In my PDE class, we are covering the method of characteristics. I have encountered the following problem
equations of the form $u_t + G(u_x,u,x,t) = 0$ can be solved by the method of characteristics, where $G(p,z,x,t)$ is a scalar function of 4 variables. The ODEs for $x(t)$,$p(t)=u_x(x(t),t)$, $q(t)=u_t(x(t),t)$, and $z(t)=u(x(t),t)$ are \begin{equation} \begin{aligned} \dot x & = G_p(p,z,x,t), \\ \dot z &= p\, G_p + q, \end{aligned} \qquad \begin{aligned} \dot p = -G_x – p\, G_z, \\ \dot q = -G_t – q\, G_z. \end{aligned}\end{equation}
We are asked to solve the equation $u_t + u/u_x = 0$ with initial conditions $u(x,0)=x^2/2$ using the method of characteristics. (Hint: for this problem, the ODE's can be solved one at a time, first for $p$, then $q$, then $z$, and finally $x$. For example, the solution for $p(t)$ is $x_0-t$, where $x_0$ is the initial location of the characteristiccurve. A formula for $u(x,t)$ is then easily derived from $x(t)$ and$z(t)$).
I am a novice in differential equations and I do not really know how to solve this type of equation using the method of characteristics. I cannot imagine how to extract the solution from the ODEs above. I am not quite certain how to proceed. May I please ask someone to help me solve this? I thank all helpers.
Best Answer
So you have $$ \frac{dx}{-z/p^2}=\frac{dt}{1}=\frac{dz}{q-z/p}=-\frac{dp}{1}=-\frac{dq}{q/p} $$ leading directly to $$p+t=p_0,~ x+q=x_0+q_0,~ q/p=q_0/p_0,~ z/p^2=z_0/p_0^2,$$ and then in combination $x+z_0/p_0^2t=x_0$.
The PDE at $t=0$ gives $q_0+z_0/p_0=0$. The initial condition evaluates to $z_0=x_0^2/2$, $p_0=u_x(x_0,0)=x_0$, $q_0=-z_0/p_0=-x_0/2$. This simplifies the equations for the characteristic so far to $$ p+t=x_0,~q+x=\tfrac12x_0,~ q/p=-\tfrac12,~z/p^2=\tfrac12,~x+\tfrac12t=x_0 $$ The solution tangent plane equation gives $$ dz = p\,dx+q\,dt = -\tfrac12p\,dt-\tfrac12p\,dt=-(x_0-t)\,dt \\~\\ z=z_0-x_0t+\tfrac12t^2=\tfrac12(x_0-t)^2=\tfrac12(x-\tfrac12t)^2 $$
Often one can condense down such exercise solutions to a much narrower set of identities.