Math 054 Partial Differential Equations - HW Assignment 1

1. Derive the discrete heat equation in one-dimension. Show all work.

2. Find δ_yδ_yu(x, y, z, t).

3. Below is a thin rod whose lateral surface has been insulated against the flow of heat. Insulating the lateral surface of the rod ensures that the heat flows axially along the rod, and if the rod is sufficiently thin, it is an acceptable approximation of reality to suppose that the temperature does not vary across the thickness of the rod. (This all means that we'll be using the 1D discrete heat equations).

Suppose that the rod is 5∈ units long. It is split up into 5 equal pieces, each ∈ units long. Each segment is labeled x_n where n = 1, 2, 3, 4, 5.

Let

t^j = t₀ + j(t₁ - t₀), u^j_n = u(x_n, t^j)   (1)

t^j is the time step. (ex. t⁰ represents t₀, t¹ = t₁ and so forth.)

u^j_n is the temperature of segment x_n at time step t^j. u³₂ is the temperature of segment x₂ at time t₃.

The 1D discrete heat equation is, u(x_n, t^j+1) = u(x_n, t^j) + β[u(x_n-1, t^j) - 2u(x_n, t^j) + u(x_n+1, t^j)].  (2)

Using (1), it can be written as

u^j+1_n = u^j_n + β[u^j_n-1 - 2u^j_n + u^j_n+1].                 (3)

You may use a computer program to compute the problems below but you must write the code. Include the code (and relevant work) with your solutions.

(a) Let β = .1, u⁰₁ = 1, u⁰₂ = 2, u⁰₃ = 3, u⁰₄ = 4, and u⁰₅ = 5. (Recognize this example???)

i. Using (3), compute u¹₂, u¹₃, and u¹₄.

ii. Explain why you cannot use (3) to find u¹₁ and u¹₅.

iii. Suppose that the temperature at the ends (x₁ and x₅) are fixed. This means that u^j₁ = 1 and u^j₅ = 5 for all j. Compute u^j_n for n = 2, 3, 4 and j = 2, 3, 4, 5. What do you notice? Does this make sense? What is a physical interpretation for fixing the ends?

iv. Suppose that you had an infinite rod. What would you expect to happen over time? Why?

(b) Let β = .1, u⁰₁ = u⁰₅ = 0, u⁰₂ = u⁰₄ = 3, u⁰₃ = 5. Suppose that the temperature at the ends (x₁ and x₅) are fixed. Compute u^j_n for n = 1, 2, 3, 4, 5 and j = 1, 2, 3, 4, 5. What do you notice? Does this make sense?