PDE Help

Ompala

Hi guys, was hoping someone would be able to look through this PDE question using separation of variables for me, see word document for the question. Rest of attachments are my efforts at it. Trying to get a headstart on it for college, and think I have most of it out (you can ignore (b) and (d), just looking for clarification on the numerical parts. You can ignore the bit above part (a) with Laplace transforms - different question altogether.

Part (c) is where I am a bit iffy, because its the first case I ever came across having more than one value of k giving a solution, but adding the solutions seemed the correct procedure. Sorry if the notation ( e.g redefine A = An etc.) I used is tedious, its just the only way I know how from the notes I was using.

Many thanks in advance

Ompala

Ran out of space for rest of attachments above

sponsoredwalk

In part c) when you worked out the case k = - p^2 and found F_n(t)G_n(t) was a solution, why did you jump to saying that the sum over all n was a solution? Because of the principle of superposition for linear homogeneous ode's and pde's, which says that if u_1 and u_2 are solutions of the pde L = 0 then u_1 + u_2 are as well, because L[u_1 + u_2] = L[u_1] + L[u_2] = 0 + 0 = 0. If you had found valid solutions v_1 & v_2 for the case k = p^2 then L[v_1 + v_2] = 0 would hold, so L[u_1 + u_2 + v_1 + v_2] =0 would also hold (you did find valid solutions, zero solutions and they are implicitly in the sum!), so it makes perfect sense to add in another solution, w, for the case k = 0 as well! Interestingly this exact idea makes the notion of a Fourier series unavoidable, check problems 1.23 to 1.25 to see this in action, also this example gives a nice geometric interpretation of the discreteness of the separation constants if you'd like some more intuition, mixing these two ideas is a nice codification of the whole meaning of a Fourier series in the first place.

Ompala

sponsoredwalk wrote: »

In part c) when you worked out the case k = - p^2 and found F_n(t)G_n(t) was a solution, why did you jump to saying that the sum over all n was a solution? Because of the principle of superposition for linear homogeneous ode's and pde's, which says that if u_1 and u_2 are solutions of the pde L = 0 then u_1 + u_2 are as well, because L[u_1 + u_2] = L[u_1] + L[u_2] = 0 + 0 = 0. If you had found valid solutions v_1 & v_2 for the case k = p^2 then L[v_1 + v_2] = 0 would hold, so L[u_1 + u_2 + v_1 + v_2] =0 would also hold (you did find valid solutions, zero solutions and they are implicitly in the sum!), so it makes perfect sense to add in another solution, w, for the case k = 0 as well! Interestingly this exact idea makes the notion of a Fourier series unavoidable, check problems 1.23 to 1.25 to see this in action, also this example gives a nice geometric interpretation of the discreteness of the separation constants if you'd like some more intuition, mixing these two ideas is a nice codification of the whole meaning of a Fourier series in the first place.

Thank you for taking the time to look through the images and respond.

Your explanation has cleared things up nicely, it did seem summing the solutions for both cases was correct, but I wanted to be certain.

I jumped to saying the sum was a solution in (c) because I was just so used to saying it, but I do understand why it is a valid statement, and see how the same thinking is applied to solving both cases for k =0 and k = -p^2.

If the solution was valid for any value of n, then n=1,2,3 etc. are all solutions, so L[n=1] = 0, L[n+2] = 0 L[n=3] =0 so there sum L[n=1] + L[n=2] + L[n=3] =0 + 0 +0 = 0 will also be a solution. Sine n can have infinte values we take the sum to infinity.