Toronto Math Forum

http://www.math.toronto.edu/courses/apm346h1/20129/HA9.html#problem-9.2

For U3, there shouldn't be a log; and for U2, it should be (1/2*pi)* log[(x^2+y^2)^(1/2)], right?

I do not quite understand step b and c(see attached) in the proof of Mean-value theorem. Did you use Green's identity or some other identity when dragging out terms from the integral?

ut−kuxx=0 for x∈(0,π)
with the boundary conditions ux(0,t)=0 and u(Ï€,t)=0 and the initial condition u(x,0)=x.

Part d) Write the solution in the form of a series.

If we use separation of variables, U=X*T, I found that T0 is t dependent. Then A0*T0 cannot not just a constant or zeor(A0 is for X).
Are there any mistakes regarding that part in both solutions poster last year?

Attached picture is a screen shot of online notes lecture 23. I was wondering if there is an addition r in front of sin(n*theta) in (12). Also, is there missing a^(-n) in calculating An and Cn?

For part c, sinx/x is an even function and its integral is Si(x). Then should we simply get positive infinity?

Build on DW's solution,part c).  For -2t < x < 2t, x+3t> 0, Ï• can be solved the same as in part a). x+2t<0, ψ is the same as x< -2t, which is a constant C, since Ï•+ψ.  The u(x,t)= 1/4exp(-x-2t)+C for -2t < x < 2t. Will this be a correct answer to finish up the question?

I think there is still problem 2/ sqrt(pi) instead of sqrt(2/pi), is this true?

It looks like I should use the result from part 3 for part 4. However if so, how should I use the initial values?