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Year 2018, Volume: 1 Issue: 1, 1 - 22, 18.05.2018

Abstract

References

  • W. Malfliet, W. Hereman, The tanh method: exact solutions of nonlinear evolution and wave equations. Phys Scripta, 1996 (54): 563-568.
  • G. M. Murphy, Ordinary Differential Equations and Their Solutions. D.Van Nostrand, New York, 1960.
  • E. J. Parkes, B. R. Duffy. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. ComputerPhysics Communications, 1996 (98): 288–300.
  • E. Fan, Extended tanh-function method and its applications to nonlinearequations. Physics Letters A, 2000 (277) (4–5): 212–218.
  • S. A. Elwakila, S. K. El-Labany, M. A. Zahran, R. Sabry, Modified extendedtanh-function method for solving nonlinear partial differential equations.Physics Letters A, 2002 (299) (2–3): 179–188.
  • Z. Yan, The extended Jacobian elliptic function expansion method andits application in the generalized Hirota–Satsuma coupled KdV system.Chaos, Solitons & Fractals, 2003 (15) (3): 575–583.
  • A.-M. Wazwaz, The sine-cosine method for obtaining solutions with compactand noncompact structures. Applied Mathematics and Computation,2004 (159) (2): 559–576.
  • E. Kamke, Differentialgleichungen: Loesungsmethoden und Loesungen.Chelsea Publishing Co., New York, 1959.
  • A. D. Polyanin, A. V. Manzhirov, Handbook of Mathematics for Engineersand Scientists. Chapman & Hall/CRC Press, Boca Raton – London, 2007.
  • E. L. Ince, Ordinary Differential Equations. Dover Publ., New York, 1964.
  • A. D. Polyanin, V. F. Zaitsev, A. I. Zhurov, Methods for the Solution ofNonlinear Equations of Mathematical Physics and Mechanics. Fizmatlit,Moscow, 2005 (in Russian).
  • N. M. Matveev, Methods of Integration of Ordinary Differential Equations.Vysshaya Shkola, Moscow, 1967 (in Russian).
  • L. E. El’sgol’ts, Differential Equations. Gordon & Breach Inc., New York,1961.
  • A. D. Polyanin, V. F. Zaitsev, Discrete-Group Methods for IntegratingEquations of Nonlinear Mechanics. CRC Press, Boca Raton, 1994.
  • N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary DifferentialEquations. John Wiley, Chichester, 1999.
  • A. D. Polyanin, V. F. Zaitsev, Handbook of Exact Solutions for OrdinaryDifferential Equations, 2nd Edition. Chapman & Hall/CRC Press, BocaRaton, 2003.
  • A. D. Polyanin, V. F. Zaitsev, Handbook of Ordinary Differential Equations,Exact Solutions, Methods, and Problems, CRC Press, Boca Raton –London, 2017.
  • A. D. Polyanin, A. I. Zhurov, The algebraic method for integration of the differential equations of nonlinear mechanics. Physics Doklady, 1994 (39)(7): 534–537.
  • A. D. Polyanin, V. F. Zaitsev, Handbook of Nonlinear Partial DifferentialEquations, 2nd Edition. Chapman & Hall/CRC Press, Boca Raton – London,2012 (see also 1st Edition, 2004).
  • N. H. Ibragimov (Editor), CRC Handbook of Lie Group Analysis of DifferentialEquations, Vol. 1, Symmetries, Exact Solutions and ConservationLaws. CRC Press, Boca Raton, 1994. 34
  • N. H. Ibragimov (Editor), CRC Handbook of Lie Group Analysis of DifferentialEquations, Vol. 2, Applications in Engineering and Physical Sciences.CRC Press, Boca Raton, 1995.
  • S. V. Meleshko, Methods for Constructing Exact Solutions of Partial DifferentialEquations. Springer, New York, 2005.
  • V. A. Galaktionov, S. R. Svirshchevskii, Exact Solutions and InvariantSubspaces of Nonlinear Partial Differential Equations in Mechanics andPhysics. Chapman & Hall/CRC Press, Boca Raton, 2006.
  • N. A. Kudryashov, Methods of Nonlinear Mathematical Physics. Izd. DomIntellekt, Dolgoprudnyi, 2010 (in Russian).
  • V. V. Stepanov, A Course of Differential Equations, 7th Edition. Gostekhizdat,Moscow, 1958 (in Russian).33

Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs

Year 2018, Volume: 1 Issue: 1, 1 - 22, 18.05.2018

Abstract

Various situations are described where the construction of exact solutions of nonlinear
ordinary and partial differential equations leads to overdetermined systems
of ODEs with parameters that are not included in the original differential equations.
A non-classical problem for ordinary differential equations with parameters
is formulated and the concept of the conditional capacity of an exact solution is
introduced. The method for investigating overdetermined systems of two ODEs
of any order on consistency, which eventually leads to algebraic equations with
parameters, is presented. A general description of the method of differential constraints
with respect to ordinary differential equations is given and many specific
examples of applying this method for obtaining exact solutions are considered. It
is shown that the use of the splitting method (and also the method based on the
use of invariant subspaces of nonlinear operators) for constructing exact generalized
separable solutions of nonlinear PDEs can lead to overdetermined systems of
ODEs with parameters. Several nonlinear partial differential equations (including
a delay PDE) of higher orders are considered, and their exact solutions are found
by analyzing the corresponding overdetermined ODE systems.

References

  • W. Malfliet, W. Hereman, The tanh method: exact solutions of nonlinear evolution and wave equations. Phys Scripta, 1996 (54): 563-568.
  • G. M. Murphy, Ordinary Differential Equations and Their Solutions. D.Van Nostrand, New York, 1960.
  • E. J. Parkes, B. R. Duffy. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. ComputerPhysics Communications, 1996 (98): 288–300.
  • E. Fan, Extended tanh-function method and its applications to nonlinearequations. Physics Letters A, 2000 (277) (4–5): 212–218.
  • S. A. Elwakila, S. K. El-Labany, M. A. Zahran, R. Sabry, Modified extendedtanh-function method for solving nonlinear partial differential equations.Physics Letters A, 2002 (299) (2–3): 179–188.
  • Z. Yan, The extended Jacobian elliptic function expansion method andits application in the generalized Hirota–Satsuma coupled KdV system.Chaos, Solitons & Fractals, 2003 (15) (3): 575–583.
  • A.-M. Wazwaz, The sine-cosine method for obtaining solutions with compactand noncompact structures. Applied Mathematics and Computation,2004 (159) (2): 559–576.
  • E. Kamke, Differentialgleichungen: Loesungsmethoden und Loesungen.Chelsea Publishing Co., New York, 1959.
  • A. D. Polyanin, A. V. Manzhirov, Handbook of Mathematics for Engineersand Scientists. Chapman & Hall/CRC Press, Boca Raton – London, 2007.
  • E. L. Ince, Ordinary Differential Equations. Dover Publ., New York, 1964.
  • A. D. Polyanin, V. F. Zaitsev, A. I. Zhurov, Methods for the Solution ofNonlinear Equations of Mathematical Physics and Mechanics. Fizmatlit,Moscow, 2005 (in Russian).
  • N. M. Matveev, Methods of Integration of Ordinary Differential Equations.Vysshaya Shkola, Moscow, 1967 (in Russian).
  • L. E. El’sgol’ts, Differential Equations. Gordon & Breach Inc., New York,1961.
  • A. D. Polyanin, V. F. Zaitsev, Discrete-Group Methods for IntegratingEquations of Nonlinear Mechanics. CRC Press, Boca Raton, 1994.
  • N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary DifferentialEquations. John Wiley, Chichester, 1999.
  • A. D. Polyanin, V. F. Zaitsev, Handbook of Exact Solutions for OrdinaryDifferential Equations, 2nd Edition. Chapman & Hall/CRC Press, BocaRaton, 2003.
  • A. D. Polyanin, V. F. Zaitsev, Handbook of Ordinary Differential Equations,Exact Solutions, Methods, and Problems, CRC Press, Boca Raton –London, 2017.
  • A. D. Polyanin, A. I. Zhurov, The algebraic method for integration of the differential equations of nonlinear mechanics. Physics Doklady, 1994 (39)(7): 534–537.
  • A. D. Polyanin, V. F. Zaitsev, Handbook of Nonlinear Partial DifferentialEquations, 2nd Edition. Chapman & Hall/CRC Press, Boca Raton – London,2012 (see also 1st Edition, 2004).
  • N. H. Ibragimov (Editor), CRC Handbook of Lie Group Analysis of DifferentialEquations, Vol. 1, Symmetries, Exact Solutions and ConservationLaws. CRC Press, Boca Raton, 1994. 34
  • N. H. Ibragimov (Editor), CRC Handbook of Lie Group Analysis of DifferentialEquations, Vol. 2, Applications in Engineering and Physical Sciences.CRC Press, Boca Raton, 1995.
  • S. V. Meleshko, Methods for Constructing Exact Solutions of Partial DifferentialEquations. Springer, New York, 2005.
  • V. A. Galaktionov, S. R. Svirshchevskii, Exact Solutions and InvariantSubspaces of Nonlinear Partial Differential Equations in Mechanics andPhysics. Chapman & Hall/CRC Press, Boca Raton, 2006.
  • N. A. Kudryashov, Methods of Nonlinear Mathematical Physics. Izd. DomIntellekt, Dolgoprudnyi, 2010 (in Russian).
  • V. V. Stepanov, A Course of Differential Equations, 7th Edition. Gostekhizdat,Moscow, 1958 (in Russian).33
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Andrei D. Polyanin

Inna K. Shingareva This is me

Publication Date May 18, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Polyanin, A. D., & Shingareva, I. K. (2018). Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs. Mathematical Advances in Pure and Applied Sciences, 1(1), 1-22.
AMA Polyanin AD, Shingareva IK. Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs. MAPAS. May 2018;1(1):1-22.
Chicago Polyanin, Andrei D., and Inna K. Shingareva. “Overdetermined Systems of ODEs With Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs”. Mathematical Advances in Pure and Applied Sciences 1, no. 1 (May 2018): 1-22.
EndNote Polyanin AD, Shingareva IK (May 1, 2018) Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs. Mathematical Advances in Pure and Applied Sciences 1 1 1–22.
IEEE A. D. Polyanin and I. K. Shingareva, “Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs”, MAPAS, vol. 1, no. 1, pp. 1–22, 2018.
ISNAD Polyanin, Andrei D. - Shingareva, Inna K. “Overdetermined Systems of ODEs With Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs”. Mathematical Advances in Pure and Applied Sciences 1/1 (May 2018), 1-22.
JAMA Polyanin AD, Shingareva IK. Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs. MAPAS. 2018;1:1–22.
MLA Polyanin, Andrei D. and Inna K. Shingareva. “Overdetermined Systems of ODEs With Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs”. Mathematical Advances in Pure and Applied Sciences, vol. 1, no. 1, 2018, pp. 1-22.
Vancouver Polyanin AD, Shingareva IK. Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs. MAPAS. 2018;1(1):1-22.