This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.

Volterra Integral Equations: An Introduction to Theory and Applications

A survey of singularly perturbed Volterra equations

This thesis studies singularly perturbed Volterra integral equations of the form
eu(t)=/(t,e)+fg(t,s,11(5)) ds, 0 0 is a small parameter The function f(t,e) is defined for 0 0. The aim is to find asymptotic approximations l to these solutions. This work is restricted to problems where there is an imtial-layer, various hypotheses are placed on g(tts,u) to exclude other behaviour. A major part of this work is that formal solutions of the nonlinear problem are determined and rigorously proved to be asymptotic approximations to the exact solutions Formal approximate solutions:

£Mi,e) NEn-0 Enu(t,E,),=un(t,e)=0(1) as e->0,

are obtained using the additive decomposition method Algorithms which improve the method used m Angell and Olmstead (1987), are presented for obtaining these solutions Assuming a stability condition in the boundary layer, it shown that there is a constant Cjv such ,that

\u(t,e)-Ujsf{t,e)\ <cN eN+1 as e 0,

uniformly for t € [0,T], thus establishing that £/jv(i,£) is an asymptotic solution SIkinner (1995) has proved similar results, but almost all the theorems here were discovered before Skinner’s work was found and are largely independent of it Lange and Smith (1988) prove results for the case g(t, syu) = k(tys)u, where k(t, s) is continuous and satisfies a stability condition in the boundary layer These results are carefully developed here and similar results for linear integrodifferential equations The problem of extending these to the class of weakly singular equations with

g(t,s,u)= k(t,s)/(t-s)BU-0</3<1

is discussed An interesting aspect of this problem and others for which the boundary 1 layer stability condition fails, is that the solutions decay algebraically rather exponentially within ithe boundary layer.

Singularly perturbed volterra integral equations

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

A uniformly valid asymptotic expansion is established for the solution to certain singularly perturbed generalized Abel integral equations. The results are applied to a problem in heat conduction.

A class of singularly perturbed singular Volterra integral equations

A uniformly valid asymptotic expansion for the solution to a class of singularly perturbed Volterra integral equations displaying exponential boundary layer behavior is established. Certain quasilinear ordinary differential equations are a noted special case, and a model for population growth with attrition is briefly discussed.

https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/1995/0002/0002/MAA-1995-0002-0002-a006.pdf

L. A. Skinner

Papers

Asymptotic solution to a class of singularly perturbed Volterra integral equations