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Beschreibung |
KursbeschreibungThis course is an introduction to what is usually known as asymptotic analysis and perturbation analysis. For problems that naturally contain a large or small parameter, these methods aim to give approximate analytic solutions (as opposed to numerical) in terms of the parameter. The approximation we obtain from such methods often are simple but insightful.
Asymptotic analysis is a standard tool in applied mathematics. Often times student can quickly learn how to substitute a formal power series, such as $x_0+\epsilon x_1+\epsilon^2 x_2+\dots$, and find coefficients by simple manipulations. However, the analytic properties of these series are not immediately known: are they convergent, asymptotic, both or neither? One goal of this course is not only to introduce to the students these tools to do calculations, but also how to correctly interpret them.
We will focus on the theory of exponential integrals and approximating solutions of differential equations. We introduce different methods to study the asymptotic expansions of integrals with exponentials, mainly the Laplace’s method, the method of steepest descent and the stationary phase method. Then we justify how we can use expansion of simple functions as solution of differential equations, as regular or singular perturbations, the latter as the WKB method. As applications, we can see the theories used in weakly viscous shock waves, asymptotic behavior of special functions (e.g. Airy function), long time behavior of nonlinear waves, semi-classical limit of Bohr-Sommerfeld quantization and boundary layer problems.
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