A control problem for an integro-differential equation is approximated by a problem with parameter for a loaded differential equation. A mutual relationship between the qualitative properties of the original and approximate problems is obtained, and estimates for the differences between their solutions are given. A numerical-approximate method for solving the control problem for the integro-differential equation is proposed, and the convergence, stability, and accuracy of the method are examined.

Numerical Solution to a Control Problem for Integro-Differential Equations

This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical examples.

A computational method for solving a problem with parameter for linear systems of integro-differential equations

The nonlinear two-point boundary value problem with parameter for systems of ordinary differential equations is investigated by the parametrization method. The method consists in partition of the interval apart, the introduction of additional parameters and reducing the original problem to the equivalent multi-point boundary value problem with parameters. The algorithms for finding a solution to the equivalent boundary value problem are constructed. The conditions for convergence of the algorithms that ensure the solvability of nonlinear two-point boundary value problem with parameter are obtained. Definition of the isolated solution to nonlinear two-point boundary value problem with parameter, which is a modification of the well-known definition for the nonlinear two-point boundary value problem without parameter, is introduced.

An Existence of an Isolated Solution to Nonlinear Two-Point Boundary Value Problem with Parameter

Abstract A problem with parameter for an integro-differential equation is approximated by a problem with parameter for a loaded differential equation. The well-posedness of a problem with parameter for the integro-differential equation is established in the terms of the well-posedness of a problem with parameter for the loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are set. A new general solution to the loaded differential equation with parameter is presented, and its properties are described. The problem with parameter for the loaded differential equation is reduced to a system of linear algebraic equations with respect to the arbitrary vectors of a general solution introduced. The system’s coefficients and right-hand sides are computed by solving the Cauchy problems for ordinary differential equations.

Well-posedness of problem with parameter for an integro-differential equation

We study the initial-boundary-value problem for an integrodifferential equation of the third order Replacing the required function and its time derivative by a combination of two new unknown functions, we reduce this problem to an equivalent problem in the form of a family of multipoint problems for a system of two Volterra integrodifferential equations of the first order and integral relations We construct algorithms for finding the solution of the equivalent problem By the method of parametrization, we establish the conditions for the unique solvability of a family of multipoint problems for the system of first-order Volterra integrodifferential equations in terms of the initial data We also establish the conditions for the existence of the unique classical solution of the initial-boundary-value problem for the integrodifferential equation of the third order in terms of the coefficients of this equation and boundary functions

Initial-Boundary-Value Problem for an Integrodifferential Equation of the Third Order

We substantiate the application of a modified projection-iterative method to the solution of boundary-value problems for weakly nonlinear integrodifferential equations with parameters.

Modified Projection-Iterative Method for Weakly Nonlinear Integrodifferential Equations with Parameters

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Modified Projection-Iterative Method for Weakly Nonlinear Integrodifferential Equations with Parameters