Applied mathematics for physicists

• Determine whether an operator and an equation is linear or nonlinear and understand the difference between the two types.
• Solve linear partial differential equations with Fourier transformation and understand the relation between a function and its Fourier spectrum.
• Solve linear ordinary differential equations with Green function.
• Derive the properties of Sturm-Liouville operators and Sturm-Liouville eigenvalue problems.
• Solve linear partial differential equations using separation of variables.
• Solve singular linear ordinary differential equations using Frobenius’ method.
• Know Bessel’s equations and be able to use Bessel functions.
• Derive the Euler-Lagrange equations and understand the concept of calculus of variations and Hamilton’s Principle for multi-dimensional systems with several variables.
• Apply the Rayleigh-Ritz variational technique to find ground state eigenfunctions and eigenvalues.
• Apply the collective coordinate variational approach to find and analyse solutions to nonlinear partial differential equations.

Fourier transformation and general integral transformations. Sturm-Liouville problems and special functions. Bessel functions. Green function. Calculus of variations.

Group work involves four extended projects, applying the introduced methods to the treatment of various physical problems, including Bose-Einstein condensates, optical fibers, electromagnetic wave propagation, nonlinear crystals, and laser beams.

01001/01003/01002/01004/01034/01035/01037/10036

13 morning sessions (8-12) with lectures and group work.

Textbook:
Arfken & Weber: Mathematical Methods for Physicists, 7th ed., Elsevier.