MA4323 Principles and Techniques of Applied Mathematics II

Continuation of MA4322. Selected topics include: calculus of variations, Hamiltonian Mechanics, distribution theory and Green's Functions in two and three dimensions, and discrete models.

Prerequisite

MA4322

Lecture Hours

4

Lab Hours

0

Course Learning Outcomes

A student who successfully completes this course will be able to:

• Classify an integral equation as either a Fredholm or Volterra integral equation of the first or second kind and know how they relate to corresponding initial or boundary value problems.
• Define a Hilbert-Schmidt kernel, be able to find the adjoint of an integral equation and know how to use the Fredholm Alternative Theorem in the solution of integral equations.
• Define a separable kernel and use its properties to solve integral equations and approximate the solution of integral equations with Hilbert-Schmidt kernels.
• Find the eigenvalues and corresponding eigenfunctions of elementary integral eigenvalue problems that have symmetric Hilbert-Schmidt kernels.
• Obtain Neumann series solutions of Fredholm or Volterra integral equations of the second kind given kernels of the Hilbert-Schmidt type.
• Define a functional, know the requirements for continuity of a functional, and properties of a normed linear space in which admissible functions for extrema of functionals may reside.
• Define the difference between strong and weak extrema of a functional.
• Understand the necessary condition for a functional to have an extrema and be able to solve Euler’s equation to find such extrema.
• Incorporate fixed, natural, or constrained boundary conditions or to allow for piecewise smooth extrema by modification of a functional.
• Define the necessary condition (Legendre Condition) and the sufficient condition for an extrema to be the minimal of a functional.
• Use Lagrange multipliers to incorporate subsidiary conditions on a functional.
• Recognize the Lagrangian of a system, Hamilton’s principle, and corresponding Hamilton’s equations for a conservative dynamical system.
• Apply the Rayleigh-Ritz method in minimizing a functional.
• recognize “big O” and “little o” notation in asymptotics.
• Use Laplace’s method, the method of stationary phase, and the saddle point method in asymptotically evaluating integrals resulting from Laplace or Fourier transform techniques.
• Define the difference between regular and singular perturbation expansions.
• recognize the concepts of matched asymptotic expansions, boundary layer theory, and multiple scale analysis.