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A hyperbolic singular perturbation of Burgers' equation
Author(s) -
Esham Benjamin F.
Publication year - 1990
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670120106
Subject(s) - mathematics , burgers' equation , uniqueness , mathematical analysis , singular perturbation , contraction mapping , contraction principle , contraction (grammar) , energy method , hyperbolic partial differential equation , partial differential equation , fixed point theorem , medicine
This paper is concerned with the effect of perturbing Burgers' equation by a small term ϵ 2 U tt . It is shown by means of an energy estimate that the solution of Burgers' equation provides a uniform O (ϵ) approximation of the solution of the full hyperbolic problem. Existence and uniqueness of classical solutions for both problems is proved. A related linear problem is first addressed using the Faedo–Galerkin method to obtain key estimates. Important for the hyperbolic problem is the introduction of an ϵ‐dependent energy in order to track the order‐ϵ behaviour of various higher‐order derivatives. Subsequent use of Schauder technique and Banach contraction mapping principle yields solutions of the semilinear problems.