SYLLABUS FORM

Course No: Math-342B Title: Methods of Applied Mathematics II

Course Prerequisite: Math-342A with a minimum of C.

COURSE DESCRIPTION

Second order ordinary differential equations, power series methods, Bessel functions, Legendre polynomials. Linear partial differential equations, separation of variables, Fourier series, Sturm-Liouville theory, orthogonal expansions, Fourier transforms. Use of computer software packages for symbolic algebra and solutions of differential equations.

Not open to students with credit in Physics 340A.

COURSE OUTLINE

1. Fourier Series (2 weeks)

· Harmonic motion, periodic functions

· Fourier series

· Average value of a function

· Fourier coefficients

· Dirichlet conditions

· Complex Fourier series

· Even and odd functions

· Parseval's theorem

2. Series solutions to differential
equations (3 weeks)

· Legendre's equation

· Orthogonal functions

· Legendre polynomials (generation, completness, orthogonality, normalization)

· Legendre series and functions

·

· Generalized power functions: Frobenius

· Bessel's equation (first & second solutions, tables, graphs, zeros, recursion,

· generation, orthogonality, other Bessel functions, approximations)

· Hermite functions, Laguerre functions, ladder operators

3. Partial Differential Equations (3 weeks)

· Laplace's equation

· Diffusion, heat equations

· Wave equation

· Vibrating string, membrane

· Steady-state temperature in a cylinder and sphere

· Poisson's equation

4. Functions of complex variable (2 weeks)

· Analytic functions

· Contour integrals

· Laurent series

· Residue theorem, definite integrals

· Points and residues at infinity

· Mappings, conformal mappings

5. Probability (2 weeks)

· Sample space

· Probability theorems

· Methods of counting

· Random variables

· Distributions (continuous, binomial, normal, Gaussian, Poisson)

6. Integral Transforms (2 weeks)

· Laplace transform, solutions to differential equations

· Fourier transforms, convolution

· Inverse Laplace transform

· Dirac Delta function

· Green functions

· Integral transform solutions to partial differential equations

7. Calculus of Variations (1 week) [if time permits]

· Euler equation

· Brachistochron problem, cycloids

· Several dependent variables, Lagrange's equation

· Isoperimetric problems

· Variational notation

Recommended Textbook: Mathematical Methods in the Physical Sciences. Mary L. Boas. Wiley, 1983.

Software recommended: Matlab and Maple