Rational solutions of Riccati-like partial differential equations. (English) Zbl 0985.34007

The rational solutions to Riccati-like partial differential equations (for Riccati equations [see W. T. Reid, Riccati differential equations, New York-London: Academic Press (1972; Zbl 0254.34003)]) are considered. These systems arise in a similar way as Riccati ODEs. The structure of rational solutions is obtained, and an algorithm, called RationalSolution (it consists of 5 steps), for finding rational solutions to an associated Riccati-like system is given. As an application of the results obtained, the RationalSolution algorithm is applied to find all rational solutions to Lie’s system \[ \begin{gathered} \partial_xu+u^2+a_1u+a_2v+a_3=0,\quad\partial_xv+uv+c_1u+c_2v+c_3=0,\\ \partial_yu+uv+b_1u+b_2v+b_3=0,\quad\partial_yv+v^2+d_1u+d_2v+d_3=0, \end{gathered} \] where \(a_k,b_k,c_k,d_k\) are rational functions of \(x,y\) and hyperexponential solutions to linear homogeneous differential systems with finite linear dimension in several unknowns.


34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
35C05 Solutions to PDEs in closed form
34A05 Explicit solutions, first integrals of ordinary differential equations
65D15 Algorithms for approximation of functions
68W30 Symbolic computation and algebraic computation
35G20 Nonlinear higher-order PDEs


Zbl 0254.34003


Full Text: DOI Link


[1] Bajaj, C.; Canny, J.; Garrity, T.; Warren, J., Factoring rational polynomials over the complex numbers, SIAM J. comput., 22, 318-331, (1993) · Zbl 0772.12001
[2] Algebraic Computation, ACM Press, Montreal, Canada, 158, 166
[3] Boulier, F.; Lazard, D.; Ollivier, F.; Petitot, M., Representation for the radical of a finitely generated differential ideal, (), 158-166 · Zbl 0911.13011
[4] Bronstein, M., Linear ordinary differential equations: breaking through the order 2 barrier, (), 42-48 · Zbl 0978.65507
[5] Bronstein, M., On solutions of linear differential equations in their coefficient fields, J. symb. comput., 13, 413-439, (1992) · Zbl 0752.34009
[6] Bronstein, M., Symbolic integration I: transcendental functions, (1997), Heidelberg, Springer · Zbl 0880.12005
[7] Faugére, J.C.; Gianni, P.; Lazard, D.; Mora, T., Efficient computation of zero-dimensional Gröbner bases by changes of ordering, J. symb. comput., 16, 377-399, (1993) · Zbl 0805.13007
[8] Geddes, K.; Czapor, S.; Labahn, G., Algorithms for computer algebra, (1992), Boston, Kluwer Academic Publishers · Zbl 0805.68072
[9] Janet, M., LES systéms d’équations aux dérivées partielles, J. de mathématiques, 83, 65-123, (1920) · JFM 47.0440.03
[10] Kaltofen, E., Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization, SIAM J. comput., 14, 469-489, (1985) · Zbl 0605.12001
[11] Kaltofen, E., Fast parallel irreducible testing, J. symb. comput., 1, 57-67, (1985) · Zbl 0599.68038
[12] Kandri-Rody, A.; Weispfenning, V., Non-commutative Gröbner bases in algebras of solvable type, J. symb. comput., 9, 1-26, (1990) · Zbl 0715.16010
[13] Kaplansky, I., An introduction to differential algebra, (1957), Hermann Paris · Zbl 0083.03301
[14] Kolchin, E., Differential algebra and algebraic groups, (1973), Academic Press New York · Zbl 0264.12102
[15] Li, Z.; Wang, D.-m., Coherent, regular and simple systems in zero decompositions of partial differential systems, Syst. sci. math. sci., 12 , Suppl., 43-60, (1873)
[16] S. Lie, Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten III, Arch. for Mathematik, VIII, 371, 458 · JFM 15.0751.03
[17] Gesammelte Abhandlungen, V, 362, 427 · Zbl 0097.30001
[18] Reid, W.T., Riccati differential equations, (1972), Academic Press New York and London · Zbl 0209.11601
[19] Ritt, J.F., Differential algebra, (1950), American Mathematical Society New York · Zbl 0037.18501
[20] Rosenfeld, A., Specializations in differential algebra, Trans. am. math. soc., 90, 394-407, (1959) · Zbl 0192.14001
[21] Schwarz, F., Janet bases for symmetry groups, (), 221-234 · Zbl 1024.34027
[22] Schwarz, F., ALLTYPES: an algebraic language and type system, (), 270-283
[23] Singer, M., Liouvillian solutions of linear differential equations with Liouvillian coefficients, J. symb. comput., 11, 251-273, (1991) · Zbl 0776.12002
[24] Trager, B.M., Algebraic factoring and rational function integration, (), 219-226
[25] Winkler, F., Polynomial algorithms in computer algebra, (1996), Springer-Verlag Wien New York · Zbl 0853.12003
[26] Wu, W.-t., On the foundation of algebraic differential geometry, Syst. sci. math. sci., 2, 289-312, (1989) · Zbl 0739.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.