zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Special symmetries to standard Riccati equations and applications. (English) Zbl 1202.34007
The general Riccati equation $$\frac{d\phi(\xi)}{d\xi}=p(\xi)\phi^2(\xi)+q(\xi)\phi(\xi)+r(\xi) \tag1$$ is studied. Here, $p,q$ and $r$ are continuous functions, defined on some interval $[a,b]\subseteq\mathbb{R}$. Using Lie group symmetry, the authors obtain new integrability conditions for the generalized Riccati equation. Using this condition, 7 families of Riccati equations in standard form (1) are obtained which are integrable by quadratures. The obtained results are applied to construct travelling wave solutions for nonlinear evolution equations.

34A05Methods of solution of ODE
34C14Symmetries, invariants (ODE)
34A34Nonlinear ODE and systems, general
Full Text: DOI
[1] Cariñena, J. F.; Marmo, G.; Nasare, J.: The non-linear superposition principle and the wei-norman method, Int. J. Mod. phys 1, 601-3627 (1998) · Zbl 0928.34025 · doi:10.1142/S0217751X98001694
[2] J.F. Cariñena, A. Ramos, Lie systems and connections in fibre bundles: applications in quantum mechanics, differential geometry and its applications, Conf. Praga 2004, Charles University, Prague (Zech Republic), 2005, pp. 437 -- 452. · Zbl 1104.81054
[3] Strelchenya, V. M.: A new case of integrability of the general Riccati equation and its application to relaxation problems, J. phys. A. math gen 24, 4965-4967 (1991) · Zbl 0757.34003 · doi:10.1088/0305-4470/24/21/010
[4] Baldwin, D.; Goktas, U.; Hereman, W.; Hong, L.; Martino, R. S.; Miller, J. C.: Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear pdfs, J. symbolic compt. 37, No. 6, 669-705 (2004) · Zbl 1137.35324 · doi:10.1016/j.jsc.2003.09.004
[5] Fan, E.; Hon, Y. C.: Generalized tanh method extended to special types of nonlinear equations, Z. naturforsch. A 57, No. 8, 692-700 (2002)
[6] Wazwaz, A. M.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. math. Comput 84-2, 1002-1014 (2007) · Zbl 1115.65106 · doi:10.1016/j.amc.2006.07.002
[7] Gomez, C. A.: Special forms of the fifth-order KdV equation with new periodic and soliton solutions, Appl. math comput. 189, 1066-1077 (2007) · Zbl 1122.65393 · doi:10.1016/j.amc.2006.11.158
[8] Yan, Z.: The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equation, Comput. phys. Comm. 152, No. 1, 1-8 (2003) · Zbl 1196.35068 · doi:10.1016/S0010-4655(02)00756-7
[9] Davis, H. T.: Introduction to nonlinear differential and integral equations, (1962) · Zbl 0106.28904
[10] Kamke, E.: Differential gleichungen, (1959)
[11] Murphy, G.: Ordinary differential equations and their solutions, (1960) · Zbl 0095.06405
[12] Rao, P. R. P.; Ukidave, V. H.: Some separable forms of the Riccati equation, Am. math. Mon. 75, No. 10, 1113-1114 (1968) · Zbl 0185.15703
[13] Haaheim, D. R.; Stein, F. M.: Methods of solution of the Riccati differential equation, Math. mag. 42, No. 5, 233-240 (1969) · Zbl 0188.15002 · doi:10.2307/2688697
[14] Allen, J. L.; Stein, F. M.: On solutions of certain Riccati differential equations, Am. math. Mon. 71, 1113-1115 (1964) · Zbl 0144.12103 · doi:10.2307/2311412
[15] Bluman, G.; Kumey, S.: Symmetries and differential equations, (1989)
[16] Olver, P. J.: Applications of Lie group to differential equations, (1980) · Zbl 0599.58050
[17] Bauman, G.: Symmetry analysis of differential equations with Mathematica, (2000)
[18] Ovsianikov, L. V.: Group analysis of differential equations, (1982)
[19] Stephani, H.: Differential equations: their solutions using symmetries, (1989) · Zbl 0704.34001
[20] J.M. Hill, Solutions of differential equations by means of one-parameter groups, University of Wollongong, vol. 1, 1982. · Zbl 0497.34002
[21] Konovalov, S. P.: Fifth-order equations with symmetries linear in the dependent variable, CRC handbook of Lie group 1 (1994)
[22] C.A. Gomez, A. Salas, Grupos de Lie y ecuaciones diferenciales, Memorias XI encuentro de geometria y sus aplicaciones, Universidad Distrital Bogotá, 2000, pp. 1 -- 22.
[23] C.A. Gomez, Ecuacion de Riccati en grupos de Lie, Memorias X encuentro de geometria y sus aplicaciones, Universidad Pedagógica Nacional Bogotá, 1999, pp. 149 -- 153.
[24] Gomez, C. A.; Salas, A.: New exact solutions for the combined sinh -- cosh Gordon equation, Lecturas math. 21, 87-93 (2006)
[25] Gomez, C. A.; Salas, A. H.: The generalized tanh -- coth method to special types of the fifth-order KdV equation, Appl. math. Comput. 203, 873-880 (2008) · Zbl 1154.65364 · doi:10.1016/j.amc.2008.05.105
[26] Salas, A. H.; Gomez, C. A.: Computing exact solutions for some fifth KdV equations with forcing term, Appl. math. Comput. 204, 257-260 (2008) · Zbl 1160.35526 · doi:10.1016/j.amc.2008.06.033