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On the approximate solution of integro-differential equations arising in oscillating magnetic fields

K. Maleknejad, M. Hadizadeh, M. Attary (2013)

Applications of Mathematics

In this work, we propose the Shannon wavelets approximation for the numerical solution of a class of integro-differential equations which describe the charged particle motion for certain configurations of oscillating magnetic fields. We show that using the Galerkin method and the connection coefficients of the Shannon wavelets, the problem is transformed to an infinite algebraic system, which can be solved by fixing a finite scale of approximation. The error analysis of the method is also investigated....

On the boundary conditions associated with second-order linear homogeneous differential equations

J. Das (2004)

Archivum Mathematicum

The ideas of the present paper have originated from the observation that all solutions of the linear homogeneous differential equation (DE) y ' ' ( t ) + y ( t ) = 0 satisfy the non-trivial linear homogeneous boundary conditions (BCs) y ( 0 ) + y ( π ) = 0 , y ' ( 0 ) + y ' ( π ) = 0 . Such a BC is referred to as a natural BC (NBC) with respect to the given DE, considered on the interval [ 0 , π ] . This observation suggests the following queries : (i)  Will each second-order linear homogeneous DE possess a natural BC ? (ii)  How many linearly independent natural BCs can a...

On the generalized boundary value problem

Boris Rudolf (2000)

Archivum Mathematicum

In the paper it is proved that the generalized linear boundary value problem generates a Fredholm operator. Its index depends on the number of boundary conditions. The existence results of Landesman-Lazer type are given as an application to nonlinear problems by using dual generalized boundary value problems.

On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations

Malkhaz Ashordia (2021)

Mathematica Bohemica

We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general...

On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order 2m with nonlocal boundary conditions

Aris Tersenov (2001)

Annales Polonici Mathematici

This paper is devoted to the solvability of the Lyapunov equation A*U + UA = I, where A is a given nonselfadjoint differential operator of order 2m with nonlocal boundary conditions, A* is its adjoint, I is the identity operator and U is the selfadjoint operator to be found. We assume that the spectra of A* and -A are disjoint. Under this restriction we prove the existence and uniqueness of the solution of the Lyapunov equation in the class of bounded operators.

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