On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

В данной статье изучается нелокальная задача для уравнения четвертого порядка в которой доказывается существование и единственность решения этой задачи. Решение построено явно в виде ряда Фурье, обоснованы абсолютная и равномерная сходимость полученного ряда и возможность почленного дифференцирования решения по всем переменным. Установлен критерий однозначной разрешимости поставленной краевой задачи. In this article, we study a nonlocal problem for a fourth-order equation in which the existence and uniqueness of a solution to this problem is proved. The solution is constructed explicitly in the form of a Fourier series; the absolute and uniform convergence of the obtained series and the possibility of term-by-term differentiation of the solution with respect to all variables are substantiated. A criterion for the unique solvability of the stated boundary value problem is established.

On the Nonlocal Problems in Time for Time-fractional Subdiffusion Equations

The nonlocal boundary value problem, dt&rho;u(t)+Au(t)=f(t) (0&amp;lt;&rho;&amp;lt;1, 0&amp;lt;t&le;T), u(&xi;)=&alpha;u(0)+&phi; (&alpha; is a constant and 0&amp;lt;&xi;&le;T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative; naturally, in the case of the Riemann - Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant &alpha; on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function &phi; in the boundary conditions are investigated.

On a Nonlocal Boundary Value Problem of a State-Dependent Differential Equation

In this paper, the existence of absolutely continuous solutions and some properties will be studied for a nonlocal boundary value problem of a state-dependent differential equation. The infinite-point boundary condition and the Riemann–Stieltjes integral condition will also be considered. Some examples will be provided to illustrate our results.

NONLOCAL BOUNDARY VALUE PROBLEM FOR ONE MIXED THIRD ORDER EQUATION

The paper considers a nonlocal boundary value problem for a mixed hyperbolic-parabolic equation of the third order. The equation is considered in a finite simply connected domain consisting of a hyperbolic and a parabolic part. The solution to the problem posed is considered for various cases of the parameter λ, which is in the original equation. In the case when (1-2m)/2&lt; &lt;λ&lt;1, the solution of the problem is reduced to a singular integral equation, which is reduced by the well-known Carleman-Vekua method to the Fredholm integral equation of the third kind. In the case when λ=(1-2m)/2, a theorem on the existence and uniqueness of a solution to the problem posed is formulated and proved. To prove the uniqueness of the solution, the method of energy integrals is used and inequalities of the type are derived on the given functions that are in the boundary condition. It is shown that the homogeneous problem corresponding to the original problem, under the conditions of the uniqueness theorem, has only a trivial solution in the entire considered domain. From which we can conclude that the original problem has only a single solution. If the obtained conditions for the given functions are violated, the problem posed does not have a unique solution. When investigating the question of the existence of a solution to the problem posed, a system of two equations is considered, consisting of the basic functional relations between the trace of the desired function and the traces of the derivative of the desired function, brought to the line of degeneration y = 0. Eliminating from the system the function τ (x) - the trace of the desired solution on the line of degeneration, we arrive at an equation for the trace of the derivative of the desired function. Under the condition of the existence and uniqueness theorem, the problem posed is equivalently reduced to the Fredholm integral equation of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problem posed.

On a semi-nonlocal boundary value problem for the three-dimensional Tricomi equation of an unbounded prismatic domain

В данной статье изучаются методами «ε-регуляризации» и априорных оценок с применением преобразования Фурье однозначная разрешимость и гладкость обобщенного решения одной полунелокальной краевой задачи для трехмерного уравнения Трикоми в неограниченной призматической области. In this article, the methods of «ε-regularization» and a priori estimates using the Fourier transform are studied the unique solvability and smoothness of the generalized solution of one semi-nonlocal boundary value problem for the three-dimensional Tricomi equation in an unbounded prismatic domain.

ASYMPTOTIC DISTRIBUTION OF EIGENVALUES AND EIGENFUNCTIONS OF A NONLOCAL BOUNDARY VALUE PROBLEM

In this work, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the second order boundary-value problem with a Bitsadze–Samarskii type nonlocal boundary condition.

A boundary value problem for a partial differential equation with fractional derivative

In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved.