Math 481 Applied PDE --- Spring 2004

Final Exam Discussion

• Question: Where is the exam?

The exam will be in 309 BH.

• Question: In problem 2, we are given the initial condition u(x,0)=sin x, when x>0. Should this be the partial with respect to x? I factored the wave equation into two first order PDEs involving w and v, where w=Ut - Ux and v=Ut + Ux. To determine w and v I need Ux. Or should I assume U is differentiable at t=0 to get Ux(x,0)=cos x?

I believe that the problem is stated correctly. You only need to use the factorization of the wave equation to conclude that the solution can be written as u(x,t)=F(x-ct)+G(x+ct). You should use this form as your starting point. You can differentiate u(x,0)=sin(x) with respect to x, if needed.

• Question: Is there a misprint for problem 1? Shouldn't the boundary condition Wx=h(x) on x=0 be replaced with Wx=h(y) on x=0?

Yes there is a misprint in question 1. The boundary condition should be h(y) since it is on the x-axis.

• Question: I have two doubts about the final questions: In problem no.1, shouldn't the boundary condition h be a function of y (h(y))? Also, problem no.6 part c is a bit vague for me. Shouldn't we construct G using eigenfunction expansion method first and then generate the solution u utilizing Lagrange identity? I guess it's not correct if we use e-function expansion method to solve for u directly since it's not homogeneous in BC's and therefore we can't differentiate Fourier series w.r.t. x. Thanks for your considerations,

The typo in problem 1 is addressed above. In question 6, I wanted you to construct the Green's function in part (a) by solving the homogeneous ODE for x not equal to x_0 using the general solution. This leads to solution defined in two parts. In part (c), I wanted you to construct the solution using eigenfunction expansion and hence get a series solution. Part (b) requires the Lagrange identity.

• In problem 4, we first use Laplace transform to solve the problem, than we need to use Green's function. Would you please talk a little bit about how to use green's function to solve that heat eqution? Does the result look the same as the one by using Laplace transform?

The second part of question 4 asks you to construct the solution using the Green's function. The Green's function solves the problem with homogeneous bc and ic, and 1 replaced by the product of delta functions. The Green's function for the semi-infinite interval problem can be constructed by the method of images. The formula is given in (11.3.34). With the Green's function, you then construct the solution u using the Lagrange-type identity for the heat equation (11.3.21), which is summarized for the one dimensional case on the top of page 530 (boundary terms only). Showing the two solutions are the same is a bit difficult and I am not expecting you to be able to that on the exam. Constructing either form of the solution is fair game.

• In question 4 you ask us to use Laplace transforms to solve the diffusion equation. However, in the book and in the notes that you did in class you only did it with Utt=Uxx and Ut=Ux. Is the way if solving them the exact same way or is it a typo? Also with the +1 at the end of the equation do you just make the transform and then us it to solve?

I believe I solved a heat equation problem 13.4.4 in class before the wine cellar problem. You apply the Laplace tranform in t to the PDE to get an ODE in x with a boundary condition at x=0 and boundedness at x=infinity. You use L[1]=1/s in the PDE.

• For number 3, I got the resulting ODE's using the method of characteristics are a. x'=-tx and b. w'=-w. I solved for a. using the integrating factor (1/2)t^2 and I solved for be by assuming the solutions was Ce^rt. and got r=-1. I used x(0)= n to solve for the constants but I don't know what to do with the boundary condition. I solved for w without it and now I don't know if my solution is correct. Any suggestions?

You need to solve the characteristic equations a. and b. but the initial conditions for a. and b. depend on whether the curves are starting on the initial curve t=0 or x=0. You must parameterize these curves differently, leading to different forms for the characteristics. I worked an example in class similar to 12.2.4.

• On problem 6 part c), I'm a little confused. Do you want us to construct u directly via eigenfunction expansion or construct the Green's function from eigenfunction expansion and then construct u?

I guess that I wanted you to construct the Green's function first and then construct u. By constructing u directly you should see the Green's function.

• Question: Just another question... This time number 4. After you apply the Laplace transform and end up with the ODE Uxx-sU= 1/s, the only way I know to solve this is solving the homog. problem and then using variation of parameters. Unfortunately, the Wronskian is ghastly. Did I make some horrible mistake, or is this what you wanted us to do?

The independent variable in the transformed ODE is x and the Laplace transform variable s is treated as a constant with respect to x. So you can just use undetermined coefficients. In other words, treat s as a constant and it has an easy solution.

• Question: I am still having trouble with number 3. Unlike 12.2.4 these aren't linear characteristics. I don't know how to find the characteristics. Normally, we looked at the solution for x. In this case, I got x=ne^(-t^2/2). This is on the t=0 line. However, I have no idea what the characteristics look like.I got the corresponding w=xe^(t^2/2)e^-t. This is true if x> than what? Looking on the x=1 line, I got x(1)=r=ce^(-1/2), then I got x=re^1/2e^(-t^2/2) But how do I solve for w(1,t). w still equals ke^-t. If I continue to solve using the fact that w(1,t)= w(1,r)=r= ke^-r, I can't get rid of the exponentials and I get e raised to xe raised to a function of t. and r is also xe^((1-t^2)/2).

The characteristics are the curves in the x-t plane and the general solution is x=ne^(t^2/2), use separation of variables. The characteristics from t=0 have an initial condition like x(0)=xi while the characteristics starting on x=1 have the initial condition x(eta)=1. The critical characteristic that separates the two types starts at t=0 and x=1. You need to develop the similar types of initial conditions for the w equation. The w solution depends on the characteristic variable xi or eta which is a function of x and t thru the x solution. Does this help. To plot the characteristics, you can graph x=ce^(t^2/2) in the t-x plane and then flip and rotate the graph to get the x-t plane.

• I am signing off for the night. I will check again in the morning.