Mixed problem for the singular partial differential equation of parabolic type

Abstract

The scheme for solving of a mixed problem is proposed for a differential equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(c(x)\frac{\partial T}{\partial x}\right) -g(x)\, T\] 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 $$p_{1}T(0,\tau)+p_{2}T^{[1]}_x (0,\tau)= \psi_1(\tau), q_{1}T(l,\tau)+q_{2}T^{[1]}_x (l,\tau)= \psi_2(\tau), $$ $$T(x,0)=\varphi(x), $$ where $p_1 p_2\leq 0$, $q_1 q_2\geq 0$ and by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $c(x)\frac{\partial T}{\partial x}$. 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 $\big(c(x)X'(x)\big)' -g(x)X(x)+ \omega a(x)X(x)=0$ 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.