Mathematical Methods

Linear Transport Equations

Pure Transport

u_t + c u_x = 0 → u(x,t) = g(x - ct) u(x,0) = g(x)

Transport with External Source

u_t + c u_x = f(x,t) → u(x,t) = g(x - ct) + ∫₀^t f(x - c(t-s), s) ds u(x,0) = g(x)

Transport with Decay

u_t + c u_x + δu = 0 → u(x,t) = g(x - ct) e^-δt u(x,0) = g(x)

Equation of the Characteristics

x = x₀ + ct

1. Identify type of equation, velocity c, initial condition
2. Write the eq. of the characteristics
3. Write the solution

Heat Equation

1. u_t = D u_xx π² u(x,0) = g(x) u(0,t) = β_in(t) u(L,t) = β₂(t)

x ∈ [0,L], t ∈ [0,T]

Separation of variables: U(x,t) = V(x) W(t) u_t = V(x) W'(t), u_xx = V''(x) W(t), u_t - D u_xx = 0 → V(x) W'(t) - D V''(x) W(t) = 0 V(x) W'(t) = D V''(x) W(t) Dividing by V(x) W(t): W'(t) / DW(t) = V''(x) / V(x) = -λ

1. V''(x) - λV(x) = 0 if β_in(t), β₂(t) = 0
2. Dirichlet BCs → u(0,t) = β_in(t) = V(0) W(t) → V(0) = 0
3. ∀t∈(0,T) → u(1,t) = β₂(t) = V(L) W(t) → V(L) = 0

V''(x) - λV(x) = 0 V(0) = 0 V(L) = 0 Eigenvalue: λ_k = - (kπ/L)² k ≥ 1

Eigenfunctions: V_k(x) = C_k sin( (kπ/L) x )

1. W'(t) - λDW(t) = 0 W_k(t) = C_k Θ^-λ(k2/L)2 t

→ U(x,t) = ∑_k≥1 C_k sin( (kπ/L) x ) . e ^{-(kπ/L)2 t}

We have to impose the initial condition \( u(x,0)=g(x) \)

\( u(x,0) = \sum_{k=1}^{\infty} C_k \sin\left(\frac{k\pi}{L} x\right) = g(x) \) Fourier expansion of \( g(x) \)

(2) Neumann BCs

• \( Cu_t - Du_{xx} = 0 \)
• \( U(x,0) = g(x) \)
• \( U(0,t) = 0 \)
• \( U_x(L,t) = 0 \)

The procedure is the same of the previous case but it is different in \( V_k(x) \):

1. \( V''(x) - \lambda V(x) = 0 \)
2. \( V'(0) = 2\ln(i) \)
3. \( V'(L) = 2\ln(g) \)

Eigenvalues \( \lambda=' \)

\( \lambda = \left(\frac{k\pi}{L}\right)^2 \)

Eigenfunctions

• \( V_0(x) = \text{const} \)
• \( V_k(x) = C_k \cos\left(\frac{k\pi}{L} x\right) \)

So the solution is:

\( U(x,t) = \sum_{k=0}^{\infty} C_k \cos\left(\frac{k\pi}{L} x\right) e^{-\left(\frac{k\pi}{L}\right)^2 t} = C_0 + \sum_{k=1}^{\infty} C_k \cos\left(\frac{k\pi}{L} x\right) e^{-\left(\frac{k\pi}{L}\right)^2 t} \)

Then impose the initial condition + Fourier expansion of \( g(x) \)

Fourier series theory

\( \tilde{g}(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}[a_k\cos(kix)+b_k\sin(m(k_i x))] \)

\( a_k = \frac{2}{\pi} \int_{O}^{J} f(x)\cos(kx)dx \quad b_k = \frac{2}{J} \int_{O}^{J} f(x)\sin(k_i x)dx \)

If \( f(x) \) is a T-periodic function

\( f(x) = \sum_{k=0}^{\infty} \left[a_k\cos\left(\frac{2\pi}{T} kx\right)+b_k\sin\left(\frac{2\pi}{T} kx\right)\right] \)

\( a_k = \frac{2}{T} \int_{-T/2}^{T/2} f(x)\cos\left(\frac{2\pi}{T} kx\right)dx \quad b_k = \frac{2}{T} \int_{-T/2}^{T/2} f(x)\sin\left(\frac{2\pi}{T} kx\right)dx \)

Even \( f(x): \tilde{g}(x) = \frac{a_0}{2} + \sum_{k=1}^{\infty} a_k \cos(k_i x) \quad a_k = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(k_i x) dx \)

Odd \( f(x): \tilde{g}(x) = \sum_{k=1}^{\infty} b_k \sin(k_i x) \quad b_k = \frac{2}{\pi} \int_{0}^{\pi} f(x) \sin(k_i x) dx \)

Dirichlet = odd expansion; Neumann = even expansion.

HEAT EQUATION - QUALITATIVE STUDY

1. Weak Maximum Principle: \( U \epsilon C^{2,1}(\overline{QT})\subset C(\overline{QT}) \) be a solution of
2. \( U_t - U_{xx} = f \quad (x,t) \epsilon (Q,L)\times(0,T) \) with \( f \leq 0 \), \( M_{ex}U = \max_{QT} U \)
3. With \( f \geq 0 \): \( \min_{QT} U = \min_{SPQT}U \)
4. If \( f = 0 \) both

The general solution of

Apply BCs: if Dirichlet

then apply superposition principle:

Impose the non-homogeneous BC, with Fourier expansion (if necessary)

Two-Dimensional Heat Equation

Separating variables

And then continue as "defense in a cube"

Pay attention that

One-Dimensional Wave Equation

Well-posed problem:

Initial profile

Initial velocity

If :

Global Cauchy problem - d'Alembert formula

If is bounded, BCs are needed (separation of variables method)

The general solution of

Pay attention when

Parabolic Boundary