Let u(x, y) be the solution of the Cauchy problem u·uₓ + uy = 0 for x ∈ ℝ, y 0, with u(x, 0) = x for x ∈ ℝ. Which of the following is the value of u(2, 3)?

Question

Let u(x, y) be the solution of the Cauchy problem u·uₓ + uy = 0 for x ∈ ℝ, y > 0, with u(x, 0) = x for x ∈ ℝ. Which of the following is the value of u(2, 3)?

Accepted Answer

C. Characteristics: dx/dy = u with u constant along them, so x = x₀ + u·y and u = x₀. Hence u = x/(1 + y), and u(2, 3) = 1/2.

Burgers characteristics

More on this topic