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Inmathematics, ahyperbolic partial differential equation of order${\displaystyle n}$ is apartial differential equation (PDE) that, roughly speaking, has a well-posedinitial value problem for the first${\displaystyle n-1}$ derivatives.^{[citation needed]} More precisely, theCauchy problem can be locally solved for arbitrary initial data along any non-characteristichypersurface. Many of the equations ofmechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is thewave equation. In one spatial dimension, this is$${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$$The equation has the property that, ifu and its first time derivative are arbitrarily specified initial data on the linet = 0 (with sufficient smoothness properties), then there exists a solution for all timet.

The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finitepropagation speed. They travel along thecharacteristics of the equation. This feature qualitatively distinguishes hyperbolic equations fromelliptic partial differential equations andparabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain.

Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration. There is a well-developed theory for lineardifferential operators, due toLars Gårding, in the context ofmicrolocal analysis. Nonlinear differential equations are hyperbolic if their linearizations are hyperbolic in the sense of Gårding. There is a somewhat different theory for first order systems of equations coming from systems ofconservation laws.

A partial differential equation is hyperbolic at a point${\displaystyle P}$ provided that theCauchy problem is uniquely solvable in a neighborhood of${\displaystyle P}$ for any initial data given on anon-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$${\displaystyle A\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+2B\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}+C\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\text{(lower order derivative terms)}=0}$$withdiscriminant$${\displaystyle B^2-AC>0}$$can be transformed to thewave 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 planarhyperbola.

The one-dimensionalwave 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.^[2]: 402

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

$${\displaystyle \frac{\mathrm{\partial }\overrightarrow{u}}{\mathrm{\partial }t}+\sum _{j=1}^d\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{\overrightarrow{f}}^j(\overrightarrow{u})=0,}$$

∗

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

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

The system (∗) ishyperbolic 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 onlyrealeigenvalues and isdiagonalizable.

If the matrix${\displaystyle A}$ hassdistinct real eigenvalues, it follows that it is diagonalizable. In this case the system (∗) is calledstrictly 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 calledsymmetric hyperbolic.

There is a connection between a hyperbolic system and aconservation 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

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

∗∗

Here,${\displaystyle u}$ can be interpreted as a quantity that moves around according to theflux 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 thedivergence 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${\displaystyle \mathrm{\Omega }}$.

• 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.

• Hyperbolic partial differential equations

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