One method to prove of existence weak solution of a mixed problem for 2D parabolic equations

In this study using the residue method the solution of the first type mixed problem for 2D linear parabolic equation in the bounded cylinder of the Euclidean space R3(x,y,t) is obtained in explicit form. When the smoothness of initial data does not permit to construct of the classical solution then it is necessary to extend the concept of classical solution. Based on the examined problem is proved that the obtained solution is a weak solution too. The use of this proof method to prove the existence of a weak solution can be applied to prove the existence of weak solutions for the more general problem with non-self-adjoint and even that contains higher-order derivative with respect to t in the boundary condition. © 2020 The Author(s

The boundary value problem for an ordinary linear half-order differential equation

This study is devoted to the study of the solution of a boundary value problem for an ordinary linear differential equation of half order with constant coefficients. Using of the fundamental solution of the main part of the considered equation, we obtained the principal relations, from which we obtain the necessary conditions for the Fredholm property of the original problem. Further, using the Mittag-Leffler function, a general solution of the homogeneous equation is obtained. Finally, the problem under consideration is reduced to an integral Fredholm equation of the second kind with a non-singular kernel, i.e., the Fredholm property of the stated problem is proved

The first basic boundary value problem of Riemann's type for bianalytical functions in a plane with slots

The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions. In the course of work there was made out a constructive method for solution of the problem given in a plane with slots. There was also found out that the solution of the problem under consideration consists of consequent solutions of two Riemann's boundary value problems for analytical functions in a plane with slots. Besides, a picture of solvability of the problem is being searched and its Noether is identified. Apie pirmojo pagrindinio kraštinio Rimano tipo uždavinio bianalizinėms funkcijoms plokštumoje su įtrūkiai sprendimą Santrauka Šiame darbe tyrinejamas uždavinys, kai ieškoma dalimis bianaliziniu funkciju, nykstančiu begalybeje, apribotu greta kontūro trūkio tašku ir šiame kontūre tenkinančiu dvi kraštines salygas. Parodoma, kad sprendžiamas uždavinys suvedamas i sprendima dvieju Rimano uždaviniu analizinems funkcijoms. First Published Online: 14 Oct 201

Numerical solution of Cauchy problem for second order nonlinear wave equation with changeable type in a class of discontinues functions

In this paper, a special numerical method for the solution of second order nonlinear wave equation with changeable type in a class of discontinues functions, which accurately describe the physical properties of the problem of interest is suggested. For this, first, an auxiliary problem having some advantages over the main problem is introduced. Since, the differentiable property of the solution of the auxiliary problem is twice higher than the differentiability of the solution of the main problem, the application of classical methods to the auxiliary problem can be easily performed. Some economic algorithms are proposed for obtaining a numerical solution of the auxiliary problem, and by using the solution of the auxiliary problem, the numerical solution of the main problem is also found. In order to show the effectiveness of the suggested algorithms some comparisons between the exact solution and the numerical solution are made. (C) 2002 Elsevier Inc. All rights reserved

A finite differences method for a two-dimensional nonlinear hyperbolic equation in a class of discontinuous functions

A numerical scheme is proposed for a scalar two-dimensional nonlinear first-order wave equation with both continuous and piecewise continuous initial conditions. It is typical of such problems to assume formal solutions with discontinuities at unknown locations, which justifies the search for a scheme that does not rely on the regularity of the solution. To this end, an auxiliary problem which is equivalent to, but has more advantages then, the original system is formulated and shown that regularity of the solution of the auxiliary problem is higher than that of the original system. An efficient numerical algorithm based on the auxiliary problem is derived. Furthermore, some results of numerical experiments of physical interest are presented. (C) 2002 Elsevier Science Inc. All rights reserved

Numerical method to solution of generalized model Buckley-Leverett in a class of discontinuous functions

A new numerical method is proposed for solving the generalized Buckley-Leverett problem, which describes the movement of two-phase mixtures of Bazhenov bed sediments in a class of discontinuous functions. To this end, we introduce an auxiliary problem that has advantages over the main problem, and using these advantages, an original finite difference method to solve of the auxiliary problem is developed. Using the suggested auxiliary problem, a solution which expresses exactly all physical characteristics of the problem is obtained