Question

Determine the unique solution to the following problem {uₜₜ = c²uₓₓ, (x,t) ∈ R⁺ X (0.T), c > 0 {u(x,0) =1, uₜ(x,0) =0, u(0,t) =0

Answer 1

To determine the unique solution to the given problem, we need to solve the partial differential equation uₜₜ = c²uₓₓ. Using the method of separation of variables, we can write the solution as: u(x,t) = A sin(kx) cos(ωt), where k = π/L and ω = ck.

Step-by-step explanation:

To determine the unique solution to the given problem, we need to solve the partial differential equation uₜₜ = c²uₓₓ. We are given the initial conditions u(x,0) = 1 and uₜ(x,0) = 0, as well as the boundary condition u(0,t) = 0.

This is a one-dimensional wave equation, and it has a well-known solution. Using the method of separation of variables, we can write the solution as:

u(x,t) = A sin(kx) cos(ωt), where k = π/L and ω = ck.