*(English)*Zbl 0831.65102

For the mixed problems described by the equation ${u}_{t}(x,t)-D\left(t\right){u}_{xx}(x,t)=0$, $0<x<p$, $t>0$ in the bounded domain ${\Omega}$ $({t}_{0},{t}_{1})=[0,p]\times [{t}_{0},{t}_{1}]$, subject to boundary conditions $u(x,0)=F\left(x\right)$ and initial conditions $u(0,t)=u(p,t)=0$ the authors construct continuous numerical solutions with prefixed accuracy. They assume that $u(x,t)$ and $F\left(x\right)$ are $r$-component vectors and $D\left(t\right)$ is a ${\u2102}^{r\times r}$ valued two-times continuously differentiable function, so that $D\left({t}_{1}\right)D\left({t}_{2}\right)=D\left({t}_{2}\right)D\left({t}_{1}\right)$ for ${t}_{2}\ge {t}_{1}>0$ and there exists a positive number $\delta $ such that every eigenvalue $z$ of the form $(D\left(t\right)+{D}^{H}\left(t\right))/2$ with $t>0$ is bigger than $\delta $.

Such coupled partial differential equations appear in many different problems, for example in magnetohydrodynamic flows, in the study of temperature distribution within a composite heat conductor, mechanics, diffusion problems, nerve conduction problems, biochemistry, armament models etc.