Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory

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Springer Science & Business Media, 2013. 3. 9. - 593ÆäÀÌÁö
The triumphant vindication of bold theories-are these not the pride and justification of our life's work? -Sherlock Holmes, The Valley of Fear Sir Arthur Conan Doyle The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that cannot be solved exactly. Our objective is to help young and also establiShed scientists and engineers to build the skills necessary to analyze equations that they encounter in their work. Our presentation is aimed at developing the insights and techniques that are most useful for attacking new problems. We do not emphasize special methods and tricks which work only for the classical transcendental functions; we do not dwell on equations whose exact solutions are known. The mathematical methods discussed in this book are known collectively as asymptotic and perturbative analysis. These are the most useful and powerful methods for finding approximate solutions to equations, but they are difficult to justify rigorously. Thus, we concentrate on the most fruitful aspect of applied analysis; namely, obtaining the answer. We stress care but not rigor. To explain our approach, we compare our goals with those of a freshman calculus course. A beginning calculus course is considered successful if the students have learned how to solve problems using calculus.
 

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2 InitialValue and BoundaryValue Problems
5
9 Differential Equations in the Complex Plane
29
Difference Equations
36
5 Nonlinear Difference Equations
53
PART II
60
2 Local Behavior Near Ordinary Points of Homogeneous Linear
66
4 Local Behavior at Irregular Singular Points of Homogeneous
76
6 Local Analysis of Inhomogeneous Linear Equations
103
Problems for Chapter 4
196
Approximate Solution of Difference Equations
205
Asymptotic Expansion of Integrals
247
5 Mathematical Structure of Perturbative Eigenvalue Problems
350
Summation of Series
368
2 Summation of Divergent Series
379
5 Convergence of Padé Approximants
400
PART III
417

8 Asymptotic Series
118
Problems for Chapter 3
136
Approximate Solution of Nonlinear Differential Equations
146
EF
157
Problems for Chapter 9
479
Perturbation Series
582
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