Programming and Computer Software

, Volume 38, Issue 3, pp 150–155

On laplace and Dini transformations for multidimensional equations with a decomposable principal symbol

Article

Abstract

Algorithms for solving linear PDEs implemented in modern computer algebra systems are usually limited to equations with two independent variables. In this paper, we propose a generalization of the theory of Laplace transformations to second-order partial differential operators in ℝ3 (and, generally, ℝ n ) with a principal symbol decomposable into the product of two linear (with respect to derivatives) factors. We consider two algorithms of generalized Laplace transformations and describe classes of operators in ℝ3 to which these algorithms are applicable. We correct a mistake in [8] and show that Dini-type transformations are in fact generalized Laplace transformations for operators with coefficients in a skew (noncommutative) Ore field. Keywords: computer algebra, partial differential equations, algorithms for solution.

Keywords

Differential Operator Laplace Transformation Partial Differential Operator Principal Symbol Order Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Darboux, G., Le-cons sur la the-orie ge-ne-rale des surfaces et les applications ge-ome-triques du calcul infinite-simal, Paris: Gauthier-Villars, 1889, vol. 2.Google Scholar
2. 2.
Zhiber, A.V. and Startsev, S.Ya., Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems, Math. Notes, 2003, vol. 74, nos. 5–6, pp. 803–811.
3. 3.
Athorne, C.A., A Z 2 × R 3 Toda System, Phys. Lett. A, 1995, vol. 206, pp. 162–166.
4. 4.
Petrén, L., Extension de la méthode de Laplace aux équations $$\sum\nolimits_{i = 0}^{n - 1} {A_{1i} \frac{{\partial ^{i + 1} z}} {{\partial x\partial y^i }}} + \sum\nolimits_{i = 0}^n {A_{0i} \frac{{\partial ^i z}} {{\partial y^i }}}$$, Lund Univ. Arsskrift, 1911, vol. 7, no. 3, pp. 1–166.Google Scholar
5. 5.
Pisati, L., Sulla estensione del metodo di Laplace alle equazioni con due variabili, Rend. Circ. Mat. Palermo, vol. 20, pp. 344–374.Google Scholar
6. 6.
Taimanov, I. and Tsarev, S., Two-Dimensional Rational Solitons and Their Blowup via the Moutard Transformation, Theor. Math. Phys., 2008, vol. 157, no. 2., pp. 1525–1541. doi 10.1007/s11232-008-0127-37.
7. 7.
Tsarev, S.P., On Darboux Integrable Nonlinear Partial Differential Equations, Proc. Steklov Inst. Math., 1999, vol. 225, pp. 372–381.
8. 8.
Tsarev, S.P., On Factorization and Solution of Multidimensional Linear Differential Equations, in Computer Algebra 2006. Latest Advances in Symbolic Algorithms, Proc. Waterloo Workshop, Canada, 10–12 April, 2006 World Scientific, 2007, pp. 181–192. http://arxiv.org/abs/cs/0609075v2
9. 9.
Tsarev, S.P., Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations, Proc. ISSAC’2005 (July 24–27, 2005, Beijing, China), ACM, 2005, pp. 325–331. http: //arxiv.org/abs/cs/0501030v1
10. 10.
Ganzha, E.I., Loginov V.M., and Tsarev S.P., Exact Solutions of Hyperbolic Systems of Kinetic Equations. Application to Verhulst Model with Random Perturbation, Math. Comput., 2008, vol. 1, no. 3, pp. 459–472. http://arxiv.org/abs/math/0612793v1
11. 11.
Kaptsov, O.V., Metody integrirovaniya uravnenii s chastnymi proizvodnymi (Integration Methods for Partial Differential Equations), Moscow: Fizmatlit, 2009.Google Scholar
12. 12.
Schwarz, F., ALLTYPES: An ALgebraic Language and TYPE System. http://www.alltypes.de/
13. 13.
Ore, O., Theory of Non-Commutative Polynomials, Ann. Math., 1933, vol. 34, pp. 480–508.
14. 14.
Ore, O., Linear Equations in Non-Commutative Fields, Ann. Math., 1931, vol. 32, pp. 463–477.
15. 15.