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I need to solve a transport equation in the form $$ \frac{\partial f}{\partial t} + a(t) \frac{\partial f}{\partial x} = b f + c$$ with $a(t) = A t$. From a reference book I took a solution $f = -c/b + e^{bt} \Phi(x - At^2/2)$ which contain an arbitrary function $\Phi(x - At^2/2)$. Unfortunately I cannot compose this function for the following conditions:

  • initially $t = 0$ the function is zero everywhere $f(0, x) = 0$, which result in $\Phi = c/b$;
  • at the boundary $x = 0$ the function is zero $f(t, 0) = 0$, which means that $\Phi = e^{-bt} c/b $.

The questions:

  1. how to choose the function $\Phi$ in the example above?
  2. is there a technique/algorithm for finding $\Phi$ which satisfy prescribed initial/boundary conditions?

1 Answers 1