# Homework Help: Help solving a PDE

1. Mar 19, 2012

### jpcjr

1. The problem statement, all variables and given/known data

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

u.sub.t + c u.sub.x = K u.sub.xx (−∞< x < ∞)

u(x, 0) = Phi(x)

where c and K are positive constants.

The advection-diffusion equation essentially combines the effects of the
transport equation and the heat equation, so that the concentration profile
is carried with speed c as it diffuses. The purpose of this problem is to
solve the advection-diffusion equation using the following three 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; then, solve the
resulting Cauchy problem for v(x, t).

(3) Use the formula for v(x, t) from Step 2 to find u(x, t), the solution of the
Cauchy problem for the advection-diffusion equation.

THANK YOU SO VERY MUCH!!!

2. Relevant equations

See above.

3. The attempt at a solution