Ultrahyperbolic equation
In the mathematical field of differential equations, the ultrahyperbolic equation is a partial differential equation (PDE) for an unknown scalar function u of 2n variables x1, ..., xn, y1, ..., yn of the form
[math]\displaystyle{ \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} - \frac{\partial^2 u}{\partial y_1^2} - \cdots - \frac{\partial^2 u}{\partial y_n^2} = 0. }[/math]
More generally, if a is any quadratic form in 2n variables with signature (n, n), then any PDE whose principal part is [math]\displaystyle{ a_{ij}u_{x_ix_j} }[/math] is said to be ultrahyperbolic. Any such equation can be put in the form above by means of a change of variables.[1]
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.[2] 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 data. [3][4]
The equation has also been studied from the point of view of symmetric spaces, and elliptic differential operators.[5] In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions.
Notes
- ↑ See Courant and Hilbert.
- ↑ Craig, Walter; Weinstein, Steven. "On determinism and well-posedness in multiple time dimensions". Proc. R. Soc. A vol. 465 no. 2110 3023-3046 (2008). http://rspa.royalsocietypublishing.org/content/465/2110/3023.
- ↑ Wang, Y; Shen, Y; Deng, D; Dinov, ID (2022). "Determinism, Well-posedness, and Applications of the Ultrahyperbolic Wave Equation in Spacekime". Partial Differential Equations in Applied Mathematics (Elsevier) 5 (100280): 100280. doi:10.1016/j.padiff.2022.100280. PMID 36159725.
- ↑ Zhang, R; Zhang, Y; Liu, Y; Guo, Y; Shen, Y; Deng, D; Qiu, Y; Dinov, ID (2022). "Kimesurface Representation and Tensor Linear Modeling of Longitudinal Data". Partial Differential Equations in Applied Mathematics (Springer) 34 (8): 6377–6396. doi:10.1007/s00521-021-06789-8. PMID 35936508. PMC 9355340. https://doi.org/10.1007/s00521-021-06789-8.
- ↑ Helgason, S (1959). "Differential operators on homogeneous spaces". Acta Mathematica (Institut Mittag-Leffler) 102 (3–4): 239–299. doi:10.1007/BF02564248.
References
- Richard Courant; David Hilbert (1962). Methods of Mathematical Physics, Vol. 2. Wiley-Interscience. pp. 744–752. ISBN 978-0-471-50439-9.
- Lars Hörmander (20 August 2001). "Asgeirsson's Mean Value Theorem and Related Identities". Journal of Functional Analysis 2 (184): 377–401. doi:10.1006/jfan.2001.3743.
- Lars Hörmander (1990). The Analysis of Linear Partial Differential Operators I. Springer-Verlag. Theorem 7.3.4. ISBN 978-3-540-52343-7.
- Sigurdur Helgason (2000). Groups and Geometric Analysis. American Mathematical Society. pp. 319–323. ISBN 978-0-8218-2673-7.
- Fritz John (1938). "The Ultrahyperbolic Differential Equation with Four Independent Variables". Duke Math. J. 4 (2): 300–322. doi:10.1215/S0012-7094-38-00423-5.
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