If you help me with this, I will give you another 2 3 4 350 KP \"silly\" questio

Question

If you help me with this, I will give you another 2 3 4 350 KP "silly" questions for a total of 1050 1400 1750 KP!!! Send me a PM if you are concerned about not getting the KP.

PLEASE show all steps.

The Cauchy problem for the advection-diffusion equation is given by

where c and K are positive constants.

The purpose of this exercise is to solve the advection-diffusion equation using the following 3 steps.

(1) Let v(x, t) = u(x+ct, t) and show that v(x, t) satisfies the heat equation.

(2) Determine the initial condition that v(x, t) must satisfy and solve the resulting Cauchy problem for v(x, t).

(3) Using the formula for v(x, t), find u(x, t), the solution of the Cauchy problem for the advection-diffusion equation.

Explanation / Answer

1) let v(x,t) = u(x+ct, t)
where u satisfies the differential equation
vt = cux + ut

vxx = uxx

the heat equation: t = kxx

v satisfies this since u satisfies the advection diffusion problem:

vt = cux + ut = kuxx = kvxx

2) v(x,t) = u(x+ct,t)

v(x,0) = u(x,0) = (x)

So you can then solve the heat equation for v.

vt = kvxx

3) Once you have v, u(x,t) = u((x-ct)+ct, t) = v(x-ct, t)

so u then solves the advection diffusion problem.

So i'm not sure how your class intends to solve the heat equation. I would use a fourier integral. Hope this was helpful. Sorry if it isn't.

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