Abstract
We propose a generalization of Laplace transformations to the case of linear partial differential operators (LPDOs) of arbitrary order in ℝn. Practically all previously proposed differential transformations of LPDOs are particular cases of this transformation (intertwining Laplace transformation, \(\mathcal{ILT}\)). We give a practical procedure of construction of \(\mathcal{ILT}\) and describe the classes of operators in ℝn suitable for this transformation.
This paper was written with partial financial support from the KSPU grant “Forming scientific collective Physics of nano- and microstructures”.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Darboux, G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. t. 2. Gautier-Villard, Paris (1887–1896)
Goursat, E.: Leçons sur l’intégration des équations aux dérivées partielles du seconde ordre a deux variables indépendants. t. 2. Gautier-Villard, Paris (1898)
Kolchin, E.R.: Differential algebra and algebraic groups. Academic Press, NY (1973)
Kaptsov, O.V.: Methods of integration of partial differential equations. Fizmatlit, Moscow (2009) (in Russian)
Ganzha, E.I., Loginov, V.M., Tsarev, S.P.: Exact solutions of hyperbolic systems of kinetic equations. Application to Verhulst Model with Random Perturbation. Mathematics of Computation 1, 459–472 (2008)
Ganzha, E.I.: On Laplace and Dini transformations for multidimensional equations with a decomposable principal symbol. Programming and Computer Software 38, 150–155 (2012)
Ore, O.: Linear equations in non-commutative fields. Ann. Math. 32, 463–477 (1931)
Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34, 480–508 (1933)
Tsarev, S.P.: An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator. In: Lakshman, Y.N. (ed.) Proc. ISSAC 1996, pp. 226–231. ACM Press (1996)
Riquier, C.: Les systèmes d’equations aux derivées partielles. Gautier-Villard, Paris (1910)
Janet, M.: Leçons sur les systèmes d’equations aux derivées partielles. Gautier-Villard, Paris (1929)
Schwarz, F.: The Riquier-Janet theory and its applications to nonlinear evolution equations. Physica D 11, 243–351 (1984)
Shemyakova, E.: Laplace transformations as the only degenerate Darboux transformations of first order. Programming and Computer Software 38, 105–108 (2012)
Shemyakova, E.: Factorization of Darboux Transformations of Arbitrary Order for Two-dimensional Schrödinger operator (2013), http://arxiv.org/abs/1304.7063
Tsarev, S.P., Shemyakova, E.: Differential transformations of parabolic second-order operators in the plane. Proc. Steklov Math. Inst. 266, 219–227 (2009)
Euler, L.: Institutionum calculi integralis. V. III, Ac. Sc. Petropoli, St. Petersburg (1770)
Le Roux, J.: Extensions de la méthode de Laplace aux équations linéaires aux derivées partielles d’ordre supérieur au second. Bull. Soc. Math. France. 27, 237–262 (1899)
Petrén, L.: Extension de la méthode de Laplace aux équations \(\sum_{i=0}^{n-1}A_{1i}\frac{\partial^{i+1}z}{\partial x\partial y^i} + \sum_{i=0}^{n}A_{0i}\frac{\partial^{i}z}{\partial y^i} = 0\). Lund Univ. Arsskrift. 7, 1–166 (1911)
Tsarev, S.P.: On factorization and solution of multidimensional linear partial differential equations. In: Kotsireas, I., Zima, E. (eds.) COMPUTER ALGEBRA 2006. Latest Advances in Symbolic Algorithms, Proc. Waterloo Workshop in Computer Algebra, Canada, April 10-12, pp. 181–192. World Scientific (2007), http://arxiv.org/abs/cs/0609075
Tsarev, S.P.: Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations. In: Labahn, G. (ed.) Proc. ISSAC 2005, pp. 325–331. ACM Press (2005), http://arxiv.org/abs/cs/0501030
Athorne, C.: A Z 2 ×R 3 Toda system. Phys. Lett. A. 206, 162–166 (1995)
Tsarev, S.P.: Factorization of linear differential operators and systems. In: MacCallum, M.A.H., Mikhailov, A.V. (eds.) Algebraic Theory of Differential Equations. LMS. Lecture Note Series, vol. 357, pp. 111–131 (2009), http://arxiv.org/abs/0801.1341
Taimanov, I.A., Tsarev, S.P.: The Moutard transformation: An algebraic formalism via pseudodifferential operators and applications (2009), http://arxiv.org/abs/0906.5141
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ganzha, E.I. (2014). Intertwining Laplace Transformations of Linear Partial Differential Equations. In: Barkatou, M., Cluzeau, T., Regensburger, G., Rosenkranz, M. (eds) Algebraic and Algorithmic Aspects of Differential and Integral Operators. AADIOS 2012. Lecture Notes in Computer Science, vol 8372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54479-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-54479-8_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54478-1
Online ISBN: 978-3-642-54479-8
eBook Packages: Computer ScienceComputer Science (R0)