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.

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  1. Mithun Bairagi: GiNaCDE: the high-performance F-expansion and First Integral Methods with C++ library for solving Nonlinear Differential Equations (2022) not zbMATH
  2. Zeidan, Dia; Chau, Chi Kin; Lu, Tzon-Tzer: On the development of Adomian decomposition method for solving PDE systems with non-prescribed data (2022)
  3. Ali, Ijaz; Seadawy, Aly R.; Rizvi, Syed Tahir Raza; Younis, Muhammad: Painlevé analysis for various nonlinear Schrödinger dynamical equations (2021)
  4. Ramírez, J.; Romero, J. L.; Muriel, C.: A new method to obtain either first- or second-order reductions for parametric polynomial ODEs (2019)
  5. Abraham-Shrauner, Barbara: Exact solutions of nonlinear partial differential equations (2018)
  6. Abraham-Shrauner, Barbara: Analytic solutions of nonlinear partial differential equations by the power index method (2018)
  7. Arafa, Anas; Elmahdy, Ghada: Application of residual power series method to fractional coupled physical equations arising in fluids flow (2018)
  8. He, Chunhua; Tang, Yaning; Ma, Jinli: New interaction solutions for the ((3+1))-dimensional Jimbo-Miwa equation (2018)
  9. Mohammed, Wael W.: Approximate solution of the Kuramoto-Shivashinsky equation on an unbounded domain (2018)
  10. Ramírez, J.; Romero, J. L.; Muriel, C.: Two new reductions methods for polynomial differential equations and applications to nonlinear pdes (2018)
  11. Bochkarev, A. V.; Zemlyanukhin, A. I.: The geometric series method for constructing exact solutions to nonlinear evolution equations (2017)
  12. Vitanov, Nikolay K.; Dimitrova, Zlatinka I.; Ivanova, Tsvetelina I.: On solitary wave solutions of a class of nonlinear partial differential equations based on the function (1/ \cosh^n(\alphax + \betat)) (2017)
  13. Ramírez, J.; Romero, J. L.; Muriel, C.: Reductions of PDEs to second order ODEs and symbolic computation (2016)
  14. Tang, Bo; Wang, Xuemin; Fan, Yingzhe; Qu, Junfeng: Exact solutions for a generalized KdV-mkdv equation with variable coefficients (2016)
  15. Xiazhi, Hao; Yinping, Liu; Xiaoyan, Tang; Zhibin, Li: A \textitMaplepackage for finding interaction solutions of nonlinear evolution equations (2016)
  16. Yin, Weishi; Xu, Fei; Zhang, Weipeng; Gao, Yixian: Asymptotic expansion of the solutions to time-space fractional Kuramoto-Sivashinsky equations (2016)
  17. Antoniou, Solomon M.: The Riccati equation method with variable expansion coefficients. III: Solving the Newell-Whitehead equation (2015)
  18. Kudryashov, N. A.: On nonlinear differential equation with exact solutions having various pole orders (2015)
  19. Kudryashov, Nikolay A.: Logistic function as solution of many nonlinear differential equations (2015)
  20. Pınar, Zehra; Öziş, Turgut: Observations on the class of “Balancing Principle” for nonlinear PDEs that can be treated by the auxiliary equation method (2015)

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