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Preliminaries:

The student is expected to be familiar with the major topics (at the advanced undergraduate, beginning graduate level) in linear algebra, advanced calculus, introductory partial differential equations, and introductory complex variables.

Exam topics may include (but are not limited to):

Dimensional Analysis and Scaling:

Buckingham Pi Theorem, characteristic scales, well-scaled problems. Perturbation Methods and Asymptotic Expansions: asymptotic sequences and series, regular perturbations, Poincare'-Lindstedt method, singular perturbations, boundary layer analysis, WKB approximations, asymptotic expansions of integrals.

Calculus of Variations:

first and second variations, Euler-Lagrange equations, first integrals, isoperimetric problems.

Integral Equations and Green's Functions:

Volterra and Fredholm integral equations, degenerate kernels, Green's functions, Fredholm Alternative.

Partial Differential Equations:

well-posed problems, maximum principles, energy argument (Lyapunov functions), orthogonal expansions, Fourier Transforms, heat kernel.

Suggested Courses:

• MATH 41021 / 51021: Theory of Matrices
• MATH 42041 / 52041: Advanced Calculus
• MATH 42045 / 52045: Introduction to Partial Differential Equations
• MATH 42048 / 52048: Introduction to Complex Variables (for preliminary material)
• MATH 62041 / 72041: Methods of Applied Mathematics I (for core material)
• MATH 62042 / 72042: Methods of Applied Mathematics II (for core material)

Suggested References:

• James P. Keener, Principles of Applied Mathematics: Transformation and Approximation, 2nd ed., Westview Press, 2000
• C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM Classics, 1988
• J. David Logan, Applied Mathematics, 4th ed., Wiley, 2013