Topic 1

Asymptotic Analysis

Week 1

Order; Asymptotic expansion

Watson's Lemma

Week 2

Method of steepest descents, stationary phase

Examples

Topic 2

Global Perturbation Methods

Week 3

Regular/singular perturbation; Poincaré-Lindstedt method

Multiple-scale analysis

Week 4

Matched asymptotic expansions

Boundary layer theory

Week 5

WKB theory

Examples

Topic 3

Calculus of Variations

Week 6

Functionals; necessary conditions for extrema; variational derivative

Euler-Lagrange equation; examples

Week 7

Higher derivatives; boundary conditions; constraints

Hamilton's Principle; canonical formulation

Examples

Topic 4

Integral Equations

Week 9

Classification; origin; relation to differential equations

Convolution kernels; integral transforms

Fredholm theory

Week 10

Degenerate kernels; Neumann and Fredholm series

Hilbert-Schmidt method

Examples

Extended Reference List

General Mathematical Methods including Course Topics

Arfken, G.B. and H.J. Weber, Mathematical Methods for Physicists, 6th Edition, Elsevier Academic Press (2005)

Courant, R. and D. Hilbert, Methods of Mathematical Physics, Wiley-Interscience, New York (1953)

Greenberg, M.D., Foundations of Applied Mathematics, Prentice-Hall Inc., Englewood Cliffs, N.J. (1978)

Logan, J.D., Applied Mathematics, 3rd Edition, Wiley-Interscience (2006)

Riley, K.F., M.P. Hobson and S.J. Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Cambridge University Press (2006)

Asymptotic Methods

Awrejcewicz, J. and V.A. Krysko, Introduction to Asypmtotic Methods, Chapman & Hall/CRC (2006)

Bender, C.M. and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Carl M. Bender, Steven A. Orszag, Springer Verlag (1999)

Bush, A.W. Perturbation Methods for Engineers and Scientists, CRC Press, Boca Raton (1992)

Hinch, E.J., Perturbation Methods, Cambridge University Press, New York (1991)

Holmes, M.H., Introduction to Perturbation Methods, Springer (1995)

Kevorkian, J. and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer (1985)

Miller, P.D., Applied Asymptotic Analysis, American Mathematical Society V. 75 (2006)

Murdock, J.A., Perturbations: Theory and Methods, SIAM (1999)

Murray, J.D., Asymptoptic Analysis, Springer-Verlag Inc., New York (1984)

Nayfeh, A.H., Introduction to Perturbation Methods, Wiley Interscience, New York (1981)

Kevorkian, J. and J.D. Cole, Perturbation in Applied Mathematics, Springer-Verlag, New York (1981)

VanDyke, M., Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA (1975)

Calculus of Variations

Brechtken-Manderschield, U., Introduction to the Calculus of Variations, Chapman and Hall, New York, (1991)

Hildebrand, F.B., Methods of Applied Mathematics, 2nd Edition, Dover (1992)

Troutman, J.L., Variational Calculus and Optimal Control, Springer (1996)

Weinstock, R., Calculus of Variations, Dover Publications, New York (1974)

Integral Equations

Hilldebrand, F.B., Methods of Applied Mathematics (Second Edition), Dover (1992)

Hochstadt, H. Integral Equations, John Wiley and Sons, Inc., New York (1983)

Jerri, A.J., Introduction to Integral Equations with Applications, 2nd ed., Wiley-Interscience (1999)

Pipkin, A.C., A Course on Integral Equations, Springer-Verlag, (1991)

Porter, D., and D.S.G. Sterling, Integral Equations, Cambridge University Press (1990)

Tricomi, F.G., Integral Equations, Dover (1983)

Wazwaz, A.-M., A First Course in Integral Equations, World Scientific (1997)

Handbooks

Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Dover Publications, New York, (1972)

Erdelyi, A., W. Magnus, F. Oberhettinger, and F.G. Tricomi, Tables of Integral Transforms, Vol I and II, McGraw Hill (1954)

Gradshteyn, I.S., and Ryzhik, I.W., Tables of Integrals, Series, and Products, 7th ed., Academic Press, Inc. (2007)