Ordinary differential equations.
When available, explicit solutions are fundamental to the understanding of differential equations, but for many problems of interest, such closed-form exact solutions can not be found. Even when they do exist, often the understanding and interpretation of explicit solutions requires computational and/or further approximation. The study of asymptotic and perturbation methods offers a systematic analysis of solutions to differential equations when tractable solutions are unavailable. The goal of perturbation methods is to replace an exact problem that does not have a tractable solution, by a series of approximate problems that do have (simpler) explicit solutions. We will cover a range of topics including asymptotic expansions for algebraic and differential equations and multiple scales for initial and boundary value problems. The methods all rely on there being a parameter in the problem that is relatively small. Such a situation is relatively common in applications and this is one of the reasons that perturbation methods are a cornerstone of applied mathematics.
After the course you will be able to study differential equations with a small parameter
using the methods of matching, multiple time scales and WBKJ method.
with multiple solutions.
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Mode of Instruction
The examination consists of homework, a written final exam, and an oral retake exam. No minimum grade is required for
the homework in order to take the exam or to pass the course. The homework counts as a practical and there is no retake
for it. The final grade based on the final exam and the retake consists of homework (40%) and the final (or retake) (60%); to pass the course,
the grade for the final exam (and the retake) should be at least 5 and the (unrounded) weighted average of the two partial grades at least 5.5.
Introduction to Perturbation Methods by M.H. Holmes
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