Bill Goodwine's Courses

Reading: Chapter 12 in the course text.

Exercises:

1. Based on the general solution to the heat equation, answer the following and explain your answer both mathematically from the solution as well as physically.
   1. If the specific heat capacity of a material is increased, will the temperature profile of a bar converge faster or slower to the steady-state solution?
   2. If the thermal conductivity of a material is decreased, will the temperature profile of a bar converge faster or slower to the steady-state solution?
   3. If the density of a material is increased, will the temperature profile of a bar converge faster or slower to the steady-state solution?
2. Solve the heat equation with homogeneous boundary conditions with alpha=1, L=4 and a uniform initial temperature of u=1. Plot the solution for various times to illustrate the nature of the solution.
3. Solve the heat equation with homogeneous boundary conditions with alpha=2, L=2 and an initial temperature distribution that is triangular, zero at both ends and with a peak of u=2 in the center. Plot the solution for various times to illustrate the nature of the solution.
4. Solve the heat equation with alpha=4, L=10, boundary conditions of T=2 on the left, T=4 on the right and a uniform initial temperature distribution of u=3. Plot the solution for various times to illustrate the nature of the solution.
5. Solve Laplace's equation using the notation from the beginning of section 12.4 with a=3, b=1 and f(x)=1 between 1 and 2 and zero elsewhere. Plot the solution in a manner similar to that in Figure 12.19.
6. Find the general solution to Laplace's equation using the notation from the beginning of section 12.4 with u(a,y)=g(y) and u(x,b)=0.

on #6 on this hw, is u(a,y)=g(x) correct? if we are looking at x=a (a constant), how would u change w.r.t. x, when x doesn't change? Should it be g(y)?

Yes, it should be g(y).

Professor,
Yesterday in class you said we could have 4 sheets; then you mentioned that we could have notes in our books too. Will we be able to use our book AND 4 sheets for reference on the exam? Thanks

wtowne wrote:Professor,
Yesterday in class you said we could have 4 sheets; then you mentioned that we could have notes in our books too. Will we be able to use our book AND 4 sheets for reference on the exam? Thanks

Yes, you can bring the book and write whatever you want in that. In addition you can bring four sheets of notes.

Prof Goodwine,

In the book, it is stated that for fourier series, all a_0 terms will be zero because sin(0)=0. However, for the solution to problem 3, HW 9, you calculate a_0 to be equal to something besides zero. How is this possible?

-Tom Banasiak

tbizzle wrote:In the book, it is stated that for fourier series, all a_0 terms will be zero because sin(0)=0. However, for the solution to problem 3, HW 9, you calculate a_0 to be equal to something besides zero. How is this possible?

Perhaps a better way to think of it is that the coefficient for the sine terms for n=0 should be zero. The same is not true for cosine.

Just to clarify: the exam covers only (and all) of the material in chapter 12? Thanks.

tgalx wrote:Just to clarify: the exam covers only (and all) of the material in chapter 12? Thanks.

Like I said in class yesterday is covers PDEs. That includes Chapter 12 as well as any additional material covered in the lectures.

In addition to the four pages of notes, will we be able to use our book during the exam?

Nick S wrote:In addition to the four pages of notes, will we be able to use our book during the exam?

Oops, disregard this post, answered it above.

Prof,

Will we need to know the specifics of the Euler-Bernoulli Beam Equation? (section 12.6) This topic hasn't been covered in lecture nor on the homeworks.

-Tom Banasiak

Professor,
I have two questions.
The first is about the b(0) (aka "b naught") coefficient in the Fourier series. The book says (on page 389) that b(0)=1/2L times the integral of f(x). Then, in the formula for the Fourier Series (equation 12.18), b(0) is divided by two. Are you supposed to divide by two both times? In the second example, 12.2.6, the 1/2L term is not used when b(0) is found, but b(0) is divided by 2 in the final answer.

My second question concerns the heat equation with inhomogeneous boundary conditions. In the book (page 403), when "f hat" is found, there is no x attached to the T2-T1/L term. That is, the formula for "f hat" = f(x) - (T2-T1)/L -T1. But in the solutions to Homework 10, Problem 4, there is an x in the formula for "f hat" (f hat = f(x) - (T2-T1)X/L - T1. Which one is correct? I have the x in the formula in my notes so is it just a typo in the book?

Thank you so much!

tbizzle wrote:Will we need to know the specifics of the Euler-Bernoulli Beam Equation? (section 12.6) This topic hasn't been covered in lecture nor on the homeworks.

I would expect you to be familiar with what is in the book on it. I believe it isn't solved in the book, so I wouldn't expect you to have worked out the solution yourself (but that wouldn't be a bad exercise...)

tlyons wrote: The first is about the b(0) (aka "b naught") coefficient in the Fourier series. The book says (on page 389) that b(0)=1/2L times the integral of f(x). Then, in the formula for the Fourier Series (equation 12.18), b(0) is divided by two. Are you supposed to divide by two both times? In the second example, 12.2.6, the 1/2L term is not used when b(0) is found, but b(0) is divided by 2 in the final answer.

The top formula on page 389 can be used to compute b_0, in which case you should NOT divide by zero again. Alternatively, you can use Equation 12.20 with n=0 to compute b_0, in which case you should divide by 2. In equation 12.18, uses 12.20 for the definition of b_0. Bottom line: don't divide by 2 twice.

My second question concerns the heat equation with inhomogeneous boundary conditions. In the book (page 403), when "f hat" is found, there is no x attached to the T2-T1/L term. That is, the formula for "f hat" = f(x) - (T2-T1)/L -T1. But in the solutions to Homework 10, Problem 4, there is an x in the formula for "f hat" (f hat = f(x) - (T2-T1)X/L - T1. Which one is correct? I have the x in the formula in my notes so is it just a typo in the book?

That is a typo. There should be a x in that part of the equation since it is the equation for a straight line.