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I am trying to solve this boundary-value problem:

$$u_x(x,y) + u_y(x,y) + u(x,y) = 0$$ $$u(0,y) = 1$$ $$u(x,0) = 1$$

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I tried to use the method of characteristics, but it seems that it is only for initial-value problems. I would thank any help

By using the method of characteristics I solve the corresponding ODEs system and get the general solution

u(x,y) = f(x-y)exp(-x)

but, when I try to obtain f for one boundary condition, the answer don't match with the other boundary condition. I don't know what is really happening

Iram Ortiz
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1 Answers1

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On a line $x-y = c$, your pde says $u(x,x-c) e^{x}$ is constant. If that line intersects both of your boundaries and the values you get there are not consistent with each other, it says your boundary value problem has no solution.

Robert Israel
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  • The line can intersect one boundary or another, not both, and depending of C. In this case, what can I say about the solution? And how can I express it? – Iram Ortiz May 05 '20 at 20:40
  • If you're interested only in $x, y \ge 0$ (which you neglected to mention) then yes, the line $x-y=c$ will intersect only one boundary and there is no inconsistency. So what's the problem? Your solution will be expressed in "piecewise" fashion: if $x-y \ge 0$ then $\ldots$ else $\ldots$. – Robert Israel May 06 '20 at 14:28
  • You will have a lack of differentiability on the line $x=y$, though. – Robert Israel May 06 '20 at 14:33
  • Thank you very much. I didn't think in express the solution in piecewise fashion, I wanted to express it be a continuous function. – Iram Ortiz May 06 '20 at 16:23