Numerical methods for singular nonlinear integrodifferential equations
type: Item
instance of: scholarly article
title: Numerical methods for singular nonlinear integrodifferential equations
zbMATH Open document ID: 0631.65142
DOI: 10.1016/0168-9274(87)90051-1
author: Ernst Hairer
author: Peter Maass
published in: Applied Numerical Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The equation which is studied here is the integrodifferential equation \[ y'(t)=f(t,y(t))+(1/\Gamma (\alpha))\int^{t}_{0}(t-s)^{\alpha - 1}K(t,s,y'(s))ds,\quad t>0, \] where \(y(0)=y_ 0\). The functions f and K are known and assumed to be smooth; \(0<\alpha <1\). It is known that the solution may not be smooth due to the presence of the weak singularity. \textit{C. Lubich} [Math. Comp. 45, 463-469 (1985; Zbl 0584.65090)] proposed a ``fractional linear multistep'' method for the numerical solution of an integral equation with a weakly singular kernel. The present authors show how this method can be adapted to deal with their equation. They prove convergence and present a stability analysis with diagrams of stability domains. The paper concludes with an examination of Basset's integrodifferential equation which describes the behaviour of a particle in a turbulent fluid. The equation is solved numerically and an associated function, the diffusion coefficient, is graphed.
reviewed by: D. Kershaw
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 76F99
zbMATH DE Number: 4026778
zbMATH Keywords: fractional linear multistep method
zbMATH Keywords: singular nonlinear integrodifferential equations
zbMATH Keywords: Volterra integrodifferential equations
zbMATH Keywords: weak singularity
zbMATH Keywords: convergence
zbMATH Keywords: stability
zbMATH Keywords: Basset's integrodifferential equation
zbMATH Keywords: turbulent fluid
MaRDI profile type: Publication
full work available at URL: 0168-9274(87)90051-1
OpenAlex ID: W2012856639
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description: scientific article; zbMATH DE number 4026778