Hyperbolic partial differential equation

A partial differential equation is hyperbolic at a point ${\displaystyle P}$ provided that the Cauchy problem is uniquely solvable in a neighborhood of ${\displaystyle P}$ for any initial data given on a non-characteristic hypersurface passing through ${\displaystyle P}$.[1] Here the prescribed initial data consist of all (transverse) derivatives of the function on the surface up to one less than the order of the differential equation.

By a linear change of variables, any equation of the form

$${\displaystyle A{\frac {\partial ^{2}u}{\partial x^{2}}}+2B{\frac {\partial ^{2}u}{\partial x\partial y}}+C{\frac {\partial ^{2}u}{\partial y^{2}}}+{\text{(lower order derivative terms)}}=0}$$

with

$${\displaystyle B^{2}-AC>0}$$

can be transformed to the wave equation, apart from lower order terms which are inessential for the qualitative understanding of the equation.[2]^: 400 This definition is analogous to the definition of a planar hyperbola.

The one-dimensional wave equation:

$${\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}-c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}=0}$$

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.[2]^: 402

The following is a system of ${\displaystyle s}$ first order partial differential equations for ${\displaystyle s}$ unknown functions ${\displaystyle {\vec {u}}=(u_{1},\ldots ,u_{s})}$, ${\displaystyle {\vec {u}}={\vec {u}}({\vec {x}},t)}$, where ${\displaystyle {\vec {x}}\in \mathbb {R} ^{d}}$:

$${\displaystyle {\frac {\partial {\vec {u}}}{\partial t}}+\sum _{j=1}^{d}{\frac {\partial }{\partial x_{j}}}{\vec {f}}^{j}({\vec {u}})=0,}$$

(∗)

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

Next, for each ${\displaystyle {\vec {f}}^{j}}$ define the ${\displaystyle s\times s}$ Jacobian matrix

$${\displaystyle A^{j}:={\begin{pmatrix}{\frac {\partial f_{1}^{j}}{\partial u_{1}}}&\cdots &{\frac {\partial f_{1}^{j}}{\partial u_{s}}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{s}^{j}}{\partial u_{1}}}&\cdots &{\frac {\partial f_{s}^{j}}{\partial u_{s}}}\end{pmatrix}},{\text{ for }}j=1,\ldots ,d.}$$

The system (∗) is hyperbolic if for all ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}\in \mathbb {R} }$ 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.

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 ${\displaystyle u=u({\vec {x}},t)}$. Then the system (∗) has the form

$${\displaystyle {\frac {\partial u}{\partial t}}+\sum _{j=1}^{d}{\frac {\partial }{\partial x_{j}}}{f^{j}}(u)=0.}$$

(∗∗)

Here, ${\displaystyle u}$ can be interpreted as a quantity that moves around according to the flux given by ${\displaystyle {\vec {f}}=(f^{1},\ldots ,f^{d})}$. To see that the quantity ${\displaystyle u}$ is conserved, integrate (∗∗) over a domain ${\displaystyle \Omega }$

$${\displaystyle \int _{\Omega }{\frac {\partial u}{\partial t}}\,d\Omega +\int _{\Omega }\nabla \cdot {\vec {f}}(u)\,d\Omega =0.}$$

If ${\displaystyle u}$ and ${\displaystyle {\vec {f}}}$ are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and ${\displaystyle \partial /\partial t}$ to get a conservation law for the quantity ${\displaystyle u}$ in the general form

$${\displaystyle {\frac {d}{dt}}\int _{\Omega }u\,d\Omega +\int _{\partial \Omega }{\vec {f}}(u)\cdot {\vec {n}}\,d\Gamma =0,}$$

which means that the time rate of change of ${\displaystyle u}$ in the domain ${\displaystyle \Omega }$ is equal to the net flux of ${\displaystyle u}$ through its boundary ${\displaystyle \partial \Omega }$. Since this is an equality, it can be concluded that ${\displaystyle u}$ is conserved within ${\displaystyle \Omega }$.

• Elliptic partial differential equation
• Hypoelliptic operator
• Parabolic partial differential equation

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

• "Hyperbolic partial differential equation, numerical methods", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
• Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.