This mathematical physics course aims to be an introduction to differential equations. Besides standard topics in ordinary and partial differential equations, nonlinear dynamical systems will be studied and nonlinear ODEs will be analysed using geometric and computational tools.
Ordinary differential equations (first order):
Overview and importance. Rate of change problems, modeling simple systems, radioactive decay, population growth, logistic curve. First order linear equations, Integrating factors. linear ODEs with constant coefficients. Phase plane, coupled systems of linear first order ODEs.
Nonlinear ODEs and Dynamical systems: phase portraits, flows and vector fields, phase plane analysis, linearisation about fixed points, stability analysis, limit cycles and bifurcations. Brief discussion of chaos.
Second order ODEs: Power series and Frobenius series solutions, some special functions (Bessel, Legendre and Hermite).
Fourier Series, orthogonal functions and Fourier integrals with physical applications.
Partial differential equations:
Linear first and second order PDEs, separation of variables, illustration through Laplace’s equation/ wave equation/ diffusion equation.
Pre-requisites: Mathematical Physics I (Mathematical and Computational Toolkit).
Books: Differential Equations with Applications and Historical Notes (G F Simmons),
Other books useful for reference and supplementary reading are:
Differential Equations and Boundary Value Problems (Edwards and Penney)
Differential equations, dynamical systems and an introduction to chaos (Smale, Hirsch and Devaney)