APPLICABILITY OF SPLINE COLLOCATION TO CORDIAL VOLTERRA EQUATIONS

Teresa Diogo,Gennadi Vainikko

doi:10.3846/13926292.2013.756072

Teresa Diogo, Gennadi Vainikko

Open Access

https://doi.org/10.3846/13926292.2013.756072

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Abstract

We study the applicability of the standard spline collocation method, on a uniform grid, to linear Volterra integral equations of the second kind with the so-called cordial operators; these operators are noncompact and the applicability of the collocation method becomes crucial in the convergence analysis. In particular, piecewise constant, piecewise linear and piecewise quadratic collocation methods are applicable under wide, quite acceptable conditions. For higher order spline collocation, it is more complicated to carry out an analytical study of the applicability of the method; however, a numerical check is rather simple and this is illustrated by some numerical examples.

Highlights

IntroductionIn the present article we study the applicability of spline collocation methods to the Volterra integral equation t μu(t) = t−1φ(t−1s)u(s) ds + f (t), 0 ≤ t ≤ T,
In the present article we study the applicability of spline collocation methods to the Volterra integral equation t μu(t) = t−1φ(t−1s)u(s) ds + f (t), 0 ≤ t ≤ T, (1.1)where φ ∈ L1(0, 1) is the core of the Volterra integral operator t (Vφu)(t) = t−1φ(t−1s)u(s) ds, 0 ≤ t ≤ T.T
We study the applicability of the standard spline collocation method, on a uniform grid, to linear Volterra integral equations of the second kind with the so-called cordial operators; these operators are noncompact and the applicability of the collocation method becomes crucial in the convergence analysis

Summary

Introduction

In the present article we study the applicability of spline collocation methods to the Volterra integral equation t μu(t) = t−1φ(t−1s)u(s) ds + f (t), 0 ≤ t ≤ T,. Since the applicability and convergence conditions of spline collocation methods for equations (1.1) and (1.4), and even for a class of related nonlinear equations, are the same in their essence [12, 13], we confine ourselves to the case of the model equation (1.1). We say that the collocation method (1.5) is applicable to equation (1.1) if the homogeneous equation μuN = PN VφuN has in SNm[0, T ] only the trivial solution uN = 0, that is, if there exist the inverses to the m-dimensional operators μI − ΠN,iVφ : SN,i → SN,i, i = 0, 1, .

Reformulation of the Applicability Condition

Applicability of Piecewise Linear Collocation

Application to Some Concrete Cordial Equations

Applicability of Piecewise Quadratic Collocation

Numerical results by cubic collocation

Numerical check of the applicability condition