diff --git a/src/differential_equations_2.md b/src/differential_equations_2.md index a3d67521eb1f13846a3022ba4d57d9287c74e4ff..9cfe99137036e793449b6724ea667b06918a29ae 100644 --- a/src/differential_equations_2.md +++ b/src/differential_equations_2.md @@ -491,6 +491,29 @@ coefficient, $$c_n:= \int^{L}_{0} dx sin(\frac{n \pi x}{L}) \psi_{0}(x). $$ - +## General recipie for seperable PDEs ## + +1. Make the separation ansatz to obtain separate ordinary differential + equations. +2. Choose which euation to treat as the eigenvalue equation. This will depend + upon the boundary conditions. Additionally, verify that the linear + differential operator $L$ in the eigenvalue equation is hermitian. +3.Solve the eigenvalue equation. Substitute the eigenvalues into the other + equations and solve those too. +4. Use the orthonormal basis functions to write down the solution corresponding + to the specified initial and boundary conditions. + +One natural question is what if the operator $L$ from setp 2 is not hermitian? +It is possible to try and make it hermitian by working on a Hilbert space +equipped with a different inner product. This means one can consider +modifications to the definition of $\langle \cdot, \cdot \rangle$ such that $L$ +is hermitian with respect to the modified inner product. This type of technique +falls under the umbrella of *Sturm-Liouville Theory*, which forms the foundation +for a lot of the analysis that can be done analytically on PDEs. + +Another question is of course what if the equation is not separable? One +possible approach is to try working in a different coordinate system. There are +a few more analytic techniques available, however in many situations it becomes +necessary to work with numerical methods of solution. \ No newline at end of file