PDEMOC2

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u_{t}+uu_{x}=0,u(x,0)=x\, This is Burgers' equation.


Let x=x(t)\, so u=u(x(t),t)\,.


u_{t}=u_{x}x'(t)+u_{t}=u_{t}+uu_{x}=0\, so u\, is constant with respect to t\, and

u(x(t),t)=u(x(0),0)=x_{0}\,

x'(t)=u\,

x(t)=ut+x_{0}\,


Since u(x,t)=x-ut\,, the solution is u(x,t)={\frac  {x}{1+t}},t>-1\,.


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