Hey I have these problems that I'm having trouble with for my PDE class. A complete solution would be nice but any sort of direction would be greatly appreciated. Also I'm not sure if I'm supposed to post multiple problems but anyways...
1. find u(r,theta) when r is equivalent to sqrt(x(1)^2+x(2)^2) less than or equal to 2
laplacian of u equals zero, in r less than or equal to 2
u(2,theta)=cos(3*theta) when 0 <theta<2pi
2.find all u(x)=u(x1,x2) in 0<x1<2; -1<x2<1 when laplacian of u equals zero
BCs are du/dx1(0,x2)=du/dx2(2,x2)=0 for previous constraints on x1, x2
3.let u(x,t); x in R^1, t>0 satisfy
u_tt - u_xx +2u_x +3u = 0; u(x,0)=f(x); u_t(x,0)=g(x)
Let uhat(xi,t)=1/sqrt(2pi) * integral from -inf to +inf (exp[-i*xi*x]*u(x,t)dx) be the fourier transform of u wrt x
i)find uhat(xi,t) in terms of fhat(xi) and ghat(xi)
ii)Give an integral for u(x,t) in terms of fhat(xi) and ghat(xi), don't evaluate integral
4. find u(x,t)=u(x1,x2,x3,t) for x in R^3, t>0 if
u_tt - laplacian u = 0; u(x,0) = 0; u_t(x,0)=aX_2(x) which is equivalent to a if |x|<2 and 0 if |x|>2.
i)give an explicit formula(s) in simplest terms for u. Evaluate any integrals.
ii)give all values of t for which u(0,0,5,t) does not equal 0
5. find the solution u(x,t), x in R^3, t>0 to
u_tt - c^2 * laplacian u = del(x1)del(x2)del(x3)del(t)
Evaluate all integrals to get the fundamental solution of this equation.
Again sorry if i'm not supposed to post more than one problem at a time, any help would be greatly appreciated.