MATH 301 Notes
Max Christie
November 15, 2017
Things To Know
Z†Zq(θ) = −∂2
θ−cot θ∂θ−1
4cot2θ+1
2sin2θ+mcos θ
sin2θ−m2
sin2θq(θ) = k2q(θ) (1)
ZZ †p(θ) = −∂2
θ−cot θ∂θ−1
4cot2θ+1
2sin2θ−mcos θ
sin2θ−m2
sin2θq(θ) = k2p(θ) (2)
1. Fourier Series and Sin/Cos Series
Solve second order DE using the characteristic equation. Solve boundary value problem of the
form y00 +λy = 0 by finding the eigenvalues/vectors and turning the solution into a Fourier
series.
2. Wave Equation and d’Alembert Solution
How to derive the d’Alembert solution, and apply it to the wave equation.
3. Separation of variables
Will focus only on the heat equation for separation of variables.
Find total energy E(t)=1/2R(ρu2
t+T u2
x)dx and show that it is conserved with dE/dt = 0 or
E(t) = E(0), or that it is always increasing/decreasing with dE/dt ≥/≤0. This will require
some integration by parts. Show that the energy solution is unique using the assumption that
there exist two solutions u1and u2whose superposition w=u1+u2is also a solution.
4. Laplace Equation
Neumann boundary conditions.
Green’s Theorem (integration by parts in two dimensions)
Iδf ·g=I(∇f·n)g− ·∇f· ∇g
Can use this theorem to prove uniqueness of the solution to the Laplace equation.
1
