M.I. Berenguer

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.

Linear Volterra integro-differential equation and Schauder bases

In this paper we present a numerical method to approximate the solution of the linear Volterra integro-differential equation. By making use of the expression of a function of the Banach space C([0,1]) in terms of a basis Schauder, we are able to define a sequence of functions which approximate (in the uniform sense) the solution of such equation. Likewise, we study the error committed in each approximation. Some advantages that this method possess are that it is not necessary to solve linear equation systems and moreover, the involved integrals are immediate.

Nonlinear Volterra Integral Equation of the Second Kind and Biorthogonal Systems

We obtain an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

The authors present a method of numerical approximation of the fixed point of an operator, specifically the integral one associated with a nonlinear Fredholm integral equation, that uses strongly the properties of a classical Schauder basis in the Banach space .

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.

Fixed point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm–Volterra integro-differential equation

Abstract In this work we approximate the solution of the first order nonlinear Fredholm–Volterra integro-differential equation, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

Galerkin schemes and inverse boundary value problems in reflexive Banach spaces

We develop the Galerkin method for a recent version of the Lax-Milgram theorem. The generation of the corresponding finite-dimensional subspaces for concrete boundary value problems leads us to consider certain biorthogonal systems in the reflexive Banach spaces in question. In addition, we present an application to the numerical solution of inverse problems involving certain elliptic boundary value problems.

Solution of systems of integro-differential equations using numerical treatment of fixed point

We approximate the solution of a system of nonlinear mixed Fredholm-Volterra integro-differential equations of the second kind, using fixed point techniques and Schauder bases in certain Banach spaces. The convergence and error are studied. Several numerical examples are given, and the obtained numerical approximations are compared with the corresponding exact solutions.

A computational method for solving a class of two dimensional Volterra integral equations

The use of the Geometric Series theorem, together with Schauder bases in certain Banach spaces, allows us to design a numerical algorithm in order to solve an important type of two-dimensional linear integral equations. A study of the convergence and error is presented here. All the calculations can be easily implemented and the efficiency of this method will be shown with numerical results.