In the
mathematical field of
differential equations, the ultrahyperbolic equation is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
(PDE) for an unknown
scalar function of
variables of the form
More generally, if is any
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
in variables with
signature , then any PDE whose
principal part is
is said to be ultrahyperbolic. Any such equation can be put in the form above by means of a change of variables.
[See Courant and Hilbert.]
The ultrahyperbolic equation has been studied from a number of viewpoints. On the one hand, it resembles the classical
wave equation. This has led to a number of developments concerning its
characteristics, one of which is due to
Fritz John: the
John equation.
In 2008, Walter Craig and Steven Weinstein proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one
hypersurface. And later, in 2022, a research team at the
University of Michigan extended the conditions for solving ultrahyperbolic wave equations to
complex-time (kime), demonstrated space-kime dynamics, and showed data science applications using tensor-based linear modeling of
functional magnetic resonance imaging
Functional magnetic resonance imaging or functional MRI (fMRI) measures brain activity by detecting changes associated with blood flow. This technique relies on the fact that cerebral blood flow and neuronal activation are coupled. When an area o ...
data.
The equation has also been studied from the point of view of
symmetric spaces, and
elliptic differential operators.
In particular, the ultrahyperbolic equation satisfies an analog of the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
for
harmonic functions.
Notes
References
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Differential operators
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