The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel

Sonia Seyed Allaei, Teresa Diogo, Magda Rebelo

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form xβ(z-x)-αg(y(x)), where α∈ (0 , 1) , β> 0 and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the L and the L2 norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.

Original languageEnglish
Pages (from-to)673-695
Number of pages23
JournalSiam Journal On Scientific Computing
Volume69
Issue number2
DOIs
Publication statusPublished - 1 Nov 2016

Keywords

  • Convergence analysis
  • Jacobi spectral collocation method
  • Nonlinear Volterra integral equation
  • Weakly singular kernel

Fingerprint Dive into the research topics of 'The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel'. Together they form a unique fingerprint.

Cite this