Berikbol T. Torebek

Representation of Green's function of the Neumann problem for a multi-dimensional ball

Representation of the Green’s function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function, we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that the Green’s function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel at and .

Representation of the Green’s function of the exterior Neumann problem for the Laplace operator

We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.

On an explicit form of the Green function of the third boundary value problem for the Poisson equation in a circle

The paper is devoted to the investigation of questions about constructing the explicit form of the Green’s function of the Robin problem. For 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 problem is presented in elementary functions.

On some integro-differential operators in the class of harmonic functions and their applications

We study properties of integro-differential operators generalizing the operators of the Riemann-Liouville and Caputo fractional differentiation in the class of harmonic functions. The properties obtained are applied to examine some local and nonlocal boundary value problems for the Laplace equation in the unit ball.

Modified Bavrin Operators and Their Applications

In the class of harmonic functions, we study the properties of fractional integro-differential operators. By way of application of these properties, we analyze the solvability of some boundary value problems for the Laplace equation in the ball and derive solvability conditions. The smoothness of the solutions in the Hölder class is studied as well.

On a new class of nonlocal problems for the Laplace operator

In this work we consider a new nonlocal boundary value problem in a disk. A Newton potential is a particular case of the problem. We establish conditions of its Noetherian property, Fredholm property and correctness. We prove self-adjointness of the problem. We construct all the eigenvalues and eigenfunctions of the problem for a correct case.

On an explicit form of the Green function of the Robin problem for the Laplace operator in a circle

Abstract The paper is devoted to investigation questions about constructing the explicit form of the Greens 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.

On an analog of Samarskii-Ionkin type boundary value problem for the Poisson equation in the disk

In this paper, we consider a Samarskii-Ionkin type boundary value problem for the Poisson equation in the disk and prove its well-posedness. The possibility of separation of variables is justified. We construct an explicit form of the Green function for this problem and obtain an integral representation of the solution.

Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana-Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana-Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.

A nonlocal fractional Helmholtz equation

In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involu- tion perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered problems are proved by spec- tral method.