### Call for Paper

CAE solicits original research papers for the May 2023 Edition. Last date of manuscript submission is **April 30, 2023**.

# The Generalized Projective Riccati Equations Method and its Applications to Nonlinear PDEs Describing Nonlinear Transmission Lines

**Year of Publication:**2015

**Publisher:**Foundation of Computer Science (FCS), NY, USA

E M E Zayed and K A E Alurrfi. Article: The Generalized Projective Riccati Equations Method and its Applications to Nonlinear PDEs Describing Nonlinear Transmission Lines. *Communications on Applied Electronics* 3(4):1-8, November 2015. Published by Foundation of Computer Science (FCS), NY, USA. BibTeX

@article{key:article, author = {E.M.E. Zayed and K.A.E. Alurrfi}, title = {Article: The Generalized Projective Riccati Equations Method and its Applications to Nonlinear PDEs Describing Nonlinear Transmission Lines}, journal = {Communications on Applied Electronics}, year = {2015}, volume = {3}, number = {4}, pages = {1-8}, month = {November}, note = {Published by Foundation of Computer Science (FCS), NY, USA} }

### Abstract

In this article, we apply the generalized projective Riccati equations method with the aid of symbolic computation to construct new exact traveling wave solutions with parameters for two nonlinear PDEs describing nonlinear transmission lines (NLTL). The first equation describes the model of governing wave propagation in the NLTL as nonlinear low-pass electrical lines. The second equation describes pulse narrowing nonlinear transmission lines. The obtained solutions include, kink and anti-kink solitons, bell (bright) and anti-bell (dark) solitary wave solutions, hyperbolic solutions and trigonometric solutions. Based on Kirchhoff

### References

- M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, New York, NY,USA, 1991.
- R. Hirota, Exact solutions of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett., 27(1971) 1192– 1194.
- J.Weiss, M. Tabor, and G. Carnevale, The Painlev´e property for partial differential equations, J. Math. Phys., 24(1983) 522–526.
- N. A. Kudryashov, Exact soliton solutions of a generalized evolution equation of wave dynamics, J. Appl. Math. Mech., 52(1988) 361–365.
- N. A. Kudryashov, Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A, 147 (1990) 287–291.
- N. A. Kudryashov, On types of nonlinear nonintegrable equations with exact solutions, Phys. Lett. A, 155 (1991) 269–275.
- M. R. Miura, B¨acklund Transformation, Springer, Berlin, Germany, 1978.
- C. Rogers and W. F. Shadwick, B¨acklund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982.
- J.-H. He and X.-H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700–708.
- E. Yusufoglu, New solitary for the MBBM equations using Exp-function method, Phys. Lett A, 372 (2008) 442–446.
- S. Zhang, Application of Exp-function method to highdimensional nonlinear evolution equations, Chaos, Solitons and Fractals, 38 (2008) 270–276.
- E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000) 212–218.
- S. Zhang and T. C. Xia, A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations, Appl. Math. E-Notes, 8 (2008) 58– 66.
- Y. Chen and Q. Wang, Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation, Chaos, Solitons and Fractals, 24 (2005) 745–757.
- S. Liu, Z. Fu, S. Liu, and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001) 69–74.
- D. Lu, Jacobi elliptic function solutions for two variant Boussinesq equations, Chaos, Solitons and Fractals, 24 (2005) 1373–1385.
- E. M. E. Zayed, New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized (G G 0 )-expansion method, J. Phys. A: Math. Theor., 42(2009) 195202, 13 pages
- M. L. Wang, X. Li, and J. Zhang, The (G G 0 )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008) 417–423.
- S. Zhang, J. L. Tong, andW.Wang, A generalized (G G 0 )- expansion method for the mKdV equation with variable coefficients, Phys. Lett. A, 372 (2008) 2254–2257.
- E. M. E. Zayed and K. A. Gepreel, The (G G 0 )-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics, J. Math. Phys., 50 (2009) 013502-013512.
- N.A. Kudryashov, A note on the (G G 0 )-expansion method, Appl. Math. Comput., 217 (2010) 1755–1758.
- E.M.E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G G 0 )- expansion method, J. Appl. Math. Informatics, 28 (2010) 383-395.
- S. Zhang, Y. N. SUN, J. M. B and L.Dong, The modified (G G 0 )-expansion method for nonlinear evolution equations, Z. Naturforsch., 66a (2011) 33-39.
- L.x. Li, Q. E. Li, and L. M Wang, The (G G 0 ; 1 G)-expansion method and its application to traveling wave solutions of the Zakharov equations, Appl Math J. Chinese. Uni., 25 (2010) 454–462.
- E. M. E. Zayed and M. A. M. Abdelaziz, The two variables (G G 0 ; 1 G)-expansion method for solving the nonlinear KdV-mKdV equation, Math. Prob. Engineering, Vol. 2012, Article ID 725061, 14 pages.
- E. M. E. Zayed and K. A. E. Alurrfi, The (G G 0 ; 1 G)- expansion method and its applications to find the exact solutions of nonlinear PDEs for nanobiosciences, Math. Prob. Engineering, Vol. 2014, Article ID 521712, 10 pages.
- E. M. E. Zayed and K. A. E. Alurrfi, The (G G 0 ; 1 G)- expansion method and its applications for solving two higher order nonlinear evolution equations, Math. Prob. Engineering, Vol. 2014, Article ID 746538, 21 pages.
- A. J. M. Jawad, M. D. Petkovic and A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217 (2010) 869-877.
- E. M. E. Zayed, A note on the modified simple equation method applied to Sharma-Tasso-Olver equation. Appl. Math. Comput., 218 (2011) 3962-3964.
- E. M. E. Zayed and S. A. Hoda Ibrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chin. Phys. Lett., 29 (2012), 060201-060204.
- W. X. Ma and Z. Zhu, Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple expfunction algorithm, Appl. Math. Comput., 218 (2012) 11871-11879.
- W. X. Ma, T.Huang and Y.Zhang, A multiple exp-function method for nonlinear differential equations and its application, Phys. Script.,82(2010) 065003.
- W. X. Ma and J. H. Lee, A transformed rational function method and exact solutions to the (3+1) dimensional Jimbo-Miwa equation, Chaos, Solitons and Fractals, 42 (2009) 1356-1363.
- A. M. Yang, X. J. Yang, and Z. B. Li, Local fractional series expansion method for solving wave and diffusion equations on cantor sets, Abst. Appl. Analy., Vol. 2013, Article ID 351057, 5 pages.
- N. Taghizadeh, M. Mirzazadeh, F. Farahrooz, Exact solutions of the nonlinear Schr¨odinger equation by the first integral method, J. Math Anal Appl., 374 (2011) 549-553.
- B. H. Q. Lu, H. Q. Zhang and F. D. Xie, Traveling wave solutions of nonlinear parial differential equations by using the first integral method, Appl. Math. Comput., 216 (2010) 1329-1336.
- E. M. E. Zayed, Y. A. Amer and R. M. A. Shohib,The improved Riccati equation mapping method for constructing many families of exact solutions for a nonlinear partial differential equation of nanobiosciences, Int. J. Phys. Sci., 8 (2013) 1246-1255.
- S. D. Zhu, The generalized Riccati equations mapping method in nonlinear evolution equation: application to (2+1)-dimensional Boiti-Lion-Pempinelle equation, Chaos, Solitons and Fractals, 37 (2008) 1335-1342.
- R. Conte and M. Musette, Link between solitary waves and projective Riccati equations, Phys. A: Math. Cen. 25 (1992) 2609-2623.
- E. M. E. Zayed and K. A. E. Alurrfi, The generalized projective Riccati equations method for solving nonlinear evolution equations in mathematical physics, Abst. Appl. Analy., Vol. 2014, Article ID 259190, 10 pages.
- E. M. E. Zayed and K. A. E. Alurrfi, The generalized projective Riccati equations method and its applications for solving two nonlinear PDEs describing microtubules, Int. J. Phys. Sci., 10 (2015) 391-402.
- G. X. Zhang, Z. B. Li and Y. S. Duan, Exact solitary wave solutions of nonlinear wave equations, Science in China A., 44 (2001), pp. 396-401.
- Z.Y. Yan, Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres, Chaos, Solitons Fractals, 16 (2003) 759-766.
- E.Yomba, The General projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations, Chin. J. Phys., 43 (2005) 991-1003.
- T. C. Bountis, V. Papageorgiou, and P. Winternitz, On the integrability of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 27 (1986), 1215-1224.
- S. Abdoulkary, T. Beda, O. Dafounamssou, E. W. Tafo and A. Mohamadou, Dynamics of solitary pulses in the nonlinear low-pass electrical transmission lines through the auxiliary equation method, J. Mod. Phys. Appl., 2 (2013) 69-87.
- Sirendaoreji, Exact traveling wave solutions for four forms of nonlinear Klein–Gordon equations, Phys. Lett. A, 363 (2007) 440-447.
- E. M. E. Zayed and K. A. E. Alurrfi, A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines, Chaos, Solitons and Fractals, (in press).
- E. Afshari and A. Hajimiri, Nonlinear transmission lines for pulse shaping in Silicon, IEEE J. Solid state circuits, 40 (2005) 744-752.
- E. M. E. Zayed and K. A. E. Alurrfi, A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing pulse narrowing nonlinear transmission lines, J. Partial Diff. Eqs., 28 (2015) 128-138.

### Keywords

Generalized projective Riccati equations method, Exact solutions, Nonlinear low-pass electrical lines, Pulse narrowing nonlinear transmission lines, Kirchhos lawsSS