From ab82077acffdb9ee13717b24e81e132f4ae74134 Mon Sep 17 00:00:00 2001
From: Scarlett Gauthier <s.s.gauthier@student.tudelft.nl>
Date: Wed, 19 Aug 2020 14:31:08 +0000
Subject: [PATCH] Add last page of lecture notes.
---
src/differential_equations_2.md | 25 ++++++++++++++++++++++++-
1 file changed, 24 insertions(+), 1 deletion(-)
diff --git a/src/differential_equations_2.md b/src/differential_equations_2.md
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--- 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.
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--
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