A Novel Approach to Solving a Multipoint Boundary Value Problem for an Integro-Differential Equation

Jul 16, 2026

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A method of parameterization is used to study a multipoint boundary value problem for a system of Fredholm integro-differential equations. The paper treats the degenerate-kernel case separately, deriving well-posedness conditions and proposing algorithms to construct approximate and numerical solutions. For the general Fredholm integro-differential multipoint problem, necessary and sufficient well-posedness conditions are established, together with algorithms that reduce the task to solving an approximating system for the degenerate-kernel case.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages15

Reading Time38 min

Introduction

Parameterization method background

Problem formulation

Multipoint boundary conditions

Fredholm integro-differential equations with degenerate kernel

arXiv :2309 . 15805v2 [math .NA] 13 Jul 2026 1 A novel approach to solving a multipoint boundary value problem for an integro-differential equation Anar T. Assanova 1 , Elmira A. Bakirova 2 , Roza E. Uteshova 3 Institute of Mathematics and Mathematical Modeling, 125, Pushkin Str., 050010, Almaty, Kazakhstan 1 e-mail: [assanova@math.kz](assanova@math.kz), 2 e-mail: [bakirova1974@mail.ru](bakirova1974@mail.ru), 3 e-mail: [ruteshova1@gmail.com](ruteshova1@gmail.com) Abstract In the present paper, we study a multipoint boundary value problem for a system of Fredholm integro-differenial equations by the method of parameterization.The case of a degenerate kernel is studied separately, for which we obtain well-posedness conditions and propose some algorithms to find approximate and numerical solutions of the problem. We then establish necessary and sufficient conditions for the well-posedness of the multipoint problem for a system of Fredholm integro-differential equations and develop some algorithms for finding its approximate solutions. These algorithms are based on the solutions of an approximating problem for the system of integro-differential equations with degenerate kernel. MSC: 45J05, 45L05; 47G20; 65Q99 . Keywords: Fredholm integro-differential equation, multipoint problem, parameterization method, algorithm, solvability criteria. 1. Introduction Various types of multipoint problems for differential and integro-differential equations have been studied by many researchers, see [1-4, 7-11, 20-25] . A number of methods have been applied to solve these problems, e.g., methods of qualitative theory of differential equations, the method of Green’s functions, the method of upper and lower solutions, numerical-analytical methods. However, the problem of establishing effective criteria for the unique solvability of multipoint problems for integrodifferential equations, as well as developing algorithms for finding their approximate and numerical solutions, still remains open. One of constructive methods of investigation and solving boundary value problems for ordinary differential equations and integro-differential equations is the method of parameterization proposed by Dzhumabaev [12] . This method was originally developed for studying and solving boundary value problems for systems of ordinary differential equations. In [12], coefficient criteria were established for the unique solvability of linear boundary value problems. An algorithm for finding their approximate solutions was developed. The method of parameterization was later extended to linear multipoint boundary value problems [20-21], for which necessary and sufficient conditions were obtained for the unique solvability in terms of initial data and an algorithm for finding their approximate solutions was proposed. In [13-15, 19], the method of parameterization was applied to two-point boundary value problems for Fredholm integro-differential equations to establish criteria for their solvability and unique solvability. For these problems, based on the method of parameterization and a new concept of general solution, novel algorithms for approximate and numerical solutions were developed, see [16-18] . The results obtained in above-mentioned papers were used to investigate a multipoint boundary value problem for loaded differential equations [4] and a boundary value problem with a parameter for Fredholm integro-differential equations [3] . Consider the multipoint boundary value problem for the system of integro-differential equations ddxt = A (t)x + Z0 T K (t,τ)x (τ)dτ + f(t), x ∈ Rn , t ∈ (0, T), (1 . 1) This paper is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement ID: 873071, project SOMPATY (Spectral Optimization: From Mathematics to Physics and Advanced Technology) . 2 m XBix(ti) = d, d ∈ Rn. (1 .2) i=0 Here x (t) = col(x1 (t), x2 (t),..., xn(t)) i

FAQs

What mathematical problem does the paper address?

It studies a multipoint boundary value problem for a system of Fredholm integro-differential equations, including the case of a degenerate integral kernel.

How does the parameterization method enter the solution strategy?

The interval is partitioned and additional parameters are introduced as the values of the solution at left endpoints, which leads to special Cauchy problems with parameters.

What results are provided for solvability and solution construction?

The paper derives well-posedness conditions, including necessary and sufficient criteria for the multipoint problem, and develops algorithms for approximate and numerical solutions based on an approximating problem for the degenerate-kernel case.

Jul 16, 2026

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