Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions. For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed. A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.

References in zbMATH (referenced in 53 articles , 1 standard article )

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  1. Abraham-Shrauner, Barbara: Exact solutions of nonlinear partial differential equations (2018)
  2. Mohammed, Wael W.: Approximate solution of the Kuramoto-Shivashinsky equation on an unbounded domain (2018)
  3. Ramírez, J.; Romero, J.L.; Muriel, C.: Two new reductions methods for polynomial differential equations and applications to nonlinear pdes (2018)
  4. Bochkarev, A.V.; Zemlyanukhin, A.I.: The geometric series method for constructing exact solutions to nonlinear evolution equations (2017)
  5. Xiazhi, Hao; Yinping, Liu; Xiaoyan, Tang; Zhibin, Li: A \itMaple package for finding interaction solutions of nonlinear evolution equations (2016)
  6. Yin, Weishi; Xu, Fei; Zhang, Weipeng; Gao, Yixian: Asymptotic expansion of the solutions to time-space fractional Kuramoto-Sivashinsky equations (2016)
  7. Antoniou, Solomon M.: The Riccati equation method with variable expansion coefficients. III: Solving the Newell-Whitehead equation (2015)
  8. Kudryashov, N.A.: On nonlinear differential equation with exact solutions having various pole orders (2015)
  9. Pınar, Zehra; Öziş, Turgut: Observations on the class of “Balancing Principle” for nonlinear PDEs that can be treated by the auxiliary equation method (2015)
  10. Kudryashov, Nikolay A.: Quasi-exact solutions of the dissipative Kuramoto-Sivashinsky equation (2013)
  11. Kudryashov, Nikolay A.: Polynomials in logistic function and solitary waves of nonlinear differential equations (2013)
  12. Elboree, Mohammed K.: Hyperbolic and trigonometric solutions for some nonlinear evolution equations (2012)
  13. El-Wakil, S.A.; Abulwafa, Essam M.; El-Shewy, E.K.; Abdelwahed, H.G.; Abd-El-Hamid, Hamdi M.: Nonlinear waveforms for ion-acoustic waves in weakly relativistic plasma of warm ion-fluid and isothermal electrons (2012)
  14. Kudryashov, Nikolai A.; Sinelshchikov, Dmitry I.: Nonlinear differential equations of the second, third and fourth order with exact solutions (2012)
  15. Kudryashov, Nikolay A.; Kochanov, Mark B.: Quasi-exact solutions of nonlinear differential equations (2012)
  16. Lakestani, Mehrdad; Dehghan, Mehdi: Numerical solutions of the generalized kuramoto-sivashinsky equation using B-spline functions (2012)
  17. Tang, Bo; He, Yinnian; Wei, Leilei; Zhang, Xindong: A generalized fractional sub-equation method for fractional differential equations with variable coefficients (2012)
  18. Seadawy, A.R.: New exact solutions for the KdV equation with higher order nonlinearity by using the variational method (2011)
  19. Gómez S., Cesar A.; Salas, Alvaro: Special symmetries to standard Riccati equations and applications (2010)
  20. Gomez Sierra, Cesar A.: New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6) (2010)

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