## HMTH330 Perturbation Theory

#### Prerequisites :HMTH232

48 lectures

Aim:

Many of the problems facing physicists, engineers, and applied mathematicians involve difficulties such as nonlinear governing equations, nonlinear boundary conditions at complex known or unknown boundaries, and variable coefficients that preclude exact solutions. Consequently, solutions are approximated using numerical techniques, analytic techniques, and combinations of both. One can use modern computers to solve the equations. However, if one needs to obtain some insight into the character of the solutions and their dependence on certain parameters, one may need to repeat the calculations for many different values of the parameters and initial conditions. Often one or more of the parameters become either very large or very small. Typically these are difficult situations to treat by straight- forward numerical procedures, so analytic methods can often provide an accurate approximation, and foremost among these methods are the systematic methods of perturbation (asymptotic expansions) in terms of a small or large parameter or coordinate.

Course Outline:

 Expansions, gauge functions, order symbols. Asymptotic sequence and asymptotic series. Uniqueness and other properties of asymptotic series. Solution of algebraic equations. 4 Asymptotic solution of the system of the first order ODE and n-th order ODE with variable coefficients. Sturm-Liouville problem and transverse vibrations of beams. Two-dimensional problems and high-frequency vibrations of rectangular plates. 12 Degeneration of boundary value problems. Vibrations of not absolutely flexible string. Regular degeneration in the case of constant coefficients. Three iterative processes. Examples. 8 Evaluation of integrals containing large (small) parameter. Integration by parts. Laplace integrals. Theorems. Laplace' methods. The method of stationary phase. Improper integrals. Saddle point (steepest descent) method. Airy's integral. 12 Classification of ordinary and singular points of ODE. General forms of the power series solutions. Different types of solutions in the neighbourhood of regular singular points. Euler type equations. Frobenius-type solutions. Convergence. Singularity at infinity. Irregular singular points. Normal and subnormal solutions. 12

Recommended Texts:

• A H Naifeh, Introduction to Perturbation Techniques (Wiley, 1981).
• W Wasow, Asymptotic Expansions for ODE (Wiley, 1965).
• E T Copson, Asymptotic Expansions (Cambridge, 1965).