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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 $u_t(x,t) - D(t) u_{xx} (x,t) = 0$, $0 < x < p$, $t > 0$ in the bounded domain $\Omega$ $(t_0, t_1) = [0, p] \times [t_0, t_1]$, subject to boundary conditions $u(x,0) = F(x)$ 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 $F(x)$ are $r$-component vectors and $D(t)$ is a $\bbfC^{r \times r}$ valued two-times continuously differentiable function, so that $D(t_1) D(t_2) = D(t_2) D(t_1)$ for $t_2 \geq t_1 > 0$ and there exists a positive number $\delta$ such that every eigenvalue $z$ of the form $(D(t) + D^H(t))/2$ 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.

65M70Spectral, collocation and related methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
35K15Second order parabolic equations, initial value problems
Full Text: DOI
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