Boundary value problems:
A boundary value problem (BVP) is a differential equation together with specific
conditions, which are not initial conditions, that the solution must satisfy.
Regular Sturm-Liouville problems:
Solve:
d
dx
[ r ( x ) y ' ] + ( q ( x ) + λp ( x ) ) y=0
Subject to:
A1 y ( a ) +B 1 y ' ( a )=0 A2 y ( b ) +B 2 y ' ( b )=0
For regular Sturm-Liouville problems:
- There exist an infinite number of real eigenvalues that can be arranged in
increasing ordered, λ 1< λ2 < λ3 <… , such than λ n → ∞ as n → ∞.
- For each eigenvalue λ n, there exists only one corresponding eigenfunction y n
(up to nonzero multiples of the same function>).
- Eigenfunctions corresponding to different eigenvalues are linearly
independent.
- The set of eigenfunctions corresponding to the set of eigenvalues is
orthogonal with respect to the weight function p(x ) on [a , b], i.e., if n ≠ m, then:
b
∫ p ( x ) y n ( x ) y m ( x ) dx=0
a
Linear second order PDE:
2 2 2
∂u ∂ u ∂ u ∂u ∂u
A 2
+B +C 2
+D +E + Fu=G
∂x ∂x ∂ y ∂y ∂x ∂y
Boundary conditions:
∂u
On a boundary, we can specify the values of u, of , the derivative of u in the
∂n
∂u
direction perpendicular to the boundary, or + hu, where h is a constant.
∂n
These conditions are called the Dirichlet condition, Neumann condition, and Robin
condition, respectively.
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