A spectral collocation method for a weakly singular Volterra integral equation of the second kind

The solution y of a weakly singular Volterra equation of the second kind posed on the interval −1 ≤ t ≤ 1 has in general a certain singular behaviour near t = −1: typically, |y′(t)|∼(1+t)−μ$|y^{\prime }(t)| \sim (1+t)^{-\mu }$ for a parameter μ ∈ (0, 1). Various methods have been proposed for the numerical solution of these problems, but up to now there has been no analysis that takes into account this singularity when a spectral collocation method is applied directly to the problem. This gap in the literature is filled by the present paper.

[1]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[2]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[3]  Tao Tang,et al.  Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel , 2010, Math. Comput..

[4]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[5]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[6]  P. Davis.  Interpolation and approximation , 1965 .

[7]  C. Lubich,et al.  Runge-Kutta theory for Volterra and Abel integral equations of the second kind , 1983 .

[8]  I. Babuska,et al.  The h , p and h-p versions of the finite element method in 1 dimension. Part II. The error analysis of the h and h-p versions , 1986 .

[9]  Jie Shen,et al.  Generalized Jacobi functions and their applications to fractional differential equations , 2014, Math. Comput..

[10]  I. Babuska,et al.  Theh,p andh-p versions of the finite element method in 1 dimension , 1986 .

[11]  Kai Diethelm,et al.  Multi-Term Caputo Fractional Differential Equations , 2010 .

[12]  Tao Tang,et al.  Supergeometric convergence of spectral collocation methods for weakly singular Volterra and fredholm integral equations with smooth solutions , 2011 .

[13]  Yuesheng Xu,et al.  A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels , 2003, SIAM J. Numer. Anal..

[14]  L. Brutman,et al.  ON THE LEBESGUE FUNCTION FOR POLYNOMIAL INTERPOLATION , 1978 .

[15]  Hermann Brunner,et al.  The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations , 1999, Math. Comput..