42,54,58], and for DIDEs, refer to Refs. [3][4][5]9,19,21,22,25,39,44,57,[70][71][72].…”

Uniform approximation to fractional derivatives of functions of algebraic singularity

“…In contrast, it appears that little literature exists on automatic quadratures for the fractional derivative except for our recent scheme [16] for D q f (s) (0 < q < 1) of a well-behaved function f (s). In practical applications, however, it is required to approximate the fractional derivatives of badly-behaved functions of various types or functions with singularities [1,8] such that f (s) = s α g(s), α > −1, where g(s) is assumed to be a well-behaved function, see [15].…”

“…Note that p n (0) = g(0) and p n (1) = g (1). Let h n−1 (t) be a polynomial of degree n − 1 defined by…”