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This is the transport equation.

Suppose x=x(t)\,. Then u=u(x(t),t)\,.

u_{t}=u_{x}x'(t)+u_{t}\, by the chain rule.

If u_{t}=u_{x}x'(t)+u_{t}=u_{t}+au_{x}=0\, then u\, is constant with respect to t\, and we have the equation x'(t)=a\, which implies x(t)=at+x_{0}\,.

Since u\, is constant with respect to t\,, u(x(t),t)=u(x(0),0)=f(x_{0})\,.

From above, x_{0}=x(t)-at=x-at\,.

The solution is u(x,t)=f(x-at)\,

Main Page : Partial Differential Equations : Method of Characteristics