A matrix approach to the analytic-numerical solution of mixed partial differential systems.

*(English)*Zbl 0839.65105The paper deals with systems of linear partial differential equations of the form

$${u}_{t}(x,t)-A{u}_{xx}(x,t)-Bu(x,t)=G(x,t),\phantom{\rule{1.em}{0ex}}0<x<p,\phantom{\rule{4pt}{0ex}}t>0,\phantom{\rule{2.em}{0ex}}\left(1\right)$$

where $u=({u}_{1},\cdots ,{u}_{m})$ is an unknown vector-function and $A$, $B$ are constant $m\times m$ complex matrices. It is supposed that every eigenvalue of $\frac{1}{2}(A+{A}^{H})$ is positive (${A}^{H}$ denotes the conjugate transpose of $A$). Assuming initial and boundary conditions (2) $u(x,0)=F\left(x\right)$, $u(0,t)=u(p,t)=0$, 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.

Reviewer: H.Marcinkowska (Wrocław)

##### MSC:

65M70 | Spectral, collocation and related methods (IVP of PDE) |

35K15 | Second order parabolic equations, initial value problems |