Linear hyperbolic partial differential equation and system
A partial differential equation (or system) of the form
for which at any point of its domain of definition one can distinguish among the real variables (if necessary, after a suitable affine transformation of the independent variables) one variable, say , in a such a way that for all points of the -dimensional Euclidean space the characteristic equation (in )
has exactly real roots. Here is a vector with non-negative integer coordinates, is the order of the differential operator , , , is the order of the system (1), is a real square matrix of order , defined in , , , is an unknown column vector, and is a vector with components, defined in .
A typical example is the wave equation
Many problems in mathematical physics reduce to linear hyperbolic partial differential equations or systems of equations.
A subset is said to be characteristic at a point if and , where
is the characteristic form of the system (1). If , then one says that the vector defines a characteristic direction or characteristic normal at the point . The surface is called a characteristic surface (or characteristic) of the system (1) if
A surface which does not have characteristic normals at any point is called a free surface. On a free surface the rank of the characteristic matrix
is equal to , while on a characteristic surface it is less than . A characteristic is said to be simple if for some and all ,
Otherwise the characteristic is said to be multiple. A characteristic is sometimes said to be simple if the rank of the matrix is .
The system (1) is said to be hyperbolic at the point with respect to the hyperplane : if the matrix is non-singular (that is, the surface is free) and if all roots , , of the characteristic equation are real for all points . The system (1) is said to be hyperbolic in the domain with respect to if it is hyperbolic with respect to at every point .
An important class of hyperbolic equations and systems consists of strictly hyperbolic equations and systems, which are sometimes called fully hyperbolic systems, or systems, hyperbolic in the narrow sense. A system (1) is called a strictly hyperbolic system if all roots of the characteristic equation are distinct for any non-zero vector . The characteristics of a strictly hyperbolic equation (or system) are simple. Strictly hyperbolic (with respect to ) systems are notable for the fact that the Cauchy problem
for them is well-posed under the single assumption of sufficient smoothness of the coefficients , of the system (1) and of the initial data (initial functions) , . There are examples of hyperbolic but not strictly hyperbolic equations of the form (1) (even with constant coefficients in front of the derivatives of order ) for which the Cauchy problem is ill-posed.
The solution of the Cauchy problem (4) for the wave equation (3) can be written out explicitly, and when , and only then, it has the property that the value of at the vertex of the characteristic cone , , depends only on the value of the initial data and and their derivatives on the base , , of this cone (the so-called Huygens principle).
For strictly hyperbolic (with respect to ) equations and systems the question of the diffusion of waves and the related question of gaps have been investigated (cf. Lacuna). An exhaustive answer has been given in the case of an equation with constant coefficients of the form
For the case of one (scalar) equation with constant coefficients the definition of strict hyperbolicity has been generalized as follows. Equation (1) is said to be hyperbolic with respect to a non-zero vector if
and if there is a real number such that for all and ,
Of all linear equations with constant coefficients, only for equations that are hyperbolic in this sense the Cauchy problem is well-posed for arbitrary sufficiently smooth initial functions defined on the hyperplane
In particular, the wave equation (3) is hyperbolic in this sense with respect to any vector for which
There are various generalizations of the definition of strict hyperbolicity of equations and systems. These are mainly equations and systems that are completely characterized by the fact that the Cauchy problem with data on a free surface is uniquely solvable for them for any sufficiently smooth initial functions, without any restrictions on the growth at infinity.
Another important class of linear hyperbolic systems of the first order is the class of symmetric hyperbolic systems. The system
where , are square matrices of order , defined in , and is an unknown vector of components, is called a symmetric hyperbolic system in if the matrices are symmetric (or are symmetrizable simultaneously by the same transformation) and if at every point there is a spatially-oriented hyperplane (or, space-like hyperplane), that is, a hyperplane whose normal has the property that the matrix is positive definite. If for a symmetric hyperbolic system (5) with sufficiently smooth coefficients the given initial functions and the right-hand side have square-integrable generalized partial derivatives of order , then there is a unique generalized solution of the Cauchy problem with the same number of square-integrable partial derivatives. Any strictly hyperbolic partial differential equation of the second order reduces to a symmetric hyperbolic system.
An equation (1) of the second order in the class of solutions regular in a domain can be written in the form
where , , , and are functions defined in . Equation (6) is hyperbolic in if at every point of all eigen values of the matrix of leading coefficients , , are non-zero, and one of these eigen values differs in sign from all others. With respect to (6), along with the characteristic surface one can distinguish two types of smooth surfaces: spatially-oriented surfaces and time-oriented surfaces (also called space-like and time-like surfaces). If the surfaces are given by an equation of the form , then on a surface of the first type , while on a surface of the second type , where
The Cauchy problem for hyperbolic partial differential equations with initial data on a time-like surface is generally not well-posed.
A hyperbolic partial differential equation
is said to be uniformly (or regularly) hyperbolic in a domain if there is a positive number such that
for all in and for any non-zero vector . For the inequality
is a necessary and sufficient condition for (6) to be uniformly hyperbolic in . The equation for the vibration of a string,
is a typical representative of a linear uniformly hyperbolic partial differential equation of the second order with two independent variables. The general solution of this equation in any convex domain of the plane is given by the d'Alembert formula:
where and are arbitrary functions.
After a non-singular real change of variables and , the hyperbolic partial differential equation (6) with reduces to the normal (canonical) form
For hyperbolic systems written in the form (7), where , and are given real square matrices of order , is a given vector and an unknown vector, both with components, the question of the well-posedness of the Cauchy problem with initial data on a non-characteristic (free) curve and the Goursat problem with data on two intersecting characteristics have been completely investigated by the Riemann method.
The main problems about hyperbolic equations are the following: the Cauchy problem, the Cauchy characteristic problem and the mixed problem (see also Mixed and boundary value problems for hyperbolic equations and systems).
In the investigation of the main problems an important role is played by fundamental solutions, which make it possible to obtain explicit (integral) representations of regular and generalized solutions and to establish their structural and qualitative properties, in particular to study the question of wave fronts and the propagation of discontinuities.
Equation (6) is called an ultra-hyperbolic equation in a domain if at every point all eigen values of the matrix are non-zero and at least two of them differ in sign from all the others, of which there are at least two. An example of an ultra-hyperbolic equation is an equation of the form
which has the following property: If is a regular solution of (8) in a domain of the Euclidean space of the variables , and if is an arbitrary point of , then the mean value of the function calculated on the sphere with centre at the point and radius , is equal to the mean value of the function calculated on the sphere with centre at the point and the same radius . This theorem is extensively used in the theory of linear hyperbolic partial differential equations of the second order with constant coefficients.
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A fully hyperbolic system, or system hyperbolic in the narrow sense, is also called a totally hyperbolic system or a system hyperbolic in the sense of Petrovskii.
For the definition of hyperbolicity in the case of scalar equations with constant coefficients, see [a1], Vol. II, Chapt. 12.
For fundamental solutions and propagation of singularities, see also [a1], Vol. III, Chapts. 13, 14.
The theorem mentioned in the last line of the main article above is called Asgeirsson's theorem, cf. , Chapt. 6.
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Linear hyperbolic partial differential equation and system. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Linear_hyperbolic_partial_differential_equation_and_system&oldid=24497