Aim:

This course aims at providing a set of powerful analytical tools for the solution of engineering problems. These methods are often necessary to obtain solutions to problems that are inaccessible to numerical computation, because of, e.g., large separation of time and length scales, or presence of singularities.

Pre-requisite:

Basic course in calculus and exposure to linear second-order ordinary differential equations

Course Contents

Definitions of asymptoticness; Asymptotic evaluation of integrals; Approximate solutions of algebraic equations; Eigenvalue problems; Regular perturbation of ODEs; Singular perturbation of ODEs: Poincare-Lindstedt, Boundary layer theory, WKB theory, Multiple scales method; Singular perturbation of PDEs; Engineering applications.

Topics with suggested lectures in parenthesis

I. Introduction to asymptotic approximations: Definitions; Convergence; Asymptoticness; Parametric expansions. (2)

II. Asymptotic analysis of integrals: Elementary examples; Integration by parts; Laplace’s method; Watson’s lemma; Method of stationary phase; Method of steepest descent. (8)

III. Solutions to algebraic equations: Regular and singular perturbations; Eigenvalue problems. (4)

IV. Regular perturbation problems in ODEs and PDEs: Initial value problems; Boundary perturbations. (3

V. Introduction to singular perturbation of ODEs. (1)

VI. Poincare-Lindstedt method. (2)

VII. Boundary layer theory. (5)

VIII. WKB Theory. (3)

IX. Multiple-scale analysis. (3)

X. Introduction to singular perturbation of PDEs (4)

XI. Engineering applications: At least one example each from fluid mechanics, solid mechanics, andvibrations. (6)

Textbooks, alternate sources and further readings:

1. Bender, C. M. and S. O. Orszag, 1999. Advanced Mathematical Methods for Scientists and Engineers, Springer-Verlag: New York, USA.
2. Hinch, E. J., 1991. Perturbation Methods, Cambridge U. Press: Cambridge, U.K.
3. Murdock, J. A., 1987. Perturbations: Theory and Methods, SIAM.
4. Van Dyke, M., 1975. Perturbation Methods in Fluid Mechanics. Parabolic Press.
5. Kevorkian, J., and J. D. Cole, 1981. Perturbation Methods in Applied Mathematics. Springer
6. Holmes, M. H. 2013. Introduction to Perturbation Methods. Springer.

Prepared by

• Ishan Sharma
• P. Wahi
• A. Gupta
• S. L. Das