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Continuous numerical solutions and error bounds for time dependent systems of partial differential equations: Mixed problems. (English) Zbl 0831.65102

For the mixed problems described by the equation ${u}_{t}\left(x,t\right)-D\left(t\right){u}_{xx}\left(x,t\right)=0$, $0, $t>0$ in the bounded domain ${\Omega }$ $\left({t}_{0},{t}_{1}\right)=\left[0,p\right]×\left[{t}_{0},{t}_{1}\right]$, subject to boundary conditions $u\left(x,0\right)=F\left(x\right)$ and initial conditions $u\left(0,t\right)=u\left(p,t\right)=0$ the authors construct continuous numerical solutions with prefixed accuracy. They assume that $u\left(x,t\right)$ and $F\left(x\right)$ are $r$-component vectors and $D\left(t\right)$ is a ${ℂ}^{r×r}$ 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 $\left(D\left(t\right)+{D}^{H}\left(t\right)\right)/2$ 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.

##### MSC:
 65M70 Spectral, collocation and related methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 35K15 Second order parabolic equations, initial value problems