(a) Differential Equations: Fourier series. Partial differential equations (PDEs): diffusion equation, wave equation, Laplace's equation. Solution by separation of variables in Cartesian and polar coordinates. Ordinary differential equations (ODEs): solution by reduction of order and variation of parameters. Series solution and the method of Frobenius. Legendre's and Bessel's equations: Legendre polynomials, Bessel functions and their recurrence relations. (b) Algorithms: An introduction to a variety of numerical methods. Solution of linear algebraic equations. Solution of nonlinear equations. Numerical integration. Numerical Solution of ODEs.