On Exact Series Solution for Strongly Coupled Mixed Parabolic Boundary Value Problems

Abstract

This paper continues with the construction of the exact solution for parabolic coupled systems of the type $u_t=A{u}_{xx}$, $A_{1}u(0,t)+B_1{u}_x(0,t)=0$, $A_{2}u(l,t)+B_2{u}_x(l,t)=0$, , $t>0$, and $u(x,0)=f(x)$, where $A_{\mathrm{1}}$, $A_{\mathrm{2}}$, $B_{\mathrm{1}}$, and $B_{\mathrm{2}}$ are arbitrary matrices for which the block matrix is nonsingular, and $A$ is a positive stable matrix. Although this problem has been solved in the literature (Soler et al., 2013), in this work we are using completely new conditions.