Partial Differential Equations
9 problems
Diffusion Equation
9 problems
- Derive the diffusion equation for an insulated rod whose specific heat, conductivity and cross-section vary along its lengthExercise 1.2.1 — 1 page
- Derive the diffusion equation for a solid sphere whose temperature depends only on radius and timeExercise 1.2.4 — 1 page
- Solve φ_t = a²φ_xx + w(x, t) on 0 < x < ℓ with φ(x, 0) = f(x) by expanding in the eigenfunctions of β″ + λβ = 0, β(0) = β′(ℓ) = 0Exercise 1.6.2(a) — 3 pages
- Show that the substitution φ − φ₀ = ψ e^{−βt/(cρA)} reduces the heat equation for a rod with lateral loss to ψ_t = a²ψ_xxExercise 1.2.3(b) — 1 page
- Derive the heat equation for a rod that loses heat through its sides at rate β(φ − φ₀) and generates heat internally at rate h(x, t)Exercise 1.2.3(a) — 1 page
- Use the Laplace transform to solve φ_t = a²φ_xx on x > 0 with φ(x, 0) = 0 and φ(0, t) = f(t)Exercise 2.2.1(a) — 1 page
- Find the separated solutions of φ_t = a²φ_xx when the separation constant is (α + iβ)²Exercise 1.4.1 — 1 page
- Find α so that φ = f(x^α t) solves φ_t = a²φ_xx, and derive the ODE for fExercise 1.4.4 — 1 page
- Choose the coefficients of a truncated sine series to best approximate the solution of φ_t = a²φ_xx with φ(0, t) = φ(ℓ, t) = 0 and φ(x, 0) = g(x)Exercise 1.4.2 — 1 page