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In this paper, we consider an inverse source problem with nonlocal boundary conditions for the heat equation involving multi-term time-fractional derivatives. We determine a source term independent of the space variable, and the temperature distribution from the energy measurement. We reduce the solution of the inverse problem to finding solutions to two problems. The well-posedness of each problem is shown using the generalized Fourier method. [ABSTRACT FROM AUTHOR]

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function p x , which belongs to the space L 2 0 , 1 . The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when p x ≡ 0 ) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions p x , for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L 2 0 , 1 , is dense within the space L 2 0 , 1 . Furthermore, it has been demonstrated that the adjoint operator shares a similar structure. [ABSTRACT FROM AUTHOR]

For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. It is proven that the difference scheme with weights is stable and its solution converges to the exact solution of the differential problem in the grid L 2 h -norm. Stability conditions are established. An estimate of the numerical solution with respect to the initial data and the right-hand side of the difference problem is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlocal boundary value problems for the Laplace operator in a ball, which are a multidimensional generalisation of the Samarskii–Ionkin problem. The well-posedness of the problems are investigated, and Fredholm property of the problems are studied. Moreover, we obtain integral representations of their solutions in explicit forms. [ABSTRACT FROM AUTHOR]

In the paper we consider a stationary diffusion problem described by the Poisson equation. The problem is considered in a model domain, chosen as a half disk. Classical Dirichlet boundary conditions are set on the arc of the circle. New nonlocal boundary conditions are set on the bottom base. The first condition means the equality of flows through opposite radii, and the second condition is the proportionality of distribution densities on these radii with a variable coefficient of proportionality. Uniqueness and existence of the classical solution to the problem are proved. An inverse problem for the solution to the Poisson equation and its right-hand part depending only on an angular variable are considered. As an additional condition we use the boundary overdetermination. Inverse problems to the Dirichlet and Neumann problems, and to problems with nonlocal conditions of the equality of flows through the opposite radii are considered. The well-posedness of the formulated inverse problems is proved. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument. [ABSTRACT FROM AUTHOR]

The spectral problem -u"(x) + αn"(-x) = λu(x), -1 < x < 1, with nonlocal boundary conditions u(-1) = βu(1), u'(-1) = u'(1), is studied in the spaces Lp(-1,1) for any α ≠ (-1,1) and β = ±1. It is proved that if r = √(1-α)/(1+α) is irrational then the system of its eigenfunctions is complete and minimal in Lp(-1,1) for any p > 1, but does not form a basis. In the case of a rational value of r, the way of supplying this system with associated functions is specified to make all the root functions a basis in Lp(-1,1). [ABSTRACT FROM AUTHOR]

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables. [ABSTRACT FROM AUTHOR]

In this article we study spectral properties of non-local boundaryvalue problem for an equation of parabolic-hyperbolic type. The non-local condition binds the solution values at points on boundaries of the parabolic and hyperbolic parts of the domain with each other. Nonlocal boundary conditions of such type are called Frankl-type conditions. This problem was first formulated by Kal'menov and Sadybekov who proved the unique strong solvability. In this article we investigate one particular case of this problem, for which we show that the problem does not have eigenvalues. [ABSTRACT FROM AUTHOR]

In this article we discuss a method for constructing trace formulae for the heat-volume potential of the time-fractional diffusion equation to lateral surfaces of cylindrical domains and use these conditions to construct as well as to study a nonlocal initial boundary value problem for the time-fractional diffusion equation. [ABSTRACT FROM AUTHOR]

In this article we propose a new formulation of boundary-value problem for a one-dimensional wave equation in a rectangular domain in which boundary conditions are given on the whole boundary. We prove the well-posedness of boundary-value problem in the classical and generalized senses. To substantiate the well-posedness of this problem it is necessary to have an effective representation of the general solution of the problem. In this direction we obtain a convenient representation of the general solution for the wave equation in a rectangular domain based on d'Alembert classical formula. The constructed general solution automatically satisfies the boundary conditions by a spatial variable. Further, by setting different boundary conditions according to temporary variable, we get some functional or functional-differential equations. Thus, the proof of the well-posedness of the formulated problem is reduced to question of the existence and uniqueness of solutions of the corresponding functional equations. [ABSTRACT FROM AUTHOR]

A local boundary value problem for an inhomogeneous biharmonic equation in a rectangular domain is considered. Boundary conditions are given on the whole boundary of the domain. It is shown that the problem turns out to be self-adjoint. And herewith the problem is ill-posed. An example is constructed demonstrating that the solution stability to the problem is violated. Necessary and sufficient conditions of the existence of a strong solution to the investigated problem are found. The idea of the method is that the solution to the problem is constructed in the form of an expansion on eigenfunctions of this self-adjoint problem. This problem has an isolated point of continuous spectrum in zero. It is shown that there exists a series (a sequence) of eigenvalues converging to zero. Asymptotics of these eigenvalues is found. Namely this asymptotics defines a reason for the ill-posedness of the investigated problem. A space ofwell-posedness for the investigated problem is constructed. [ABSTRACT FROM AUTHOR]

The paper is devoted to investigation questions about constructing the explicit form of the Green's function of the Robin problem in the unit ball of ℝ2. In constructing this function we use the representation of the fundamental solution of the Laplace equation in the form of a series. An integral representation of the Green function is obtained and for some values of the parameters the Green function is given in terms of elementary functions. [ABSTRACT FROM AUTHOR]