Solve a first order partial differential equation with the boundary condition $u(x,x+x^2)=\sin(x)$ instead of a initial value.

Question

The problem I want to solve is the following one: $$ \begin{cases} u_x + u_t + u = 1,\\ u(x,x+x^2)=\sin(x). \end{cases} $$ I know how to use method of characteristic only if a initial value condition is given in $t=0$: loosely speaking I know how to deal with boundary conditions in the form $u_0(x)=u(x,0)$. What is the solution in the above case? Thanks!

Answer 1

The characteristic curves satisfy the ODEs \begin{align} \dot{x}(s)&=1, \tag{1} \\ \dot{t}(s)&=1, \tag{2} \\ \dot{u}(s)&=1-u. \tag{3} \end{align} Solving them, with initial conditions $x(0)=x_0, t(0)=t_0,$ and $u(0)=u_0$, we obtain \begin{align} x&=x_0+s, \tag{4} \\ t&=t_0+s, \tag{5} \\ u&=1+(u_0-1)e^{-s}. \tag{6} \end{align} The boundary condition $u(x,x+x^2)=\sin(x)$ is equivalent to \begin{align} t_0&=x_0+x_0^2, \tag{7} \\ u_0&=\sin(x_0). \tag{8} \end{align} Plugging $(7)$ into $(5)$ and solving $(4)$-$(5)$ for $x_0$ and $s$, we obtain \begin{align} x_0&=\sqrt{t-x}, \tag{9} \\ s&=x-\sqrt{t-x}. \tag{10} \end{align} Finally, plugging $(8)$-$(10)$ into $(6)$, we obtain $$ u(x,t)=1+e^{-x+\sqrt{t-x}}\left(\sin(\sqrt{t-x})-1\right). \tag{11} $$