Share This Article:

Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

Abstract Full-Text HTML Download Download as PDF (Size:766KB) PP. 63-103
DOI: 10.4236/apm.2012.22013    6,580 Downloads   12,365 Views   Citations

ABSTRACT

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. Ludkovsky, "Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE," Advances in Pure Mathematics, Vol. 2 No. 2, 2012, pp. 63-103. doi: 10.4236/apm.2012.22013.

References

[1] V. S. Vladimirov, “Equations of Mathematical Physics,” Nauka, Moscow, 1971.
[2] M. A. Lavretjev and B. V. Shabat, “Methods of Functions of the Complex Variable,” Nauka, Moscow, 1987.
[3] B. van der Pol and H. Bremmer, “Operational Calculus Based on the Two-Sided Laplace Integral,” Cambridge University Press, Cambridge, 1964.
[4] S. V. Ludkovsky, “The Two-Sided Laplace Transformation over the Cayley-Dickson Algebras and Its Applications,” Journal of Mathematical Sciences, Vol. 151, No. 5, 2008, pp. 3372-3430. doi:10.1007/s10958-008-9038-y
[5] S.V. Lüdkovsky and F. van Oystaeyen, “Differentiable Functions of Quaternion Variables,” Bulletin des Sciences Mathématiques, Vol. 127, No. 9, 2003, pp. 755-796. doi:10.1016/S0007-4497(03)00063-0
[6] S. V. Ludkovsky, “Differentiable Functions of Cayley- Dickson Numbers and Line Integration,” Journal of Mathematical Sciences, Vol. 141, No. 3, 2007, pp. 1231-1298. doi:10.1007/s10958-007-0042-4
[7] W. R. Hamilton, “Selected Works. Optics. Dynamics. Quaternions,” Nauka, Moscow, 1994.
[8] J. C. Baez, “The Octonions,” Bulletin of the American Mathematical Society, Vol. 39, No. 2, 2002, pp. 145-205. doi:10.1090/S0273-0979-01-00934-X
[9] I. L. Kantor and A. S. Solodovnikov, “Hypercomplex Numbers,” Springer-Verlag, Berlin, 1989. doi:10.1007/978-1-4612-3650-4
[10] A. G. Kurosh, “Letures on General Algebra,” Nauka, Mos- cow, 1973.
[11] H. Rothe, “Systeme Geometrischer Analyse,” In: Encyklop?die der Mathematischen Wissenschaften, Geometrie, Teubner, Leipzig, Vol. 3, 1914-1931, pp. 1277-1423.
[12] G. Emch, “Méchanique Quantique Quaternionienne et Relativit’èrestreinte,” Helvetica Physica Acta, Vol. 36, 1963, pp. 739-788.
[13] F. Gürsey and C.-H. Tze, “On the Role of Division, Jordan and Related Algebras in Particle Physics,” World Scientific Publishing Co., Singapore, 1996.
[14] H. B. Lawson and M.-L. Michelson, “Spin Geometry,” Princeton University Press, Princeton, 1989.
[15] M. A. Solovjev, “A Structure of a Space of Non-Abelian Gauge Fields,” Proceeding of Lebedev Physical Institute, No. 210, 1993, pp. 112-155.
[16] S. V. Ludkovsky, “Differential Equations over Octonions,” Advances in Applied Clifford Algebras, Vol. 21, No. 4, 2011, pp. 773-797. doi:10.1007/s00006-011-0286-4
[17] E. H. Spanier, “Algebraic Topology,” Academic Press, New York, 1966.
[18] L. I. Kamynin, “Course of Mathematical Analysis,” Moscow State University Press, Moscow, 1993.
[19] V. A. Zorich, “Mathematical Analysis,” Nauka, Moscow, Vol. 2, 1984.
[20] I. Rubinstein and L. Rubinstein, “Partial Differential Equations in Classical Mathematical Physics,” Cambridge University Press, Cambridge, 1998.
[21] I. M. Gelfand and G. E. Shilov, “Generalized Functions and Operations with Them,” Fiziko-Mathematicheskaya Literatura, Moscow, 1958.
[22] S. V. Ludkovsky, “Feynman Integration over Octonions with Application to Quantum Mechanics,” Mathematical Methods in the Applied Sciences, Vol. 33, No. 9, 2010, pp. 1148-1173.
[23] S. V. Ludkovsky and W. Sproessig, “Ordered Representations of Normal and Super-Differential Operators in Quaternion and Octonion Hilbert Spaces,” Advances in Applied Clifford Algebras, Vol. 20, No. 2, 2010, pp. 321-342. doi:10.1007/s00006-009-0180-5
[24] S. V. Ludkovsky, “Algebras of Operators in Banach Spaces over the Quaternion Skew Field and the Octonion Algebra,” Journal of Mathematical Sciences, Vol. 144, No. 4, 2008, pp. 4301-4366. doi:10.1007/s10958-007-0273-4
[25] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, “Vector-Valued Laplace Transforms and Cauchy Problems,” Birkh?üuser, Basel, 2001.
[26] L. Berg, “Einfürung in Die Operatorenrechnung,” VEB Deutscher Verlag der Wissenschaften, Berlin, 1965.
[27] U. Graf, “Applied Laplace Transform for Scientists and Engineers,” Birkh?user, Basel, 2004. doi:10.1007/978-3-0348-7846-3
[28] J. Leray, “Un Prologement de la Transformation de Laplace qui Transforme la Solution Unitaire d’un op e’rateur Hyperbolique en sa Solution e’L?mentaire,” Bulletin de la Société Mathématique de France, Vol. 90, 1962, pp. 39-156.
[29] S. V. Ludkovsky, “Functions of Several Cayley-Dickson Variables and Manifolds Over them,” Journal of Mathematical Sciences, Vol. 141, No. 3, 2007, pp. 1299-1330. doi:10.1007/s10958-007-0043-3
[30] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, “Integrals and Series,” Nauka, Moscow, 1981.

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.