Hyperbolic partial differential equation

Examples

By a linear change of variables, any equation of the form $${\displaystyle A\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+2B\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}+C\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\text{(lower order derivative terms)}=0}$$ with discriminant $${\displaystyle B^2-AC>0}$$ can be transformed to the wave equation, apart from lower order terms which are inessential for the qualitative understanding of the equation. This definition is analogous to the definition of a planar hyperbola.

The one-dimensional wave equation: $${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}-c^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}=0}$$ is an example of a hyperbolic equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE. This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.

Hyperbolic systems of first-order equations

The following is a system of first-order partial differential equations for ${\displaystyle s}$ unknown functions ${\displaystyle \overrightarrow{u}=(u_1,\dots ,u_s)}$, ${\displaystyle \overrightarrow{u}=\overrightarrow{u}(\overrightarrow{x},t)}$, where ${\displaystyle \overrightarrow{x}\in {\mathbb{R}}^d}$:

where ${\displaystyle {\overrightarrow{f}}^j\in C^1({\mathbb{R}}^s,{\mathbb{R}}^s)}$ are once continuously differentiable functions, nonlinear in general.

Next, for each ${\displaystyle {\overrightarrow{f}}^j}$ define the ${\displaystyle s\times s}$ Jacobian matrix $${\displaystyle A^j:=\left(\begin{array}{ccc}\frac{\mathrm{\partial }{f}_1^j}{\mathrm{\partial }{u}_1}& \cdots & \frac{\mathrm{\partial }{f}_1^j}{\mathrm{\partial }{u}_s}\\ ⋮& \ddots & ⋮\\ \frac{\mathrm{\partial }{f}_s^j}{\mathrm{\partial }{u}_1}& \cdots & \frac{\mathrm{\partial }{f}_s^j}{\mathrm{\partial }{u}_s}\end{array}\right),\text{ for }j=1,\dots ,d.}$$

The system (∗) is hyperbolic if for all ${\displaystyle {\alpha }_1,\dots ,{\alpha }_d\in \mathbb{R}}$ the matrix ${\displaystyle A:={\alpha }_1{A}^1+\cdots +{\alpha }_d{A}^d}$ has only real eigenvalues and is diagonalizable.

If the matrix ${\displaystyle A}$ has s distinct real eigenvalues, it follows that it is diagonalizable. In this case the system (∗) is called strictly hyperbolic.

If the matrix ${\displaystyle A}$ is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system (∗) is called symmetric hyperbolic.

Hyperbolic system and conservation laws

There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function ${\displaystyle u=u(\overrightarrow{x},t)}$. Then the system (∗) has the form

Here, ${\displaystyle u}$ can be interpreted as a quantity that moves around according to the flux given by ${\displaystyle \overrightarrow{f}=(f^1,\dots ,f^d)}$. To see that the quantity ${\displaystyle u}$ is conserved, integrate (∗∗) over a domain ${\displaystyle \mathrm{\Omega }}$ $${\displaystyle {\int }_{\mathrm{\Omega }}\frac{\mathrm{\partial }u}{\mathrm{\partial }t}d\mathrm{\Omega }+{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \overrightarrow{f}(u)d\mathrm{\Omega }=0.}$$

If ${\displaystyle u}$ and ${\displaystyle \overrightarrow{f}}$ are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and ${\displaystyle \mathrm{\partial }/\mathrm{\partial }t}$ to get a conservation law for the quantity ${\displaystyle u}$ in the general form $${\displaystyle \frac{d}{dt}{\int }_{\mathrm{\Omega }}ud\mathrm{\Omega }+{\int }_{\mathrm{\partial }\mathrm{\Omega }}\overrightarrow{f}(u)\cdot \overrightarrow{n}d\mathrm{\Gamma }=0,}$$ which means that the time rate of change of ${\displaystyle u}$ in the domain ${\displaystyle \mathrm{\Omega }}$ is equal to the net flux of ${\displaystyle u}$ through its boundary ${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$. Since this is an equality, it can be concluded that ${\displaystyle u}$ is conserved within ${\displaystyle \mathrm{\Omega }}$.