A.I. Garralda-Guillem

Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.

Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and .

Analytical Techniques for a Numerical Solution of the Linear Volterra Integral Equation of the Second Kind

In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.

About some numerical approaches for mixed integral equations

Abstract In this paper we tackle on mixed Volterra–Fredholm integral equations, as in linear as in non linear cases. To the aim to obtain numerical solutions of these models, the authors propose in the linear case the direct collocation method using a p-order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by Schauder tensor bases. The advantages of both methods are outlined and their convergence is studied. Numerical results confirm the theoretical statements.

Numerical approaches for systems of Volterra-Fredholm integral equations

In this work we introduce two numerical methods for solving systems of Volterra-Fredholm integral equations. In the nonlinear case we suggest a fixed point method, where the iterations are perturbed in a suitable way according to a Schauder basis in the Banach space of continuous functions C[a,b]^2. In the linear case we propose a collocation method based on a particular class of approximating functions. In both methods, convergence analysis and/or low computational cost are analysed, taking into account the properties of the basis under consideration. Numerical results confirm the theoretical study.

Ranges of operators and convex variational inequalities

The main topic in this work is the analysis of the range of a continuous and linear operator between Banach spaces, approached here through non-linear techniques, more precisely, by means of the classical Fan minimax theorem. We characterise the elements in the range of such an operator as those satisfying a certain variational inequality and provide a numerical scheme of Galerkin type to determine approximately the preimage of an element that lies in that range. The passage from the theoretical setting to its numerical realisation is done by means of the use of bases in adequate spaces. In addition we deal with the extension of the previous results to the case of systems of variational inequalities.