PDEMOC1

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u_t + a u_x = 0, u(x,0)=f(x)\,

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 + a u_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

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