TY - JOUR
AU - Makhnei, O.V.
PY - 2017/06/08
Y2 - 2024/05/23
TI - Boundary problem for the singular heat equation
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 9
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.9.1.86-91
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1450
SP - 86-91
AB - <p>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 thegeneralized 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 $$\left\{ \begin{array}{l}p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau),\end{array}\right.$$ 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 thereduction 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 oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundaryconditions 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 andexpansions in eigenfunctions of some boundary value problem forthe 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.</p>
ER -