Mathematical Methods in Physics

Syllabus

• Review of solution methods of ODE’s
  • Separable equations, Integrating factors, Homogeneous equations
  • Clairot, Ricatti & Bernoulli equations
  • Qualitative difference between linear and nonlinear ODE’s
  • The Wronskian
  • Boundary value problems
  • Green’s function
  • Eigenvalue problems (Sturm-Liouville problem)
• Approximation methods
  • Approximation of integrals - Steepest descent
  • Approximation for solving boundary problems - WKB
  • Solving linear equations with small nonlinearities
  • (Poincare-Lindstedt, Multiple time scales)
• Review of solution methods for linear PDE’s
  • Diffusion equation: Separation of variables, Green’s function
  • Self-similar solutions and Green’s function
• Basic notions on continuous groups
  • Pauli matrices, SU(2), SU(3)
  • Rotation group
  • Simple examples of discrete groups
• Fundamentals of integral equations
• Basic notions on fractals



Sources:

1. Wilf, Mathematics for the Physical Sciences (Dover)
2. Kahn, Mathematical Methods for Scientists and Engineers (Wiley)
3. Byron & Fuller, Mathematical Methods of Classical & Quantum Physics (Dover)
4. Mathews & Walker, Mathematical Methods (Addison-Wesley)
5. Bender & Orszag, Advanced Mathematical Methods for Scientists & Engineers (McGraw-Hill)
6. Arfken & Weber, Mathematical Methods for Physicists (Academic Press)
7. Collins- Mathematical Methods for Physicists and engineers (Dover)
8. Ince - Ordinary Differential Equations (Dover)
9. King & Billingham - Differential Equations (Cambridge)



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