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This book presents an introduction to singular-perturbation
problems, problems which depend on a parameter in such a way that
solutions behave non-uniformly as the parameter tends toward some
limiting value of interest. The author considers and solves a
variety of problems, mostly for ordinary differential equations. He
constructs (approximate) solutions for oscillation problems, using
the methods of averaging and of multiple scales. For problems of
the nonoscillatory type, where solutions exhibit 'fast dynamics' in
a thin initial layer, he derives solutions using the
O'Malley/Hoppensteadt method and the method of matched expansions.
He obtains solutions for boundary-value problems, where solutions
exhibit rapid variation in thin layers, using a multivariable
method. After a suitable approximate solution is constructed, the
author linearizes the problem about the proposed approximate
solution, and, emphasizing the use of the Banach/Picard fixed-point
theorem, presents a study of the linearization. This book will be
useful to students at the graduate and senior undergraduate levels
studying perturbation theory for differential equations, and to
pure and applied mathematicians, engineers, and scientists who use
differential equations in the modelling of natural phenomena.
This book presents an introduction to singular-perturbation
problems, problems which depend on a parameter in such a way that
solutions behave non-uniformly as the parameter tends toward some
limiting value of interest. The author considers and solves a
variety of problems, mostly for ordinary differential equations. He
constructs (approximate) solutions for oscillation problems, using
the methods of averaging and of multiple scales. For problems of
the nonoscillatory type, where solutions exhibit 'fast dynamics' in
a thin initial layer, he derives solutions using the
O'Malley/Hoppensteadt method and the method of matched expansions.
He obtains solutions for boundary-value problems, where solutions
exhibit rapid variation in thin layers, using a multivariable
method. After a suitable approximate solution is constructed, the
author linearizes the problem about the proposed approximate
solution, and, emphasizing the use of the Banach/Picard fixed-point
theorem, presents a study of the linearization. This book will be
useful to students at the graduate and senior undergraduate levels
studying perturbation theory for differential equations, and to
pure and applied mathematicians, engineers, and scientists who use
differential equations in the modelling of natural phenomena.
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