Mixed problem for the singular partial differential equation of parabolic type
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
Abstract
The scheme for solving of a mixed problem is proposed for a differential equation a(x)∂T∂τ=∂∂x(c(x)∂T∂x)−g(x)T with coefficients a(x), g(x) that are the generalized derivatives of functions of bounded variation, c(x)>0, c−1(x) is a bounded and measurable function. The boundary and initial conditions have the form p1T(0,τ)+p2T[1]x(0,τ)=ψ1(τ),q1T(l,τ)+q2T[1]x(l,τ)=ψ2(τ), T(x,0)=φ(x), where p1p2≤0, q1q2≥0 and by T[1]x(x,τ) we denote the quasiderivative c(x)∂T∂x. A solution of this problem seek by the reduction method in the form of sum of two functions T(x,τ)=u(x,τ)+v(x,τ). This method allows to reduce solving of proposed problem to solving of two problems: a quasistationary boundary problem with initial and boundary conditions for the search of the function u(x,τ) and a mixed problem with zero boundary conditions for some inhomogeneous equation with an unknown function v(x,τ). The first of these problems is solved through the introduction of the quasiderivative. Fourier method and expansions in eigenfunctions of some boundary value problem for the second-order quasidifferential equation (c(x)X′(x))′−g(x)X(x)+ωa(x)X(x)=0 are used for solving of the second problem. The function v(x,τ) is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.
Authors and Affiliations
O. Makhnei
Keywords
Related Articles
- EP ID EP532827
- DOI 10.15330/cmp.10.1.165-171
- Views 77
- Downloads 0
k-bitransitive and compound operators on Banach spaces
In this this paper, we introduce new classes of operators in complex Banach spaces, which we call k-bitransitive operators and compound operators to study the direct sum of diskcyclic operators. We create a set of suffic...
Permanence of a discrete predator-prey system with nonmonotonic functional responses and endless delay
A discrete-time analogue of predator-prey model with nonmonotonic functional responses and endless delay is considered in the paper. We investigate the question of obtaining conditions of permanent behavior of the dynami...
The vertex Zagreb indices of some graph operations
Recently, Tavakoli et al. introduced a new version of Zagreb indices, named as vertex Zagreb indices. In this paper explicit expressions of different graphs operations of vertex Zagreb indices are presented and also as a...
The nonlocal problem for the 2n differential equations with unbounded operator coefficients and the involution
We study a problem with periodic boundary conditions for a 2n-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces...
On convergence (2,1,...,1)-periodic branched continued fraction of the special form
(2,1,...,1)-periodic branched continued fraction of the special form is defined. Conditions of convergence are established for 2-periodic continued fraction and (2,1,...,1)-periodic branched continued fraction of the spe...