- #1
Ratpigeon
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Homework Statement
Solve u_t -k u_xx +V u_x=0
With the initial condition, u(x,0)=f(x)
Use the transformation y=x-Vt
Homework Equations
The solution to the equation u_t - k u_xx=0 with the initial condition is
u(x,t)=1/Sqrt[4[itex]\pi[/itex] kt] [itex]\int[/itex] e^(-(x-y)^2 /4kt)f(y) dy
The Attempt at a Solution
I really just need help subbing in the change in variable.
I think it's something like
u_y= u_t dt/dy +u_x dx/dy with dx/dy=1/(dy/dx)=1, dt/dy=1/V
=-1/V u_t +u_x
But this doesn't put the equation into a useful form...
the other thing I thought of was
u_x=u_t dt/dx +u_y dy/dx =0+u_y
and u_t=u_x dx/dt+u_y dy/dt =-Vu_y
And then we have that u_t+V u_x=-V u_y +V u_y=0, so the DE is just k u_xx=0; which I'm guessing isn't right either - because then it's just straight integration; (with the constants as functions of y?)...
Anyway, I'm fairly sure that the change of variables will result in either u_y=u_t+V u_x or possibly some multiple.