The image presents an initial boundary value problem (IBVP) which is outlined as follows:

IBVP:

$u_t + u_x + u = 0, \quad x > 0, \, t > 0$ with the initial and boundary conditions: $u(0,t) = \sin(t), \quad u(x,0) = 0$

The solution provided is: $u(x,t) = e^{-x} \sin(t - x) H(t - x)$ where $H(t - a)$ is the unit step function, which ensures that the solution is defined only for $t > x$.

Explanation of the solution:

This is a first-order linear partial differential equation (PDE) and can be solved using the method of characteristics. The general steps are:

1. Transforming the PDE into a form suitable for characteristic lines.
2. Finding the characteristic equations, which describe how solutions propagate along certain curves.
3. Solving along the characteristics, while satisfying the initial and boundary conditions.

In this case, the unit step function $H(t - x)$ ensures that the solution is only valid for $t > x$, which is a standard approach in problems involving wave propagation or transport equations.

Do you want more details on the step-by-step method used to solve this PDE?

Related Questions:

1. What is the method of characteristics in solving PDEs?
2. How does the unit step function $H(t - x)$ affect the solution of PDEs?
3. Can you explain how boundary conditions are applied in such problems?
4. What are characteristic curves, and why are they important?
5. How does the solution change if the initial or boundary conditions are modified?

Tip:

When solving PDEs like this, always check the domain of validity, as solutions often depend on conditions such as $t > x$, handled by functions like the unit step.