Perturbation Techniques

Code: MA753 | L-T-P-C: 4-0-0-8

MA753 Perturbation Techniques L-T-P-C [4-0-0-8]

Asymptotic expansion and approximation, asymptotic solution of algebraic and transcendental equations, regular and singular perturbations for first and second-order ordinary differential equations, physical examples, initial-value problems, multiple scales, two-scale asymptotic approximation, averaging technique, composite asymptotic expansions, initial layers - matching by Van Dyke rules. Two-point boundary-value problems: Boundary layers -exponential and cusp layers, matched asymptotic expansions, composite asymptotic expansions, WKB (Wentzel, Kramers, Brillouin) expansion method, conditions for validity of the WKB approximation, patched asymptotic approximations, WKB solution of inhomogeneous ordinary differential equations. Perturbation methods for linear eigenvalue problems, Rayleigh-Schrodinger theory, singularity structure of eigenvalues as functions of complex perturbing parameter, level crossing. Nonlinear eigenvalue problems, direct error estimation, oscillatory phenomena - free conservative and free self-sustained oscillations, harmonic resonance, shock and transition layers.

Texts/References:

1. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, New York, 1999.
2. W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979.
3. J. Kevorkian, J.D. Cole, Perturbation Methods in Applied Mathematics, Springer- Verlag, New York, 1981.
4. P.A. Lagerstrom, Matched Asymptotic Expansions, Springer-Verlag, New York, 1988.
5. J. A. Murdock, Perturbations -Theory and Methods, SIAM -Classics in Applied Mathematics, Vol. 27, SIAM, Philadelphia, 1999.
6. A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.
7. R.E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.