A few important PDEs
constant coefficients = 1
variable coefficients = c(x)
∇ ⋅ ( c∇u ) = f
∇ 2u = f
Poisson’s equation:
example: f = charge density,
u = –electric potential
example: f = heat source/sink rate
u = steady-state temperature
example: f = solute source/sink rate,
u = steady-state concentration
example: f ~ force on stretched string/drum
u = steady-state displacement
c = thermal conductivity
c = diffusion coefficient
c ~ “springy-ness”
∇ ⋅ ( c∇u ) = 0
∇ 2u = 0
Laplace’s equation:
c = permittivity ε
examples: as for Poisson, but no sources
∂u
= ∇ 2u
∂t
Heat/diffusion equation:
∂u
= ∇ ⋅ (c∇u)
∂t
examples: u = temperature
u = solute concentration
c = thermal conductivity
c = diffusion coefficient
∂ 2u
= ∇ ⋅ (c∇u)
∂t 2
∂ 2u
= ∇ 2u
∂t 2
Scalar wave equation:
examples: u = displacement of stretched string/drum
u = density of gas/fluid
c2 = 1 / wave speed
+ many, many others…
Maxwell (electromagnetism)
Schrödinger (quantum mechanics)
Navier–Stokes / Stokes / Euler (fluids)
Black-Scholes (options pricing)
Lamé–Navier (linear elastic solids)
beam equation (bending thin solid strips)
advection-diffusion (diffusion in flows)
reaction-diffusion (diffusion+chemistry)
minimal-surface equation (soap films)
nonlinear wave equation (e.g. solitary ocean waves)
unknowns:
18.06
18.303
finite-dimensional linear algebra
linear algebra w/ functions & derivatives
Ω
vector space of

column vectors x (or x ) in n (or  n),
or possibly x(t) [time-dependent]
vector space of real-valued (or complex)
functions u(x) [for x in some domain Ω],
or possibly u(x,t) [time-dependent],
…
possibly restricted by some boundary conditions
at the boundary ∂Ω [e.g. u(x) = 0 on ∂Ω]
…
possibly with vector-valued u(x) [vector fields]
vector space:
we can add, subtract, &
multiply by constants
without leaving the space
linear operators:
∂Ω
matrices A
linear operators on functions Â,
[ Âu = function ]
using partial derivatives. examples:
Â1u = ∇2u [ Laplacian operator ]
Â2u = 3u [ mult. by constant ]
Â3u |x = a(x) u(x) [ mult. by function ]
 = 4Â1 + Â2 + 7Â3 [ linear comb. of ops. ]
linearity:
A(αx+βy) = αAx + βAy
Â(αu+βv) = αÂu + βÂv
dot product
and transpose:
complex x:
x ⋅ y = xTy = Σi xiyi
xT → xT = x*
T
T
x ⋅ Ay = x Ay = (Ax) y
⇔ (A)Tij = Aji [swap rows/cols]
basis:
set of vectors bi with span = whole space
⇔ any x = Σi ci bi for some coefficients ci
… if orthonormal basis, then ci = biTx
⎛ ∂ ⎞T
⎜ ⎟ = ???
⎝ ∂x ⎠
[ e.g.
Fourier series! ]
u(x) ⋅ v(x) = 〈u,v〉 = ???????? [inner product]
[= some integral]
〈u,Âv〉 = 〈Â*u,v〉
⇒ Â* = ???????? (= † in physics) [adjoint]
∞ set of functions bi(x) with span = whole space
⇔ any u(x) = Σi ci bi(x) for some coefficients ci
… if orthonormal basis, then ci = 〈bi, u〉
linear equations:
solve Ax = b for x
solve Âu = f for u(x)
existence
& uniqueness:
Ax = b solvable if b in column space of A.
Solution unique if null space of A = {0},
or equivalently if eigenvalues of A are ≠ 0.
Âu = f solvable if f(x) in col. space (image) of Â.
Solution unique if null space (kernel) of  = {0},
or equivalently if eigenvalues of  are ≠ 0.
eigenvalues/vectors:
solve Ax = λx for x and λ.
For this x, A acts just like a number (λ).
[e.g. Anx = λnx, eAx = eλx.]
solve Âu = λu for u(x) [eigenfunction] and λ.
For this u, Â acts just like a number (λ).
example:
[e.g. Ânu = λnu, eÂu = eλu.] ∂ 2
2
∂x 2
sin(kx) = (−k ) sin(kx)
time-evolution
initial-value
problem:
solve dx/dt = Ax for x(0)=b [system of ODEs]
⇒ x = eAt b [ if A constant ]
… expand b in eigenvectors, mult. each by eλt
solve ∂u/∂t = Âu for u(x,0)=f(x)
⇒ u(x,t) = eÂt f(x) [ if  constant ]
… expand f in eigenfunctions, mult. each by eλt
symmetric:
A = AT
⇒ real λ, orthogonal eigenvectors, diagonalizable
 = ÂT
[??????]
⇒ real λ, orthogonal eigenvectors (???)
diagonalizable (???)
positive definite
/ semi-definite:
A = AT, xTAx > 0 for any x ≠ 0 / xTAx ≥ 0
⇔ real λ>0/≥0, A=BTB for some B
 = Â*, 〈u,Âu〉>0 / ≥0 for u ≠ 0 (???)
ˆ ˆ
ˆ
⇔ real λ>0/≥0, Â=B*B for some B (???)
important fact: –∇2 is symmetric positive definite or semi-definite!
inverses:
orthogonal /
unitary:
A-1 A = A A-1 = 1 [if it exists]
⇒ Ax=b solved by x = A-1b
⎛ ∂ ⎞−1
⎜ ⎟ = ???
⎝ ∂x ⎠
… some kind of integral?
A-1 = AT ⇔ (Ax) ⋅ (Ax) = x ⋅ x for any x
⇒ |λ|=1, orthogonal eigenvectors, diagonalizable
Â-1 = ??????
⇒ Âu = f solved by f = –1u ???
[…delta functions
& Green’s functions]
Â-1 = Â* ⇔ 〈Âu,Âu〉 = 〈u,u〉 for any u
⇒ |λ|=1, orthogonal eigenvectors (???)
diagonalizable (???)