Blow-up behavior of Hammerstein-type delay Volterra integral equations

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations (DVIEs). Two types of delays, i.e., vanishing delay (pantograph delay) and non-vanishing delay (constant delay), are considered. With the same assumptions of Volterra integral equations (VIEs), in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. The blow-up behaviors of DVIEs with non-vanishing delay vary with different initial functions and the length of the lag, while DVIEs with pantograph delay own the same blow-up behavior of VIEs. Some examples and applications to delay differential equations illustrate this influence.     

The Numerical Solution of Two-Dimensional Volterra Integral Equations by Collocation and Iterated Collocation

While the numerical solution of one-dimensional Volterra integralequations of the second kind with regular kernels is now wellunderstood there exist no systematic studies of the approximatesolution of their two-dimensional counterparts. In the presentpaper we analyse the numerical solution of such equations bymethods based on collocation and iterated collocation techniquesin certain polynomial spline spaces. The analysis focuses onthe global convergence and local superconvergence propertiesof the approximating spline functions.     

Piecewise Polynomial Collocation for Volterra-type Integral Equations of the Second Kind

The exact solution of a given integral equation of the secondkind of Volterra type(with regular or weakly singular kernel)is projected into the space of (continuous) piecewise polynomialsof degree m 1 and with prescribed knots by using collocationtechniques. It is shown that a number of discrete methods forthe numerical solution of such equations based on product integrationtechniques or on finite-difference methods are particular discreteversions of collocation methods of the above type. The errorbehaviour of approximate solutions obtained by collocation (includingtheir discretizations) is discussed.     

Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels

In this paper we investigate the attainable order of (global)convergence of collocation approximations in certain polynomialspline spaces for solution of Volterra integrodifferential equationswith weakly singular kernels. While the use of quasi-uniformmeshes leads, due to the nonsmooth nature of these solutions,to convergence of order less than one, regardless of the degreeof the approximating spline function, collocation on suitablygraded meshes will be shown to yield optimal convergence rates.