On integro-differential equations with the highest-order derivative in the integral term
DOI:
https://doi.org/10.31489/2026m2/48-64Keywords:
integro-differential equations, highest-order derivative in the integral term, two-point condition, continuous coefficients, Dzhumabaev parametrization method, parametrized problem, functional term, existence and uniqueness, explicit solutionAbstract
In this article, a two-point boundary value problem for an integro-differential equation in which the highestorder derivatives appear in the integral term is considered. The Dzhumabaev parametrization method is applied to solve the problem. The original problem is reduced to an equivalent problem for an integrodifferential equation with parameters. The resulting problem includes an integro-differential equation with parameters, an initial condition, and an additional relation. Conditions for the existence and uniqueness of a solution to the integro-differential equation with parameters are established in terms of the coefficients and kernels of the equation, as well as the boundary functions. An explicit representation of the solution in terms of the parameters is constructed. The unique solvability of the original two-point boundary value problem is established in terms of the initial data. A special case of the integro-differential equation with the highest-order derivative appearing in the integral term, subject to two-point boundary conditions, is also investigated. The Dzhumabaev parametrization method is used to solve the problem. An explicit form of the solution is obtained.
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