Solving PDE: Help with Advection-Diffusion Equation

[jpcjr]

Homework Statement

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!

Homework Equations

See above.

The Attempt at a Solution

See https://www.physicsforums.com/showthread.php?t=588387

 

[mighty2000]

" for a thorough explanation of the solution to this problem. Essentially, we can use the substitution v(x, t) = u(x + ct, t) to transform the advection-diffusion equation into the heat equation. This allows us to use the well-known solution to the heat equation to find v(x, t), which can then be used to find u(x, t) using the formula from Step 3.

As for the initial condition, we can use the definition of v(x, t) to determine that v(x, 0) = u(x, 0) = Phi(x). This means that the initial condition for v(x, t) is also Phi(x).

Overall, by following these steps, we can solve the Cauchy problem for the advection-diffusion equation and find the concentration profile u(x, t) at any given time. This solution can have important applications in fields such as fluid dynamics, chemical engineering, and atmospheric sciences.