Question

Solve the IBVP
ut + 2ux = 0, x > 0, t > 0
u(x, 0) = e−x, u(0, t) = (1 + t2)−1

Answer 1

To solve the IBVP (Initial Boundary Value Problem) put + 2ux = 0, with x > 0 and t > 0, separate the variables x and t, integrate both sides of the equation, and solve to find u(x,t). The initial condition is given as u(x,0) = e^-x, and the boundary condition is u(0,t) = (1 + t^2)^-1.

Step-by-step explanation:

The given problem is a partial differential equation known as the IBVP (Initial Boundary Value Problem). The equation to be solved is put + 2ux = 0, with x > 0 and t > 0. The initial condition is u(x,0) = e^-x, and the boundary condition is u(0,t) = (1 + t^2)^-1. To solve this equation, we need to separate the variables x and t and integrate, from time t = 0 to an arbitrary time t.

Step 1: Separate the variables x and t, and rewrite the equation as (1/u)du = -2xdx.

Step 2: Integrate both sides of the equation, with respect to the respective variables, from x = 0 to an arbitrary x, and t = 0 to an arbitrary t.

Step 3: Solve the resulting equation to find u(x,t).