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Journal of Scientific Computing

Erschienen in:

30.04.2016

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

verfasst von: Sonia Seyed Allaei, Teresa Diogo, Magda Rebelo

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2016

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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 \( x^{\beta }\, (z-x)^{-\alpha } \, g(y(x))\), where \(\alpha \in (0,1), \beta >0\) 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 \(\displaystyle L^{\infty }\) and the \(L^2\) norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.

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Titel: The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel
verfasst von: Sonia Seyed Allaei
Teresa Diogo
Magda Rebelo
Publikationsdatum: 30.04.2016
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-016-0213-x

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