Bill Goodwine's Courses

Reading: you should have read all of chapters 11 and 12.

Exercises: 11.9 (number 3 only), 11.12, 11.17. 11.19 and 12.20.

The exercises in red, 11.17 and 11.19, are optional extra credit.

I am confused on how to proceed with 11.9 since u(x,0)=f(x)
when you plug in 0 for y I get that u=0 no matter what. How then could you solve for the c_n values? Thanks!

Jessie wrote:I am confused on how to proceed with 11.9 since u(x,0)=f(x)
when you plug in 0 for y I get that u=0 no matter what. How then could you solve for the c_n values? Thanks!

It mirrors the development of the solution to Laplace's equation in the book. Basically, even though f(x) is not specified, you can determine a formula for the constants. It will look much like the "boxed" equations for the coefficients in a Fourier series, but where f(x) is not specified, e.g., Equation 11.44.

So are you saying we would use the same equation just ignoring the part that contains f(x)?

Jessie wrote:So are you saying we would use the same equation just ignoring the part that contains f(x)?

No, I'm saying you can find an expression for all the infinite number of coefficients in terms of a generic f(x).

Professor,

For Problem 11.2, part 2, can we use ode45 to plot the numerical solution, or should we write a program using one of the numerical methods?

sschwane wrote:Professor,

For Problem 11.2, part 2, can we use ode45 to plot the numerical solution, or should we write a program using one of the numerical methods?

I would use your own because it's more reliable. All you have to do is change the two functions in any other 4th order R-P program you wrote.

for 11.2 i found a fourier series that models f(t) but i do not know how to then use undetermined coefficients

Jessie wrote:for 11.2 i found a fourier series that models f(t) but i do not know how to then use undetermined coefficients

I would suggest thinking of it as a big collection of individual terms.

Jessie wrote:I am confused on how to proceed with 11.9 since u(x,0)=f(x)
when you plug in 0 for y I get that u=0 no matter what. How then could you solve for the c_n values? Thanks!

I'm also running into the same problem. Did you come up with any solutions?

astewar9 wrote:

Jessie wrote:I am confused on how to proceed with 11.9 since u(x,0)=f(x)
when you plug in 0 for y I get that u=0 no matter what. How then could you solve for the c_n values? Thanks!

I'm also running into the same problem. Did you come up with any solutions?

I have the same thing and I'm stuck.

pat wrote:

astewar9 wrote:

Jessie wrote:I am confused on how to proceed with 11.9 since u(x,0)=f(x)
when you plug in 0 for y I get that u=0 no matter what. How then could you solve for the c_n values? Thanks!

I'm also running into the same problem. Did you come up with any solutions?

I have the same thing and I'm stuck.

I have the same thing as well.

Although, I did try to somehow force my way through it, by taking Y(y) and considering y(0) = f(x), and rearranging to get k1 = f(x) - k2, and then following along the same procedure as the book (substituting into the Y function), rearranging and eventually getting f(x) = k1(1 - exp[2*n*pi*Ly/Lx]). Continuing through the rest of the solution gave me a final u(x,y) and c_n that were the same as in the book except with the (1 - exp[]) term from the previous line instead of the (exp[] - exp[]) term. But I have no idea whether this was the correct procedure (and judging from my previous homeworks, it's almost certainly not). What are we supposed to do after considering the y(0) term?

#epsilon_zen wrote:

pat wrote:

Jessie wrote:I am confused on how to proceed with 11.9 since u(x,0)=f(x)
when you plug in 0 for y I get that u=0 no matter what. How then could you solve for the c_n values? Thanks!

I'm also running into the same problem. Did you come up with any solutions?

I have the same thing as well.

Although, I did try to somehow force my way through it, by taking Y(y) and considering y(0) = f(x), and rearranging to get k1 = f(x) - k2, and then following along the same procedure as the book (substituting into the Y function), rearranging and eventually getting f(x) = k1(1 - exp[2*n*pi*Ly/Lx]). Continuing through the rest of the solution gave me a final u(x,y) and c_n that were the same as in the book except with the (1 - exp[]) term from the previous line instead of the (exp[] - exp[]) term. But I have no idea whether this was the correct procedure (and judging from my previous homeworks, it's almost certainly not). What are we supposed to do after considering the y(0) term?

This sounds right (well, pretty close)! It's hard to follow in words, but basically observe there are three sides with 0 as the BC and one side that is not. Take the variable with two zeros (X, I believe) and do the usual stuff, which will give lambda. Then you using that in the remaining equation (Y, I think) get the general solution. The sub in the value for y that corresponds to the 0 side (this is where it's different from the book). That gives you a relationship between the two constants (it's not as simple as the example, but you get something simpler later, for what it's worth). Then your remaining constant can be different for each value of n. Sum together everything and plug in the value for y corresponding to where f(x) it. You can do the multiply by sin(m pi x/L) and integrate from 0 to L (it's either L_x or L_y, depending on the side, which I can't remember off hand) which gives you a way to solve for the infinite number of coefficients.

This problem basically forces you to understand the derivation and then to adapt it to the case where the nonzero BC is on another side. A good problem, IMHO.

For 12.20, the insulated end computer program, how do we account for the insulated end in Fortran?

pat wrote:For 12.20, the insulated end computer program, how do we account for the insulated end in Fortran?

You need to force the deravative to be zero. After you compute the last node value just change the boundary value to be the same as that.