PDE - Solve heat equation with convection

[Ratpigeon]

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\pi kt] \int 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.

 

[tiny-tim]

Hi Ratpigeon!

(try using the X₂ button just above the Reply box )

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

no, you have two sets of variables: (x,t) and (y,t') with t' = t

your u_t should have u_y and u_t' parts, not u_y and u_x parts

 

[Ratpigeon]

So is s=x+Vt
Then is it ux= uy +us, uxx=uyy+2uys+uss
ut=-Vuy+Vus and the total equaton is 2Vuy+uyy+2uys+uss?

 

[tiny-tim]

So is s=x+Vt

what on Earth is "s" ?

you have two sets of variables: (x,t) and (y,t') with y = x - Vt and t' = t

your u_t should have u_y and u_t' parts, not u_y and u_x parts ​

 

[Ratpigeon]

Okay so i define h(x-Vt,t')=u(x,t) and get
ht=ut-Vuy and hy=ux for y=x-Vt
So:
ht-khyy=ut-kuxx+Vux=0
And the initial conditon u(x,0)=f(x) becomes h(x-Vt,0)=f(x-Vt); and since t=0 this is just h(x,0)=f(x)
and then i have an integral of...

exp(-(x-Vt-y)^2/4kt) f(y)dy (with the square root scalar out the front) and that's the solution for h and hence also u?

 

[tiny-tim]

sorry, this is too difficult to read

but anyway wouldn't it be easier to find u_x and u_t, since they're actually mentioned in the question?

 

[Ratpigeon]

Im about 90% sure i got it. thankyou so much for your help :) i just wasnt getting the variable change until you explained it... :S