Applied Partial Differential Equations (3rd Edition), …tmk5/courses/m333/F16/Homework1.pdfLast...

Last updated 9/9/2016 Math 333 Homework Problems Math 333 Homework Problems #1 Applied Partial Differential Equations (3rd Edition), by J.D. Logan 1. Linear and homogeneous PDEs with constant coeﬃcients have plane wave solutions, u(x,t )= Ae i(kx-ωt) , where A is the amplitude, k is the wave number, and ω is the (temporal) frequency. The solution requires that ω = ω(k), which is known as the dispersion relation. Find the dispersion relation for the following PDEs: (a) ∂ t u = D∂ 2 x u (b) ∂ 2 t u = c 2 ∂ 2 x u (c) i∂ t u + ∂ 2 x u =0 (d) ∂ t u + ∂ 3 x u - c∂ x u =0 2. Consider the heat equation ∂ t u = k∂ 2 x u ∂ x u(0,t )= ∂ x u(1,t )=0. Set E 1 (t )= Z 1 0 u(x,t )dx, E 2 (t )= Z 1 0 u(x,t ) 2 dx. (a) Provide a physical interpretation for E 1 (t ) and E 2 (t ). (b) Show that E 1 (t )= E(0). What is a physical interpretation of this result? (c) Show that E 2 (t ) ≤ E 2 (0). What is a physical interpretation of this result? 3. Consider the heat equation ∂ t u = ∂ 2 x u - r 2 u u(0,t )=1, u(1,t )=1. Here r> 0 is a constant which describes the rate at which the bar loses its heat across the lateral boundary. The steady-state temperature, U (x), is a time-independent solution to the heat equation, i.e., a solution to the ODE U 00 - r 2 U = 0; U (0) = U (1) = 1. Graph the solution to this boundary value problem, and describe the manner in which the heat is distributed in the bar. 4. Consider the random walk derivation given in Section 1.4. (a) Suppose the probability of moving left is p ‘ , and the probability of moving right is p r =1 - p ‘ . Find a governing equation as Δx, Δt → 0. (b) Let p ‘ be the probability of moving left, p r the probability of moving right, and p s the probability of staying. The probabilities sum to one. Find a governing equation as Δx, Δt → 0. 1 of 1

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Transcript of Applied Partial Differential Equations (3rd Edition), …tmk5/courses/m333/F16/Homework1.pdfLast...

Last updated 9/9/2016 Math 333 Homework Problems

Math 333 Homework Problems #1Applied Partial Differential Equations (3rd Edition), by J.D. Logan

1. Linear and homogeneous PDEs with constant coefficients have plane wave solutions, u(x, t) = Aei(kx−ωt), whereA is the amplitude, k is the wave number, and ω is the (temporal) frequency. The solution requires that ω = ω(k),which is known as the dispersion relation. Find the dispersion relation for the following PDEs:

(a) ∂tu =D∂2xu

(b) ∂2t u = c2∂2xu

(d) ∂tu +∂3xu − c∂xu = 0

2. Consider the heat equation

∂tu = k∂2xu

∂xu(0, t) = ∂xu(1, t) = 0.

Set

E1(t) =∫ 1

0u(x, t)dx, E2(t) =

∫ 1

0u(x, t)2dx.

(a) Provide a physical interpretation for E1(t) and E2(t).

(b) Show that E1(t) = E(0). What is a physical interpretation of this result?

3. Consider the heat equation

∂tu = ∂2xu − r2uu(0, t) = 1, u(1, t) = 1.

Here r > 0 is a constant which describes the rate at which the bar loses its heat across the lateral boundary. Thesteady-state temperature, U (x), is a time-independent solution to the heat equation, i.e., a solution to the ODE

U ′′ − r2U = 0; U (0) =U (1) = 1.

Graph the solution to this boundary value problem, and describe the manner in which the heat is distributed in thebar.

4. Consider the random walk derivation given in Section 1.4.

(a) Suppose the probability of moving left is p`, and the probability of moving right is pr = 1 − p`. Find agoverning equation as ∆x,∆t→ 0.

(b) Let p` be the probability of moving left, pr the probability of moving right, and ps the probability ofstaying. The probabilities sum to one. Find a governing equation as ∆x,∆t→ 0.

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