faheem.

Differential Equations

Separation, Green's functions, Sturm-Liouville theory. Mostly about what the boundary conditions are quietly doing.

  1. 01Separation of VariablesAssuming the solution factorises works far more often than it has any right to. The coordinate system is why.
  2. 02Green's FunctionsSolve the problem for a point source, then build every other source out of it by superposition.
  3. 03Sturm–Liouville TheoryThe structure shared by every well-posed boundary value problem. Its orthogonal basis is not an accident.
  4. 04Fourier Series and the TransformThe eigenfunction expansion Sturm-Liouville promised. What becomes of it when the interval opens up to infinity.
  5. 05Series Solutions and Special FunctionsFrobenius at a regular singular point. Legendre and Bessel functions turn up here, and not by coincidence.