Abstract
The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.
Highlights
The theory of control problems for a system of ordinary differential equations and for a system of integro-differential equations in partial derivatives, with parameters, is rapidly developing and used in various fields of applied mathematics, biophysics, biomedicine, chemistry, etc
We extend the methods and results of [14, 15] to a boundary value problem for integro-differential equations with parameters
We study the properties of δm(θ) general solution and obtain solvability criteria for loaded differential equations with a parameter
Summary
The theory of control problems for a system of ordinary differential equations and for a system of integro-differential equations in partial derivatives, with parameters, is rapidly developing and used in various fields of applied mathematics, biophysics, biomedicine, chemistry, etc. Called as boundary value problems with parameters and parameter identification problems for systems of ordinary differential and integro-differential equations with parameters, are intensively studied by many authors [3, 4, 8, 9, 17, 18, 19, 20, 24, 25]. Note that the problems of determining effective criteria for unique solvability and constructing numerical algorithms for finding optimal solutions to control problems for systems of ordinary differential and integro-differential equations with parameters are still relevant. We extend the methods and results of [14, 15] to a boundary value problem for integro-differential equations with parameters. The solvability conditions for boundary value problems for a system of integro-differential equations with parameters are established. A numerically approximate method for solving the investigated boundary value problem is constructed, and its convergence, stability, and accuracy are investigated
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