PDEs/method of characteristics
Hi everyone,
I have some PDEs problems that I just cannot solve. Any help would be appreciated.
1. Consider IVP u_x + au_t = f(x,t), where a is a positive constant and
u(x,0) = 0,
f(x,t) = {1 if x >= 0
0 else}
Show that the solution is
u(x,t) = {0 if x =< 0
x/a if x-at =<0 and x >= 0
t if x-at >= 0 and x >= 0
I'm not really sure how to solve this because of the f(x,t). Also, it seems like the solution given by the book should be wrong because for the part of the solution with x/a, u(x,0) = x/a, is clearly not equal to 0 for any x != 0.
2. Solve IVP for u_t + (1/(1+(cos(x)/2)))*u_x = 0
Show the solution is given by u(t,x) = u_0(z), where z is unique solution of z + sin(z)/2 = x + sin(x)/2 - t.
I know how to set up the problem, but I can't seem to arrive at that answer!
Any help would be greatly appreciated. Also, if anyone knows any online texts on method of characteristics, I would really appreciate that too! Thanks so much.
I have some PDEs problems that I just cannot solve. Any help would be appreciated.
1. Consider IVP u_x + au_t = f(x,t), where a is a positive constant and
u(x,0) = 0,
f(x,t) = {1 if x >= 0
0 else}
Show that the solution is
u(x,t) = {0 if x =< 0
x/a if x-at =<0 and x >= 0
t if x-at >= 0 and x >= 0
I'm not really sure how to solve this because of the f(x,t). Also, it seems like the solution given by the book should be wrong because for the part of the solution with x/a, u(x,0) = x/a, is clearly not equal to 0 for any x != 0.
2. Solve IVP for u_t + (1/(1+(cos(x)/2)))*u_x = 0
Show the solution is given by u(t,x) = u_0(z), where z is unique solution of z + sin(z)/2 = x + sin(x)/2 - t.
I know how to set up the problem, but I can't seem to arrive at that answer!
Any help would be greatly appreciated. Also, if anyone knows any online texts on method of characteristics, I would really appreciate that too! Thanks so much.