Method of characteristics

Branko Ćurgus


A first order partial differential equation

Here we consider the following first order partial differential equation: \begin{equation} \label{eq:pde} A(x,y,u) \, u_x(x,y) + B(x,y,u) \, u_y(x,y) = C(x,y,u) \end{equation} subject to the initial condition \begin{equation} \label{eq:ic} u(x,0) = f(x), \quad x \in \mathbb R. \end{equation} For simplicity we assume that the given functions $A$, $B$ and $C$ are smooth functions defined on ${\mathbb R}^3$ with the values in $\mathbb R$ and that the given function $f$ is a smooth function defined on $\mathbb R$ with values in $\mathbb R$.
A geometric interpretation of \eqref{eq:pde}


The related system of ordinary differential equations


Mathematica notebooks


Example 1

Consider the following first order PDE \begin{equation*} y \, u_x + 3 \, u_y = -u \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = \cos(x), \quad \text{for} \quad x \in \mathbb R. \end{equation*}
Example 2

Consider the following first order PDE \begin{equation*} 2\, u_x + u_y = u^2 \quad \text{in} \quad U \subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = \cos(x), \quad x \in \mathbb R. \end{equation*} Notice that the domain $U$ above is not specified. As a part of your solution you should determine the largest "rectangular" box $U$ (whose boundary is the $x$-axis) on which the problem has a solution.
Example 3

Consider the following first order partial differential equation \begin{equation*} u_t + u\, u_x = 0 \quad \text{in} \quad U \subseteq \bigl\{(x,t) \in \mathbb R^2 : t \gt 0 \bigr\} \end{equation*} subject to the initial condition \begin{equation*} u(x,0) = f(x), \quad x \in \mathbb R. \end{equation*} You should consider the following three specific functions $f(x) = \arctan x$, $f(x) = -\arctan x$ and $f(x) = \exp(-x^2)$.
Notice that the domain $U$ above is not specified. As a part of your solution, for each specific $f$ you should determine the largest "rectangular" box $U$ (whose boundary is the $x$-axis) on which the problem has a solution.
Problems

  1. Solve the following first order PDE \begin{equation*} y\, u_x + u_y = 0 \quad \text{in} \quad \mathbb R^2 \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \gt 0. \end{equation*} You can try $f(x) = x$, $f(x) = x^2$, $f(x) = \cos x$ as some specific examples.
  2. Solve the following first order PDE \begin{equation*} u_x - x u_y = 0 \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \gt 0. \end{equation*} You can try $f(x) = x$, $f(x) = x^2$, $f(x) = \cos x$ as some specific examples.
  3. Solve the following first order PDE \begin{equation*} y\, u_x - x u_y = 0 \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \gt 0. \end{equation*} You can try $f(x) = \sin x$, $f(x) = \cos x$, $f(x) = \exp(-x^2)$ as some specific examples.
  4. Solve the following first order PDE \begin{equation*} u_x + x u_y = u \quad \text{in} \quad \mathbb R^2 \end{equation*} subject to the boundary condition \begin{equation*} u(0,y) = g(y) \quad \text{for} \quad x \in \mathbb R. \end{equation*} You can try $g(y) = y$, $g(y) = \cos y$ as some specific examples.
  5. Solve the following first order PDE \begin{equation*} y\, u_x - x u_y = u \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \geq 0. \end{equation*} You can try $f(x) = \sin x$, $f(x) = \cos x$ as some specific examples.
  6. Solve the following first order PDE \begin{equation*} x u_x + y\, u_y = u \quad \text{in} \quad \mathbb R^2 \end{equation*} subject to the condition \begin{equation*} u\bigl(\cos \theta, \sin \theta\bigr) = 1 \quad \text{for} \quad 0 \leq \theta \lt 2 \pi. \end{equation*}
  7. Solve the following first order PDE \begin{equation*} y u_x + u_y = x \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = x^2 \quad \text{for} \quad x \in \mathbb R. \end{equation*}
  8. Solve the following first order PDE \begin{equation*} x^2 u_x + u_y = 0 \quad \text{in} \quad U \subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \in \mathbb R. \end{equation*} Notice that the domain $U$ is not given in the problem.
    1. For an arbitrary differentiable function $f$ find the solution $u(x,y)$ of the above problem and determine its maximum domain $U\subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}$.
    2. Under which condition on $f$ the solution $u(x,y)$ will be defined on $\bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}$.
  9. Consider the following first order PDE \begin{equation*} u_x - u\, u_y = 0 \quad \text{in} \quad \boxed{U \subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}} \end{equation*} subject to the boundary condition \begin{equation*} u (x,0) = x \quad \text{for} \quad \boxed{x \in \mathbb R}. \end{equation*}
    1. Does there exist an open set $U \subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}$ whose boundary is the $x$-axis such that the above problem has a solution on $U$?
    2. Choose smaller specific sets in the boxed formulas in such a way that the corresponding modified problem has a solution. Prove your claim by providing a solution of the modified problem.
  10. Solve the following first order PDE \begin{equation*} (1+x^2) u_x + 2 x y \, u_y = 0 \quad \text{in} \quad {\mathbb R}^2 \end{equation*} subject to the condition \begin{equation*} u(0,y) = g(y) \quad \text{for all} \quad x \in \mathbb R. \end{equation*}
  11. Solve the following first order PDE \begin{equation*} (1+x^2) u_x - 2 x y \, u_y = 0 \quad \text{in} \quad {\mathbb R}^2 \end{equation*} subject to the condition \begin{equation*} u(0,y) = g(y) \quad \text{for all} \quad x \in \mathbb R. \end{equation*}
  12. Solve the following first order PDE \begin{equation*} x \, u_x + y \, u_y = 2 u \ln u \quad \text{in} \quad {\mathbb R}^2 \end{equation*} subject to the condition \begin{equation*} u(x, 1) = e^{x^2-1} \quad \text{for all} \quad x \in \mathbb R. \end{equation*}