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 80
- Downloads 0
Geometry of hypersurfaces of a quarter symmetric non metric connection in a quasi-Sasakian manifold
The purpose of the paper is to study the notion of CR-submanifold and the existence of some structures on a hypersurface of a quarter symmetric non metric connection in a quasi-Sasakian manifold. We study the existence o...
Representation of spectra of algebras of block-symmetric analytic functions of bounded type
The paper contains a description of a symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\ell_1$-sum of the space $\mathbb{C}^2$. We show that the specrum of such algebra does...
Paley-Wiener-type theorem for polynomial ultradifferentiable functions
The image of the space of ultradifferentiable functions with compact supports under Fourier-Laplace transformation is described. An analogue of Paley-Wiener theorem for polynomial ultradifferentiable functions is proved.
Wick calculus on spaces of regular generalized functions of Levy white noise analysis
Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based...
On the growth of a composition of entire functions
Let $\gamma$ be a positive continuous on $[0,\,+\infty)$ function increasing to $+\infty$ and $f$ and $g$ be arbitrary entire functions of positive lower order and finite order. In order to $$ \lim\limits_{r\to+\infty}...