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I have encountered a problem here. Solve the partial differential equation, $$\frac{\partial u(t,x)}{\partial t}+\frac{\partial u(t,x)}{\partial x}+u(t,x)=1$$ with the inflow boundary value $$u(0,t)=t^2$$ I know how to solve this with an initial condition, just don't know how to implement the boundary condition.

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Hint: method of characteristics.

EDIT: Using the method of characteristics, you get a formula for $u(t,x)$ on every characteristic curve $x-t=c$, involving an arbitrary constant. Substitute the boundary condition to find that arbitrary constant.

You may be confused because of an unfortunate notational conflict. In the PDE your function is given as $u(t,x)$, but in the boundary value the second variable is given as $t$ instead of $x$. Since you call this a "boundary value" rather than "initial condition", I assume that this $t$ is supposed to be the "time" variable. So either replace $u(t,x)$ by $u(x,t)$ in the PDE, or replace $u(0,t)$ by $u(t,0)$ in the boundary condition. $u(t,0)$.

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  • $\begingroup$ Thank you for your answer. I know it's the method of characteristics, I just don't know how to implement the boundary condition. $\endgroup$ – Sekots Reivan Feb 5 '15 at 5:40
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    $\begingroup$ I don't know why this was downvoted, it was posted before the OP edited his post to include the last sentence.. +1 from me. $\endgroup$ – Mattos Feb 5 '15 at 6:07
  • $\begingroup$ Thank you, Dr. Israel. I still have to look into that. My textbook does not have a similar example. $\endgroup$ – Sekots Reivan Feb 5 '15 at 6:58

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