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Perturbation techniques with the emphasis on developing asymptotic solutions: approximation of integrals Laplace's method, stationary phase method, method of steepest descent , regular perturbations of ODEs, singular perturbations to ODEs matched asymptotic expansions, WKB approximation , multiple scale methods.
For most equations that arise in modeling applications it is unlikely that exact solutions can be found. Even convergent series approximations are often not available, or they are of limited use if they converge very slowly. Instead, asymptotic expansions can yield good approximations. They are typically divergent if summed to infinity but a few terms can often give excellent and well defined approximations.
This course will introduce the basic ideas and show how they can be applied to algebraic and differential equations, and to the evaluation of integrals. Usually some parameter or some coordinate value is small or large , which leads to an expansion of a solution in this parameter.
These perturbation expansions can be well behaved regular if the perturbation parameter goes to zero, or they can become singular. Most emphasis is placed on the latter, singular perturbations. Practical problems are used as illustrations. These techniques are especially useful when accurate numerical solutions are hard, or impossible, to obtain.
In other words, ordinary differential equations and complex variables. You will have a choice of proposing your own project or taking one of the projects offered by the instructor. Projects should be chosen by Wed, Feb 28th. Projects are due on Wed, April 25th. You should consult the instructor regarding your choice of own project as soon as possible and not later than Wed, Feb 21st.
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All printed materials disseminated in class or on the web are protected by Copyright laws. One photocopy or download from the web is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited. Email: berko math. Orszag, Advanced mathematical methods for scientists and engineers , McGraw-Hill , reprinted by Springer. Hinch, Perturbation methods , Cambridge University Press, Definitions and terminology.
Local approximations to linear ODEs; irregular singular points. Approximation of integrals; Laplace's method, stationary phase method; method of steepest descents. Regular perturbations of ODEs, eigenvalue problems. Dimensionless variables and scaling as aids to approximation; balances between terms. Singular perturbations that lead to boundary layers; matched asymptotic expansions. Singular perturbations that lead to highly oscillatory functions; WKB approximation.
The method of multiple scales for finding uniformly valid perturbation expansions. Singular perturbations of partial differential equations.
Asymptotic and Perturbation Methods