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Abstract initial boundary value problems

Published online by Cambridge University Press:  14 November 2011

C. Palencia  and
I. Alonso Mallo
C. Palencia
Affiliation:
Departamento de Matematica Aplicada y Computatión, Universidad de Valladolid, Vallodolid, Spain email:palencia@cpd.uva.es
I. Alonso Mallo
Affiliation:
Departamento de Matematica Aplicada y Computatión, Universidad de Valladolid, Vallodolid, Spain email:palencia@cpd.uva.es
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Extract

We consider abstract initial boundary value problems in a spirit similar to that of the classical theory of linear semigroups. We assume that the solution u at time t is given by u(t) = S(t) ξ + V(t)g, where ξ and g are respectively the initial and boundary data and S(t) and V(t) are linear operators. We take as a departing point the functional equations satisfied by the propagators S and V. We discuss conditions under which a pair (S, V) describes the solution of an abstract differential initial boundary value problem. Several examples are provided of parabolic and hyperbolic problems that can be accommodated within the abstract theory. We study the backward Euler's method for the time integration of the problems considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 5 , 1994 , pp. 879 - 908
Copyright
Copyright © Royal Society of Edinburgh 1994

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