Derive the discrete heat equation in one-dimension

Assignment Help Engineering Mathematics

Reference no: EM131103673

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).

x₁

x₂

x₃

x₄

x₅

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. t⁰represents t₀, t¹ = t₁ and so forth.)

u^j_nis the temperature of segment x_n at time step t^j. u³₂is 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_nfor 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_nfor n = 1, 2, 3, 4, 5 and j = 1, 2, 3, 4, 5. What do you notice? Does this make sense?

Reference no: EM131103673

Questions Cloud

Where does the incidence of the tax lie : Assume that the government places a 50 percent tax on widgets. Neither the demand for widgets or supply of widgets is perfectly elastic or inelastic. Draw a graph showing how the tax will affect the market. Where does the incidence of the tax lie? Ex..

People without prior knowledge of game theory : Write a paper that uses game theory to to set up a game designed to help a consumer decide whether to buy life insurance or not. To keep the game relatively simple, assume the life insurance being considered is term life, i.e. insurance without an ac..

How did you create x so that the plot looked good : How did you create x so that the plot looked good. Where a is the size of the semi-major axis (along the x-axis) and e is the eccentricity. Plot ellipses using this formula, ensuring that the curves are smooth by selecting an appropriate number of po..

Example of permanent supply shock : An example of a permanent supply shock is:

Derive the discrete heat equation in one-dimension : Math 054 Partial Differential Equations - HW Assignment 1. Derive the discrete heat equation in one-dimension. Show all work. Find δyδyu(x, y, z, t)

Property rights security in russian deprivatization : Property Rights Security in Russian Deprivatization. What impact will the prospect of deprivatization have on investment by managers of privatized firms? What effect will deprivatization have on foreign investment in Russia? Who gains from deprivatiz..

Closing over under applied overhead into cost of goods sold : At the end of the year, overhead applied was $3,272,961. Actual overhead was $3,742,718. Closing over/under applied overhead into cost of goods sold would cause net income to increase/decrease by?

If its required return is 15 what is the stocks expected : A stock is expected to pay a dividend of $3.00 at the end of the year (i.e., D1 = $3.00), and it should continue to grow at a constant rate of 10% a year. If its required return is 15%, what is the stock's expected price 1 years from today? Round you..

If the debt to total assets ratio is 20 percent what is the : The King Arts Company has a return on assets ratio of 5 percent. If the debt-to-total assets ratio is 20 percent, what is the return on equity? If the firm had no debt, what would the return-on-equity ratio be?

User Account

All Pages