PH-308 Methods of Mathematical and Computational Physics      (Pre-req: PH-301)

1. Vector spaces, basis vectors, linear independence, function spaces.
2. Review of differentiation and integration, continuity and differentiability, functions of many variables.
3. First order differential equations, general solution by integration, uniqueness property.
4. Second order differential equations with constant coefficients, Euler linear equations, singular points, series solution by Frobenius' method, solution space, Wronskian, uniqueness.
5. Special functions: gamma and beta functions, Stirling's series, Legendre equation, Associated Legendre functions, Hermite equation and polynomials, Laguerre equation and associated polynomials, Bessel’s equation and solutions, spherical Bessel functions.
6. Use of Mathematica for items 2-5 above: students will be given practice questions to solve using a computer.
7. Second order linear partial differential equations, Laplace equation, wave equation, solution of Poisson equation
8. Definition of probability, simple properties, random variables, binomial distribution, Poisson and Gaussian distributions, central limit theorem, statistics.
9. Numerical methods: interpolation, root finding, numerical integration, matrix manipulation, numerical solutions of ODE’s, least square fit.

Recommended texts:

• Mathematical Methods for Physicists, by Arfken & Weber, publisher: Academic Press; 6th Edition, (2005).
• Mathematical Methods for Physicists, by Tai L. Chow, publisher: Cambridge University Press (2000).
• Numerical Analysis, by J. Douglas Faires and Richard L. Burden, publisher: Brooks, 8th Edition (2005).