### Important equations

#### Heat equation

• Equation:

where

• u is the temperature
• c is the specific heat
• ρ is the mass density
• K0 is the thermal conductivity
• Q is a heat source

#### Wave equation

• Equation:

where

• u is the magnitude of the wave (such as the vertical displacement of a string)
• c is the group velocity of the wave

#### Laplace’s equation

• Equation:

• Mean value theorem: u at any point is the average of u along any circle of radius r0 (lying inside the system boundary) centered at that point
• Maximum principle: assuming no sources, in steady state u cannot attain extrema in the interior, unless u is constant everywhere. In steady state the extrema occur at the boundary.
• The solution of Laplace’s equation is unique.
• Solvability (or compatibility) condition: If on the boundary u is specified instead of u, Laplace’s equation may have no solutions.

### Overall methodology

• Principle of superposition: If u1 and u2 satisfy a linear homogeneous equation, then an arbitrary linear combination of them also satisfies the equation.
• Conditions to consider in an eigenvalue problem:
• λ > 0
• λ = 0
• λ < 0
• λ complex (if of SL type, this does not exist)
• Often IV problems are unique, so it doesn’t matter how one determines the solution.
• To generally include an initial condition, so the most general work first: first create a general solution that’s a linear combination of all solutions.
• It’s thus wise to expand the initial condition in the same basis as that of the general solution.
• The integral over any number of complete periods of the square of sine or cosine is one-half the length of the total interval.
• General solutions to homogeneous BVPs:
 Boundary conditions (2 each)→ ϕ(0) = ϕ(L) = 0 (0) = (L) = 0 Eigenvalues λn 2; n = 1,2,… 2; n = 0,1,2,… 2; n = 0,1,2,… Eigenfunctions sin cos sin and cos Series f(x) = ∑ n=1∞Bn sin f(x) = ∑ n=0∞An cos Coefficients Bn = ∫ 0Lf(x)sindx
• Linear homogeneous boundary conditions:
• First kind or Dirichlet condition: ϕ = 0
• Second kind or Neumann condition: = 0
• Third kind or Robin condition: = ±(+ left end; - →right end)…for this to be a physical condition, I believe h > 0
• Periodicity condition: example of mixed type:
• Singularity condition: <
• Sturm-Liouville eigenvalue problems:
• General SL ODE:

• Theorems:
• All the eigenvalues λ are real.
• There exist an infinite number of eigenvalues:

• There is a smallest eigenvalue, usually denoted λ1.
• There is not a largest eigenvalue and λn →∞ as n →∞.
• Corresponding to each eigenvalue λn, there is an eigenfunction, denoted ϕn(x) (which is unique to within an arbitrary multiplicative constant). ϕn(x) has exactly n - 1 zeros for a < x < b.
• The eigenfunctions ϕn(x) form a “complete” set, meaning that any piecewise smooth function f(x) can be represented by a generalized Fourier series of the eigenfunctions:

Furthermore, this infinite series converges to 2 for a < x < b (if the coefficients an are properly chosen).

• Eigenfunctions belonging to different eigenvalues are orthogonal relative to the weight function σ(x). In other words,

• Any eigenvalue can be related to its eigenfunction by the Rayleigh quotient:

where the BCs may somewhat simplify this expression.

• Lagrange’s identity: For any two functions u and v (not necessarily eigenfunctions), simply calculate the quantity
This is Lagrange’s identity. Integrating, we have Green’s formula:

• Theorem: If u and v are any two functions satisfying the same set of homogeneous BCs (of the regular SL type), the Green’s formula equals zero. Further, L, along with the corresponding BCs, is called self-adjoint.
• We can use the Rayleigh quotient to determine/eliminate the sign of possible eigenvalues.
• The minimum value of the RQ for all continuous functions satisfying the BCs (but not necessarily the diffeq) is the lowest eigenvalue, λ1:

• If the BCs are not homogeneous (but the PDE is):
• Transform to BCs that are…we can always do this, by either:
• Determining the equilibrium solution to the BVP, or
• Determining ANY solution that satisfies the BCs.
• The resulting solution to either of the above options is deemed the reference solution.
• Then solve by using the displacement from the reference solution.
• If the PDE is not homogeneous (but the BCs are [and are of self-adjoint type]), use the Fredholm alternative:
 # of solutions Orthogonality of forcing func w/homog soln (weight = 1): ∫ abf(x)ϕh(x)dx ϕh = 0 (λ≠0) 1 0 (orthogonal) ϕh≠0 (λ = 0) ∞ 0 (orthogonal) ϕh≠0 (λ = 0) 0 ≠0 (not orthogonal)
• To use the Fredholm alternative:

1. Solve the homogeneous problem, obtaining ϕh(x).
2. If ϕh(x) = 0 is the only homogeneous solution (alternatively, if λ = 0 is not an eigenvalue), you know there is a single unique solution to the non-homogeneous problem.
3. If ϕh(x)0 (alternatively, if λ = 0 is an eigenvalue), enact abf(x)ϕh(x)dx. If it’s zero, there is an infinite number of solutions; otherwise, there are no solutions.

### Particular methods

• Separation of variables
• When to use: PDE and BC’s are linear and homogeneous
• To choose the correct sign of the individual constant λ, mentally solve the time ODE to ensure the solution does not blow up, if that’s not physical.
• Method of eigenfunction expansion
• When to use: PDE is linear; BC’s are linear and homogeneous
• Assume the solution is a linear combination of the solutions to the related homogeneous problem, simply make the constants time-dependent.
• Plug and chug.
• Detailed method for the linear, general, nonhomogeneous problem:

1. Homogenize BC’s (this can always be done):

1. Do this by either determining the equilibrium solution (the solution to the time-independent problem) or just determining ANY solution that satisfies the BC’s. This equilibrium solution can depend on x and t, and it’s okay if it makes the PDE messier, since Q(x,t) is already messy itself!
2. Thus, the new problem becomes
2. Determine the eigenfunctions ϕn(x) and eigenvalues λn of the related homogeneous problem (probably using separation of variables)
3. Make the solution a generalized Fourier series of the eigenfunctions. A generalized Fourier series simply has time-dependent coefficients:

4. NOW satisfy the ICs:
5. Now the ICs are satisfied (via the an(0)) and the BCs are satisfied (via r(x,t)). The degrees of freedom that remain are the an(t) (note the time-dependence), and we exercise these by plugging and chugging:
It will probably by necessary to utilize the fact that the eigenfunctions satisfy

to simplify things. This leads to many equations (one for each eigenfunction coefficient an(t)) in t using Fourier’s trick, with one side equaling the generalized Fourier series of Q(x,t), which is just a general function of t, qn(t).

6. Solve this by regular ODE methods, such as using an integrating factor or the variation of parameters. Integrating from 0 to t will yield the full solution for an(t), and we’re done.
• Green’s functions for time-independent problems
• When to use: PDE and PC’s are linear
• Notes:
• A Green’s function exists for every BVP and is the same regardless of the IC’s.
• Fundamental interpretation of a Green’s function: it is the response at x due to a concentrated source at x0.
• If zero is an eigenvalue to the problem (the full problem, even if inhomogeneous), a Green’s function does not exist.
• Method:
• Determine the Green’s function:
• Set up the defining equation for the Green’s function:

• Use these facts to determine the complete Green’s function:
@ It always satisfies the related homogeneous BC’s
@ It’s continuous at the source point x0
@ To get the final condition integrate the defining equation above.
• Once the Green’s function for the BVP is determined:
• If BC’s are homogeneous, use

• If BC’s are inhomogeneous, use

where the BC’s were

• Fourier transform:
• When to use: Infinite domain is present or higher order derivatives are present
• Method:
• Take the FT of the PDE according to the following rules:
• The spatial FT of a time derivative equals the time derivative of the FT:
• The spatial FT of the nth spatial derivative equals (ik)n times the FT, assuming that f(x,t) 0 sufficiently fast as x →±∞:
• Solve the ODE
• Apply the IC’s, determining the initial FT
• Go back into position space by either:
• Using the convolution theorem: To take the FT of H(k) = F(k)G(k), use
• Simply taking the IFT if possible
• Method of characteristics
• When to use: quasi-linear (wave?) equations, have a first-order derivative in both space and time
• Theory is this: under a certain condition, say, along a characteristic curve relating the independent variables, the PDE can be fastly simplified (i.e., the desired variable has a much simpler ODE under this condition) (the “characteristic” is this special condition)
• Examples:
• For the PDE

if = c, the equation simplifies to = 0.

• For the PDE

if = c(f,x,t), the equation simplifies to = Q(f,x,t).