Question

Find the solution of the following pure convection problem

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

Answer 1

The PDE ut = -2ux describes pure convection and can be solved using the method of characteristics. The solution indicates that u is constant along characteristic lines; without initial conditions, the precise solution form remains unspecified.

Step-by-step explanation:

The solution of the partial differential equation (PDE) ut = -2ux, where ut is the time derivative of u and ux is the spatial derivative of u, involves applying the method of characteristics. The given PDE is a first-order linear PDE describing pure convection, commonly seen in fluid dynamics and wave propagation scenarios. The equation suggests that the quantity u is being transported along x without changing shape or dissipation.

To solve this equation, one approach is:

1. Express the solution u(x, t) through the characteristics, which are straight lines in the (x,t)-plane due to the linear nature of the PDE. The characteristic curves are determined by the equation dx/dt = -2, meaning that x changes with t at a constant rate of -2.
2. On these characteristic lines, the value of u remains constant since there is no change in shape or amplitude.
3. Generally, an initial condition u(x, 0) = f(x) is required to determine a specific solution. The solution will be u(x, t) = f(x + 2t).

However, if no initial condition is given, we cannot specify the exact form of the solution. The problem statement does not include initial or boundary conditions, so a more specific solution cannot be determined without additional information.