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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.

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