Extension Operator Method for the Exact Solution of Integro-Differential Equations

Parasidis, I. N.; Providas, E.

doi:10.1007/978-3-319-31317-7_23

Cited by 12 publications

(8 citation statements)

References 26 publications

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“…Loaded differential equations of the form (2.10) with various other types of initial-boundary conditions, as well as the corresponding inverse problems, optimal control problems in various settings have been studied by many authors [4,5,10]. For them, the necessary conditions for the existence and uniqueness of the solution were obtained [7], [16]- [18] and various numerical solution schemes were proposed [3,6,8]. Considering a significant increase in the order of the original problem (2.1) and (2.2), the use of such approaches to the study and solution of practical problems of the form (2.1) and (2.2) and related inverse problems, optimal control problems, causes serious difficulties [4,5,10].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Tamarkin [9,20]. Later, existence and uniqueness conditions for solutions were obtained for various classes of these problems [12,13,21], and problems is associated with obtaining constructive, necessary and sufficient conditions for the existence and uniqueness of the solution easily verified directly from the data of the problem, and its qualitative properties [7], [16]- [19].…”

Section: Introductionmentioning

confidence: 99%

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To the solution of integro-differential equations with nonlocal conditions

2021

Turk J Math

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We investigate linear integro-differential equations with ordinary derivatives. The kernels of the integrands depend only on the variable of integration, and the conditions involve the terms with the point and integral values of the unknown function. We drive necessary and sufficient conditions for the existence and uniqueness of the solution of the problem, which can be used both for analytical and numerical solutions. We present the results of solving an illustrative test problem.

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Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

To the solution of integro-differential equations with nonlocal conditions

2021

Turk J Math

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“…Methods for solving Volterra IE and Fredholm IE exactly are given in [104], which describes the method for integral transforms, the method for differentiation and the direct method for integral equations in both the Urysohn form and the Hammerstein form. Motivated by this work and based on our previous experience with the closed-form solution of the Fredholm-type integro-differential equations [105][106][107][108], we consider the VFIE of the second kind in the form…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

Providas

2022

Algorithms

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In this paper, a direct operator method is presented for the exact closed-form solution of certain classes of linear and nonlinear integral Volterra–Fredholm equations of the second kind. The method is based on the existence of the inverse of the relevant linear Volterra operator. In the case of convolution kernels, the inverse is constructed using the Laplace transform method. For linear integral equations, results for the existence and uniqueness are given. The solution of nonlinear integral equations depends on the existence and type of solutions ofthe corresponding nonlinear algebraic system. A complete algorithm for symbolic computations in a computer algebra system is also provided. The method finds many applications in science and engineering.

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“…In this setting, problems are often too complex to be solved analytically and therefore numerical solutions are proposed [5,6]. Analytical solutions of a class of boundary value problems, involving the governing differential equation or slight generalizations of the one considered in (1), but with different and/or simpler differential operators, have been proposed in [7][8][9][10]. Existence theorems for nonlocal multipoint boundary value problem and IDEs via analytical methods are given in [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

An exact solution method for Fredholm integro-differential equations

Tsilika

2019

ICS

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Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

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Extension Operator Method for the Exact Solution of Integro-Differential Equations

Cited by 12 publications

References 26 publications

To the solution of integro-differential equations with nonlocal conditions

To the solution of integro-differential equations with nonlocal conditions

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

An exact solution method for Fredholm integro-differential equations

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