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Solve Burgers' equation $$u_t + uu_x =0,$$ with $u=u(x,t)$ and the side condition $u(x,0)=x$.

I am not sure on how to find the solution $u(x,t)$. I have learned the method of characteristics. I am neither sure on how to use the side condition in Burgers' equation.

EditPiAf
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Sapna Kumar
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1 Answers1

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This PDE is called the (inviscid) Burgers' equation. The characteristic equations are the system of ODEs $${dx\over dt}=u, \quad {du\over dt}=0,$$ which has solution $$u=C_1, \quad x=ut+C_2,$$ where $C_1,C_2$ are constants. But $C_1$ is a function of $C_2$, so $$u=C_1\implies u=C_1(C_2)=C_1(x-ut).$$ To determine $C_1$, note that the side condition requires $$u(x,0)=C_1(x-u(x,0)\cdot 0)=x\implies C_1(x)=x.$$ Thus, $u=x-ut$ and we conclude $$u(x,t)={x\over 1+t}.$$

JohnD
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    I saw and answered the newer version of this same question here first. Didn't know if I should attempt to close this one as a duplicate since it came first. I defer to others on the board as to how to handle this. – JohnD Feb 18 '13 at 02:33
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    I voted to close the newer question as a duplicate of this one. You should feel free to vote to close as well, since you have 3K. –  Feb 18 '13 at 02:48
  • I was wondering how you can deal with this solution at t=-1? My lecturer suggested introducing a shock, but I have only met shocks in the context of data curves with discontinuities, and am not sure how to apply that here... Could you help? – user3709 Oct 21 '21 at 14:42