We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.
We give a definition of Green’s function of the general boundary value problems for non-self-adjoint second order differential equation with involution. The sufficient conditions for the basis property of system of eigenfunctions are established in the terms of the boundary conditions. Uniform equiconvergence of spectral expansions related to the second-order differential equations with involution:−y″(x)+αy″(−x)+qxyx=λyx,−1<x<1, with the boundary conditions y′−1+b1y−1=0,y′1+b2y1=0, is obtained. As a corollary, it is proved that the eigenfunctions of the perturbed boundary value problems form the basis in L2(−1,1) for any complex-valued coefficient q(x)∈L1(−1,1).
In this paper the question on unconditional basicity of the system of eigenfunctions of the involutive perturbed Sturm-Liouville operator is investigated. The Green's function of the operator under consideration in the case of constant coefficients is constructed. The estimates of the Green's functions are obtained. The existence of the Green's function is shown in the case when the operator under consideration has a variable coefficient. The theorem on the equiconvergence of expansions with respect to the eigenfunctions of the indicated operators is proved with the help of the Green's function. The basicity of the eigenfunctions of the operator under consideration in the class L2 (−1, 1) is proved. It is established that the basis from the eigenfunctions of the involutive perturbed Sturm-Liouville operator is the unconditional basis.
The paper is devoted to finding a solution and restoring the right-hand side of the heat equation with reflection of the argument in the second derivative, with a complex-valued variable coefficient. We prove a theorem on the Riesz basis property for eigenfunctions of the second-order differential operator with involution in the second derivative. We establish the existence and uniqueness of the solution of the studied problems by the method of separation of variables
В настоящей работе мы изучаем спектральную задачу для дифференциального оператора второго порядка с инволюцией и с краевыми условиями типа Дирихле. Построена функция Грина изучаемой краевой задачи. Получены равномерные оценки функций Грина рассматриваемых краевых задач. Установлена равносходимость разложений произвольной функции из класса L 1 (−1, 1) по собственным функциям двух дифференциальных операторов второго порядка с инволюцией с краевыми условиями типа Дирихле. Мы используем интегральный метод, основанный на функции Грина дифференциального оператора второго порядка с инволюцией и со спектральным параметром. Как следствие из доказанной теоремы о равносходимости разложений по собственным функциям, мы доказываем базисность в пространстве L 2 (−1, 1) собственных функций спектральной задачи с непрерывным комплекснозначным коэффициентом q(x). Ключевые слова: дифференциальное уравнение с инволюцией, функция Грина, разложения по собственным функциям, базис.
In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved.
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