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A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation

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Abstract

A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.

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Funding

This research is supported by the Ministry of Education and Science of the Republic of Kazakhstan, grant no. AP 05132486 and the Award ‘‘Best University Teacher 2019.’’

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Correspondence to D. S. Dzhumabaev, K. Zh. Nazarova or R. E. Uteshova.

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(Submitted by E. K. Lipachev)

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Dzhumabaev, D.S., Nazarova, K.Z. & Uteshova, R.E. A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation. Lobachevskii J Math 41, 1791–1800 (2020). https://doi.org/10.1134/S1995080220090103

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