{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:45:34Z","timestamp":1772293534873,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T00:00:00Z","timestamp":1434672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin\u2013Bona\u2013Mahony equation and the (2 + 1)-dimensional cubic Klein\u2013Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw two- and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9.<\/jats:p>","DOI":"10.3390\/e17064255","type":"journal-article","created":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T10:23:33Z","timestamp":1434709413000},"page":"4255-4270","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics"],"prefix":"10.3390","volume":"17","author":[{"given":"Haci","family":"Baskonus","sequence":"first","affiliation":[{"name":"Department of Computer Engineering, Tunceli University, Tunceli, 62100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hasan","family":"Bulut","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Firat, Elazig, 23119, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2505","DOI":"10.7498\/aps.54.2505","article-title":"Trial Equation Method and its Applications to Nonlinear Evolution Equations","volume":"54","author":"Liu","year":"2005","journal-title":"Acta Phys. Sin."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1088\/0253-6102\/45\/3\/003","article-title":"A New Trial Equation Method its Applications, Communications in Theoretical Physics","volume":"45","author":"Liu","year":"2006","journal-title":"Commun. Theor. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1088\/0253-6102\/45\/2\/005","article-title":"Trial Equation Method for Nonlinear Evolution Equations with Rank Inhomogeneous: Mathematical Discussions and Applications","volume":"45","author":"Liu","year":"2006","journal-title":"Commun. Theor. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"636802","DOI":"10.1155\/2013\/636802","article-title":"The Modified Trial Equation Method for Fractional Wave Equation and Time-Fractional Generalized Burgers Equation","volume":"2013","author":"Bulut","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2259","DOI":"10.1016\/j.amc.2011.07.043","article-title":"Exact Solutions for the Nonlinear Schr\u00f6dinger Equation with Variable Coefficients Using the Generalized Extended Tanh-Function, the Sine-Cosine and the Exp-function Methods","volume":"218","author":"Zayed","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.chaos.2004.09.122","article-title":"The Tanh Method: Solitons and Periodic Solutions for Dodd\u2013Bullough\u2013Mikhailov and Tzitzeica\u2013Dodd\u2013Bullough Equations","volume":"25","author":"Wazwaz","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.asej.2013.07.007","article-title":"Traveling Wave Solutions of the (2 + 1)-Dimensional Zoomeron Equation and the Burgers Equations via the MSE Method and the Exp-function Method","volume":"5","author":"Khan","year":"2014","journal-title":"Ain Shams Eng. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1356","DOI":"10.1016\/j.chaos.2009.03.043","article-title":"A Transformed Rational Function Method and Exact Solutions to the 3 + 1 Dimensional Jimbo\u2013Miwa Equation","volume":"42","author":"Ma","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"065003","DOI":"10.1088\/0031-8949\/82\/06\/065003","article-title":"A Multiple Exp-function Method for Nonlinear Differential Equations and its Application","volume":"82","author":"Ma","year":"2010","journal-title":"Phys. Scr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"11871","DOI":"10.1016\/j.amc.2012.05.049","article-title":"Solving the (3+1)-Dimensional Generalized KP and BKP Equations by the Multiple Exp-function Algorithm","volume":"218","author":"Ma","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_11","first-page":"140","article-title":"Generalized Bilinear Differential Equations","volume":"2","author":"Ma","year":"2011","journal-title":"Stud. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/0020-7462(95)00064-X","article-title":"Explicit and Exact Solutions to a Kolmogorov\u2013Petrovskii\u2013Piskunov Equation","volume":"31","author":"Ma","year":"1996","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.physleta.2006.11.048","article-title":"Partial Differential Equations Possessing Frobenius Integrable Decompositions","volume":"364","author":"Ma","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Khan, K., Ali Akbar, M., and Rayhanul Islam,, S.M. (2014). Exact Solutions for (1 + 1)-Dimensional Nonlinear Dispersive Modified Benjamin\u2013Bona\u2013Mahony Equation and Coupled Klein\u2013Gordon Equations. SpringerPlus, 3.","DOI":"10.1186\/2193-1801-3-724"},{"key":"ref_15","first-page":"74","article-title":"Exact Solutions of the (2+1)-Dimensional Cubic Klein\u2013Gordon Equation and the (3+1)-Dimensional Zakharov\u2013Kuznetsov Equation Using the Modified Simple Equation Method","volume":"15","author":"Khan","year":"2014","journal-title":"J. Assn. Arab Univ. Basic Appl. Sci."},{"key":"ref_16","unstructured":"Benjamin\u2013Bona\u2013Mahony Equation. Available online: http:\/\/en.wikipedia.org\/wiki\/Benjamin%E2%80%93Bona%E2%80%93Mahony_equation."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1016\/j.physleta.2007.07.062","article-title":"New Solitary Solutions for the MBBM Equations Using Exp-function Method","volume":"372","author":"Yusufoglu","year":"2008","journal-title":"Phys. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"768573","DOI":"10.1155\/2010\/768573","article-title":"Applications of an Extended (G\u2019\/G)-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics","volume":"2010","author":"Eme","year":"2010","journal-title":"Math. Probl. Eng."},{"key":"ref_19","unstructured":"Available online: https:\/\/en.wikipedia.org\/wiki\/Klein%E2%80%93Gordon_equation."},{"key":"ref_20","unstructured":"Evans, L.C. Available online: https:\/\/math.berkeley.edu\/~evans\/entropy.and.PDE.pdf."},{"key":"ref_21","unstructured":"J\u00fcngel, A. Available online: http:\/\/www.asc.tuwien.ac.at\/~juengel\/scripts\/Bielefeld1203.pdf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.jcp.2015.03.026","article-title":"Entropy Stable Wall Boundary Conditions for the Three-Dimensional Compressible Navier\u2013Stokes Equations","volume":"292","author":"Parsani","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.physa.2015.02.068","article-title":"Crowd Macro State Detection Using Entropy Model","volume":"431","author":"Zhao","year":"2015","journal-title":"Physica A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/S0252-9602(15)30020-5","article-title":"Relative Entropy and Compressible Potential Flow","volume":"35","author":"Elling","year":"2015","journal-title":"Acta. Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.ijheatmasstransfer.2015.04.023","article-title":"Analysis of Entropy Generation in Double-Diffusive Natural Convection of Nano fluid","volume":"87","author":"Chen","year":"2015","journal-title":"Int. J. Heat Mass Tran."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.jcp.2015.04.026","article-title":"Entropy-Bounded Discontinuous Galerkin Scheme for Euler Equations","volume":"295","author":"Lv","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"365","DOI":"10.3390\/e10030365","article-title":"Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form","volume":"10","author":"Broadbridge","year":"2008","journal-title":"Entropy"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2248","DOI":"10.1016\/j.cnsns.2011.10.016","article-title":"One Method for Finding Exact Solutions of Nonlinear Differential Equations","volume":"17","author":"Kudryashov","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s12043-013-0520-9","article-title":"Exact Travelling Wave Solutions for some Important Nonlinear Physical Models","volume":"80","author":"Lee","year":"2013","journal-title":"Pramana"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3965","DOI":"10.1016\/j.amc.2011.09.027","article-title":"Application of the Kudryashov Method for Finding Exact Solutions of the High Order Nonlinear Evolution Equations","volume":"218","author":"Ryabov","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_31","first-page":"901540","article-title":"Generalized Kudryashov Method for Time-Fractional Differential Equations","volume":"2014","author":"Demiray","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_32","unstructured":"\u00c7imen, T. (September, January 28). On the Existence of Solutions Characterized by Riccati Equations to Infinite-Time Horizon Nonlinear Optimal Control Problems. Milano, Italy."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0022-0396(90)90092-4","article-title":"Existence Theorems for Nonlinear Second Order Boundary Value Problems","volume":"85","author":"Baxley","year":"1990","journal-title":"J. Differ. Equ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1017\/S0305004100055572","article-title":"Euler Operators Conservation Laws of the BBM Equation","volume":"85","author":"Olver","year":"1979","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1892","DOI":"10.1016\/j.physd.2010.06.011","article-title":"On the Stability of Periodic Solutions of the Generalized Benjamin\u2013Bona\u2013Mahony Equation","volume":"239","author":"Johnson","year":"2010","journal-title":"Physica D"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0022-0396(02)00061-X","article-title":"Existence and Stability of Solitary-Wave Solutions of Equations of Benjamin\u2013Bona\u2013Mahony Type","volume":"188","author":"Zeng","year":"2003","journal-title":"J. Differ. Equ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1137\/0508062","article-title":"Existence and Uniqueness for Periodic Solutions of the Benjamin\u2013Bona\u2013Mahony Equation","volume":"8","author":"Medeiros","year":"1977","journal-title":"SIAM J. Math. Anal."},{"key":"ref_38","first-page":"499","article-title":"Normal Forms and Global Existence of Solutions to a Class of Cubic Nonlinear Klein\u2013Gordon Equations in one Space Dimension","volume":"10","author":"Moriyama","year":"1997","journal-title":"Differ. Integr. Equ."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1007\/s00030-012-0196-8","article-title":"Unconditional Uniqueness in the Charge Class for the Dirac\u2013Klein\u2013Gordon Equations in two Space Dimensions","volume":"20","author":"Selberg","year":"2013","journal-title":"Nonlinear Differ. Equ. Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/4255\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:48:10Z","timestamp":1760215690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/4255"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,19]]},"references-count":39,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["e17064255"],"URL":"https:\/\/doi.org\/10.3390\/e17064255","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,19]]}}}