A matrix approach to the analytic-numerical solution of mixed partial differential systems
The paper deals with systems of linear partial differential equations of the form \[ u_t(x, t)- Au_{xx}(x, t)- Bu(x, t)= G(x, t),\quad 0< x< p,\;t> 0,\tag{1} \] where \(u= (u_1,\dots, u_m)\) is an unknown vector-function and \(A\), \(B\) are constant \(m\times m\) complex matrices. It is supposed that every eigenvalue of \({1\over 2} (A+ A^H)\) is positive (\(A^H\) denotes the conjugate transpose of \(A\)). Assuming initial and boundary conditions (2) \(u(x, 0)= F(x)\), \(u(0, t)= u(p, t)= 0\), the authors seek the solution of (1), (2) in the form of a Fourier series with respect to \(x\), whose coefficients depend on \(t\). An approximate solution is obtained by truncating this series and by suitable approximation of its coefficients. The estimate of the error is proved.
scientific article; zbMATH DE number 825386
scientific article