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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 Ω (t 0 ,t 1 )=[0,p]×[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 r×r valued two-times continuously differentiable function, so that D(t 1 )D(t 2 )=D(t 2 )D(t 1 ) for t 2 t 1 >0 and there exists a positive number δ such that every eigenvalue z of the form (D(t)+D H (t))/2 with t>0 is bigger than δ.

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