Question

Solve the following initial-value problem using the independent variable (coordinate) transformation method of Chapter 15:

PDE: uₜ = uₓₓ - 2uₓ, -[infinity] < x < [infinity]

Answer 1

To solve the initial-value problem using the independent variable (coordinate) transformation method, we can use the transformation ξ = x - 2t to simplify the equation and find the solution for u(x, t).

Step-by-step explanation:

To solve the initial-value problem using the independent variable (coordinate) transformation method, we have the following equation: uₜ = uₓₓ - 2uₓ, with -∞ < x < ∞.

We can use the transformation ξ = x - 2t to simplify the equation. By applying the chain rule, we can rewrite the equation in terms of the new variable ξ and the new dependent variable v(ξ). After solving for v(ξ), we can use the inverse transformation to find u(x, t).

Let's go through these steps in detail:

1. Apply the chain rule to express uₓ in terms of v(ξ).
2. Substitute the expressions for uₓₓ and uₓ into the given equation.
3. Solve the resulting ordinary differential equation for v(ξ).
4. Apply the inverse transformation to find u(x, t) from v(ξ).