1. Consider the PDE
auₜ + buuₓ + cuₓₓₓ = 0 (1)
for u(x,t) with positive constants (a, b, c). Find a rescaling of x and t that transforms
(1) into the standard KdV equation
uₜ + 6uuₓ + uₓₓₓ = 0. (2)

2. Recall the Cole-Hopf transformation for the viscous Burgers equation
uₜ + 6uuₓ + uₓₓₓ = 0, (3)
which relates u(x,t) to a function ϕ(x,t) > 0 that solves the heat equation
ϕₜ = kϕₓₓ (4)
What kind of solution does the Green's function on the real line of (4) correspond to in terms of u?

3. Find an integral expression for the general solution to the initial-value problem of (3) on the real line by combining the Cole-Hopf transformation with the Green's function for (4).

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Equation Problems
    $30.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Differential Equations Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats