Solution for PDE $f f_x = a f_t + b$

Question

Does anyone know the solution to this PDE, with $f = f(x,t)$

$$f f_x = a f_t + b$$

the boundary conditions are: $f(0,t) = 0$, $f(L,t) = \text{const}_1$, $f(x,0) = \text{const}_2$

Answer 1

$ff_x=af_t+b$

$af_t-ff_x=-b$

Hint:

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dt}{ds}=a$ , letting $t(0)=0$ , we have $t=as$

$\dfrac{df}{ds}=-b$ , letting $f(0)=f_0$ , we have $f=f_0-bs=f_0-\dfrac{bt}{a}$

$\dfrac{dx}{ds}=-f=bs-f_0$ , letting $x(0)=g(f_0)$ , we have $x=\dfrac{bs^2}{2}-f_0s+g(f_0)=g\left(f+\dfrac{bt}{a}\right)-\dfrac{ft}{a}-\dfrac{bt^2}{2a^2}$