Boundary problem for the singular heat equation
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 1
Abstract
The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation $$ a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(\lambda(x)\frac{\partial T}{\partial x}\right) $$ with coefficient $a(x)$ that is the generalized derivative of a function of bounded variation, $\lambda(x)>0$, $\lambda^{-1}(x)$ is a bounded and measurable function. The boundary conditions have the form $$ \left\{\arraycolsep=0pt \begin{array}{l} p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\ p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau), \end{array}\right. $$ where by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $\lambda(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by the reduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. 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 $u(x,\tau)$ and a mixed problem with zero boundary conditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. 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 $(\lambda(x)X'(x))'+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ 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 EP325082
- DOI 10.15330/cmp.9.1.86-91
- Views 82
- Downloads 0
Green-Rvachev's quasi-function method for constructing two-sided approximations to positive solution of nonlinear boundary value problems
A homogeneous Dirichlet problem for a semilinear elliptic equations with the Laplace operator and Helmholtz operator is investigated. To construct the two-sided approximations to a positive solution of this boundary valu...
Some classes of dispersible dcsl-graphs
A distance compatible set labeling dcsl of a connected graph G is an injective set assignment $f : V(G) \rightarrow 2^{X},$ $X$ being a non empty ground set, such that the corresponding induced function $f^{\oplus} :E(G)...
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...
New approach to derivation of quantum kinetic equations with initial correlations
We propose a new approach to the derivation of kinetic equations from dynamics of large particle quantum systems, involving correlations of particle states at initial time. The developed approach is based on the descript...
Wiener weighted algebra of functions of infinitely many variables
In this article we consider a weighted Wiener type Banach algebra of infinitely many variables. The main result is a description of the spectrum of this algebra.