On the solvability of a mixed problem for partial differential equations of parabolic type with involution

Mixed partial differential equation of parabolic type with involution is considered. The sufficient conditions of the existence and uniqueness of the solution for the parabolic type equations with involution is obtained. The Fourier method is used to find a classical solution of the mixed problem for the transformed parabolic type equation with involution.


1.
Introduction Many works [1]- [6] are dedicated to the study of partial differential equations (PDE) of parabolic type. Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instrument. This paper is devoted to investigating the problems of existence and uniqueness of the solution for the parabolic type equations with involution. We quote here only some latest papers on the problems with involution. The fractional analogue of Helmholtz equation with an involution perturbation in a rectangular domain is considered in [7]. The spectral problem for the second-order differential operators with involution and boundary conditions of Dirichlet type is studied in [8], where the Green's function related to the boundary problem is constructed and uniform estimates of the latter are obtained. The equi-convergence of eigenfunction expansions of two second-order differential operators with involution and boundary conditions of Dirichlet type for any function in 2 (−1,1) is established. The solvability of the main boundary value problems for the nonlocal Poisson equation is studied in [9], where necessary and sufficient conditions for existence and uniqueness for the considered problems are obtained. In this paper, we study boundary value problems for heat equation with involution and obtained the necessary and sufficient condition for existence and uniqueness of a solution of the considered problems by using the spectral method. Let   . , We deal with the following problem: find a function and initial condition and boundary conditions

2.
On the eigenfunctions and eigenvalues of a spectral problem with involution In this section we solve a spectral problem (1)-(4) by using separation (Fourier's) method. If the solution of the problem (1)-(4) is represented as  For the proof we refer to [10]. We proceed to the study of the problem (7), (8). , w x y is eigenfunction, and  corresponding eigenvalue, then a function Proof. Let

( )
, w x y and  is an eigenfunction and an eigenvalue of the problem (7), (8). By putting in xy, we obtain the system of equations which can be written in the matrix form as follows hj == The following result generates the method of construction of the eigenfunctions and eigenvalues of the problem (7), (8).
is eigenfunction of the problem (7), (8) (7). It is not difficult to show that this function also satisfies the boundary condition (8), which completes the proof of the theorem.
Here we give one useful property of the eigenfunctions of the Laplace operator, which can be obtained by the using the ideas of the paper [11]. To prove the theorem, we use the statement of Theorem 2. First, we note that the system (17) can be rewritten in the following matrix form , where It is easy to show that the inverse of the matrix to the matrix H has a form 1 1 2 Then the eigenfunctions and the corresponding eigenvalues of the problem (7), (8) form a complete orthonormal system in 2 ( ). LG

3.
On the existence and uniqueness of the regular solution Let consider a linear operator      Conclusion The necessary and sufficient conditions for regular solution to exist and to be unique are obtained by using the properties of the orthonormal system eigenfunction of the mixed heat problem with involution in all arguments. The properties of the second order partial differential operators with involution are studied and applied to solve mixed heat equation with involution.