Teresa Diogo

An extrapolation method for a Volterra integral equation with weakly singular kernel

Abstract In this work we consider second kind Volterra integral equations with weakly singular kernels. By introducing some appropriate function spaces we prove the existence of an asymptotic error expansion for Eulers method. This result allows the use of certain extrapolation procedures which is illustrated by means of some numerical examples.

Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter µ, is considered. Although for certain values of µ this equation possesses an infinite set of solutions, we have been able to prove that Eulers method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.

A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.

Fully discretized collocation methods for nonlinear singular Volterra integral equations

We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.

Applicability of Spline Collocation to Cordial Volterra Equations

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.

The Coiflet-Galerkin method for linear Volterra integral equations

This paper deals with the application of the Wavelet-Galerkin method based on Coiflets as a basis for solving linear Volterra integral equations (VIEs). The main contribution of this work is that some new connection coefficients are introduced and a suitable algorithm is developed for their solution; once they have been computed they can be stored and applied to any linear VIE. The convergence properties of the Coiflet-Galerkin method are analyzed. Some test examples are presented to illustrate the performance of the method with respect to the error norms and CPU time.

A Mathematical Treatment of the Fluorescence Capillary-Fill Device

A mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labeled antigen for antibody sites on a cell wall; boundary conditions are such that the problem is both nonlinear and nonlocal. This is then recharacterized first as a pair of coupled singular integro-differential equations and then as a system of four Volterra integral equations. The latter permits a proof of existence and uniqueness of the solution of the original problem. Small and large time asymptotic solutions are derived and, from the first characterization, a regular perturbation solution is obtained. Numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin.

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

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

High-Order Methods for Weakly Singular Volterra Integro-Differential Equations

x^{\beta }\, (z-x)^{-\alpha } \, g(y(x))