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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 \(\mathbb{C}^{r \times r}\) 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:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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