Research Article

Year 2022,
Volume: 12 Issue: 2, 162 - 176, 30.12.2022
### Abstract

### Keywords

### References

This study obtains some wave solutions of the B-type Kadomtsev Petviashvili equation by applying the modified exponential function method (MEFM). Thanks to this method, the exact solutions of the non-linear partial differential equations will be obtained and there will be an opportunity to examine the physical structure of these solutions. Due to the nature of MEFM, two different cases are presented here that have been analyzed to obtain more solutions in this structure. More wave solutions can be obtained by analyzing different situations. When the resulting solutions are analyzed, hyperbolic, trigonometric, and rational functions are observed. It has been checked whether the solution functions found with Wolfram Mathematica software provide the B type Kadomtsev Petviashvili equation and graphs simulating the wave solution behavior with the determined appropriate parameters are presented.

Modified Exponential Function Method (MEFM) B-type Kadomtsev Petviashvili Equation Wave Solutions Geliştirilmiş Üstel Fonksiyon Metodu (GÜFM) B tipi Kadomtsev Petviashvili Dalga Çözümleri

- [1] Elwakil, S.A., El-Labany, S.K., Zahran, M.A., Sabry, R., Modified extended tanh- function method for solving nonlinear partial differential equations, Physics Letters A, 299 (2- 3), 179-188, 2002.
- [2] Zheng, X., Chen, Y., Zhang, H., Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Physics Letters A, 311 (2-3), 145-157, 2003.
- [3] Liu, C.S., Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications, Communications in Theoretical Physics, 45 (2), 219-223, 2006.
- [4] Bulut, H., Baskonus, H.M., Pandir Y., The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, In Abstract and Applied Analysis Hindawi, Vol. 2013, 2013.
- [5] Gurefe, Y., Misirli, E., Sonmezoglu, A., Ekici, M., Extended trial equation method to generalized nonlinear partial differential equations, Applied Mathematics and Computation, 219 (10), 5253-5260, 2013.
- [6] Yang, X.F., Deng, Z.C., Wei, Y.A., Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Advances in Difference Equations, 2015 (1), 1- 17, 2015.
- [7] Baskonus, H.M., Bulut, H., Regarding on the prototype solutions for the nonlinear fractional-order biological population model, In AIP Conference Proceedings AIP Publishing LLC, 1738, 2016.
- [8] Abdelrahman, M.A., A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations, Nonlinear Engineering, 7 (4), 279-285, 2018.
- [9] Hosseini, K., Gholamin, P., Feng’s first integral method for analytic treatment of two higher dimensional nonlinear partial differential equations,Differential Equations and Dynamical Systems, 23 (3), 317-325, 2015.
- [10] Kutluay, S., Karta, M., Yağmurlu, N.M., Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau–KdV equation, Numerical Methods for Partial Differential Equations, 35, 2221–2235, 2019.
- [11] Yağmurlu, N.M., Karakaş, A.S., Numerical solutions of the equal width equation by trigonometric cubic B‐spline collocation method based on Rubin–Graves type linearization, Numerical Methods for Partial Differential Equations, 36 (5), 1170-1183, 2020.
- [12] Özer, S., Yağmurlu, N.M., Numerical solutions of nonhomogeneous Rosenau type equations by quintic B-spline collocation method, Mathematical Methods in the Applied Sciences, 45 (9), 5545–5558, 2022.
- [13] He, J.H., Wu, X.H., Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30 (3), 700-708, 2006.
- [14] Baskonus, H.M., Askin, M., Travelling wave simulations to the modified Zakharov- Kuzentsov model arising in plasma physics, In Litteris et Artibus, Lviv Polytechnic Publishing House. 2016.
- [15] Gurefe, Y., Misirli, E., Exp-function method for solving nonlinear evolution equations with higher order nonlinearity, Computers & Mathematics with Applications, 61 (8), 2025-2030, 2011.
- [16] Misirli, E., Gurefe, Y., The Exp-function method to solve the generalized Burgers- Fisher equation, Nonlinear Science Letters A, 323-328, 2010.
- [17] Misirli, E., Gurefe, Y., Exact solutions of the Drinfel’d–Sokolov–Wilson equation using the exp-function method, Applied Mathematics and Computation, 216 (9), 2623-2627, 2010.
- [18] Gao, X., Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid mechanics, Ocean Engineering, 245–247, 2015.
- [19] Cheng, L., Zhang, Y., Multiple wave solutions and auto-Bäcklund transformation for the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation, Computers and Mathematics with Applications, 70(5), 765-775, 2015.
- [20] Ma, W.X., Zhu, Z.N., Solving the (3+1)-dimensional generalized KP and BKP equations by the exp-function algorithm, Applied Mathematics and Computation, 218, 11871– 11879, 2012.
- [21] Nisar, K.S., Ilhan, O.A., Abdulazeez, S.T., Manafian, J., Mohammed, S.A., Osman, M.S., Novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method, Results in Physics 21,103769, 2021.
- [22] Asaad, M.G., Ma, W.X., Pfaffian solutions to a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation and its modified counterpart, Applied Mathematics and Computation 218, 5524–5542, 2012.
- [23] Cao, X., Lump Solutions to the (3+1)-Dimensional Generalized b-type Kadomtsev- Petviashvili Equation, Advances in Mathematical Physics, 5, 7843498, 2018.
- [24] Wazwaz, A.M., Distinct Kinds of Multiple-Soliton Solutions for a (3+1)-Dimensional Generalized B-type Kadomtsev-Petviashvili Equation, Physica Scripta, 84, 5, 055006, 2011.
- [25] Yokuş, A., Durur, H., Duran, S., Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism, Computational and Applied Mathematics, 41, 174, 2022.
- [26] Duran, S., Yokuş, A., Durur, H., Kaya, D., Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid dynamics, Modern Physics Letters B, Vol. 35, 26, 2150363, 2021.
- [27] Yokuş, A., Simulation of bright–dark soliton solutions of the Lonngren wave equation arising the model of transmission lines, Modern Physics Letters B,Vol. 35, 32, 2150484, 2021.
- [28] Duran, S., Kaya, D., Breaking analysis of solitary waves for the shallow water wave system in fluid dynamics, The European Physical Journal Plus, 136, 980, 2021.
- [29] Yokuş, A., Durur, H., Duran, S., Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation, Optical and Quantum Electronics 53, 402, 2021.

Year 2022,
Volume: 12 Issue: 2, 162 - 176, 30.12.2022
### Abstract

### References

Bu çalışmada, modifiye edilmiş üstel fonksiyon metodu uygulanarak B tipi Kadomtsev Petviashvili denkleminin bazı dalga çözümleri elde edilmiştir. Modifiye edilmiş üstel fonksiyon yönteminin doğası gereği, bu yapıdaki çözümlerden daha fazla elde etmek için incelenilmiş olan iki farklı durum burada sunulmuştur. Farklı durumlar da incelenerek daha fazla dalga çözümü elde edilebilir. Ortaya çıkan çözümler analiz edildiğinde hiperbolik, trigonometrik ve rasyonel fonksiyonlar gözlemlenmiştir. Wolfram Mathematica yazılımı ile bulunan çözüm fonksiyonlarının B tipi Kadomtsev Petviashvili denklemini sağlayıp sağlamadığı kontrol edilmiş ve belirlenen uygun parametrelerle dalga çözümünün üç boyutlu kontur, yoğunluk ve iki boyutlu grafiklerin analizi sunulmuştur.

- [1] Elwakil, S.A., El-Labany, S.K., Zahran, M.A., Sabry, R., Modified extended tanh- function method for solving nonlinear partial differential equations, Physics Letters A, 299 (2- 3), 179-188, 2002.
- [2] Zheng, X., Chen, Y., Zhang, H., Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Physics Letters A, 311 (2-3), 145-157, 2003.
- [3] Liu, C.S., Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications, Communications in Theoretical Physics, 45 (2), 219-223, 2006.
- [4] Bulut, H., Baskonus, H.M., Pandir Y., The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, In Abstract and Applied Analysis Hindawi, Vol. 2013, 2013.
- [5] Gurefe, Y., Misirli, E., Sonmezoglu, A., Ekici, M., Extended trial equation method to generalized nonlinear partial differential equations, Applied Mathematics and Computation, 219 (10), 5253-5260, 2013.
- [6] Yang, X.F., Deng, Z.C., Wei, Y.A., Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Advances in Difference Equations, 2015 (1), 1- 17, 2015.
- [7] Baskonus, H.M., Bulut, H., Regarding on the prototype solutions for the nonlinear fractional-order biological population model, In AIP Conference Proceedings AIP Publishing LLC, 1738, 2016.
- [8] Abdelrahman, M.A., A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations, Nonlinear Engineering, 7 (4), 279-285, 2018.
- [9] Hosseini, K., Gholamin, P., Feng’s first integral method for analytic treatment of two higher dimensional nonlinear partial differential equations,Differential Equations and Dynamical Systems, 23 (3), 317-325, 2015.
- [10] Kutluay, S., Karta, M., Yağmurlu, N.M., Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau–KdV equation, Numerical Methods for Partial Differential Equations, 35, 2221–2235, 2019.
- [11] Yağmurlu, N.M., Karakaş, A.S., Numerical solutions of the equal width equation by trigonometric cubic B‐spline collocation method based on Rubin–Graves type linearization, Numerical Methods for Partial Differential Equations, 36 (5), 1170-1183, 2020.
- [12] Özer, S., Yağmurlu, N.M., Numerical solutions of nonhomogeneous Rosenau type equations by quintic B-spline collocation method, Mathematical Methods in the Applied Sciences, 45 (9), 5545–5558, 2022.
- [13] He, J.H., Wu, X.H., Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30 (3), 700-708, 2006.
- [14] Baskonus, H.M., Askin, M., Travelling wave simulations to the modified Zakharov- Kuzentsov model arising in plasma physics, In Litteris et Artibus, Lviv Polytechnic Publishing House. 2016.
- [15] Gurefe, Y., Misirli, E., Exp-function method for solving nonlinear evolution equations with higher order nonlinearity, Computers & Mathematics with Applications, 61 (8), 2025-2030, 2011.
- [16] Misirli, E., Gurefe, Y., The Exp-function method to solve the generalized Burgers- Fisher equation, Nonlinear Science Letters A, 323-328, 2010.
- [17] Misirli, E., Gurefe, Y., Exact solutions of the Drinfel’d–Sokolov–Wilson equation using the exp-function method, Applied Mathematics and Computation, 216 (9), 2623-2627, 2010.
- [18] Gao, X., Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid mechanics, Ocean Engineering, 245–247, 2015.
- [19] Cheng, L., Zhang, Y., Multiple wave solutions and auto-Bäcklund transformation for the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation, Computers and Mathematics with Applications, 70(5), 765-775, 2015.
- [20] Ma, W.X., Zhu, Z.N., Solving the (3+1)-dimensional generalized KP and BKP equations by the exp-function algorithm, Applied Mathematics and Computation, 218, 11871– 11879, 2012.
- [21] Nisar, K.S., Ilhan, O.A., Abdulazeez, S.T., Manafian, J., Mohammed, S.A., Osman, M.S., Novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method, Results in Physics 21,103769, 2021.
- [22] Asaad, M.G., Ma, W.X., Pfaffian solutions to a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation and its modified counterpart, Applied Mathematics and Computation 218, 5524–5542, 2012.
- [23] Cao, X., Lump Solutions to the (3+1)-Dimensional Generalized b-type Kadomtsev- Petviashvili Equation, Advances in Mathematical Physics, 5, 7843498, 2018.
- [24] Wazwaz, A.M., Distinct Kinds of Multiple-Soliton Solutions for a (3+1)-Dimensional Generalized B-type Kadomtsev-Petviashvili Equation, Physica Scripta, 84, 5, 055006, 2011.
- [25] Yokuş, A., Durur, H., Duran, S., Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism, Computational and Applied Mathematics, 41, 174, 2022.
- [26] Duran, S., Yokuş, A., Durur, H., Kaya, D., Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid dynamics, Modern Physics Letters B, Vol. 35, 26, 2150363, 2021.
- [27] Yokuş, A., Simulation of bright–dark soliton solutions of the Lonngren wave equation arising the model of transmission lines, Modern Physics Letters B,Vol. 35, 32, 2150484, 2021.
- [28] Duran, S., Kaya, D., Breaking analysis of solitary waves for the shallow water wave system in fluid dynamics, The European Physical Journal Plus, 136, 980, 2021.
- [29] Yokuş, A., Durur, H., Duran, S., Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation, Optical and Quantum Electronics 53, 402, 2021.

There are 29 citations in total.

Primary Language | English |
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Subjects | Mathematical Sciences, Applied Mathematics |

Journal Section | Mathematics |

Authors | |

Publication Date | December 30, 2022 |

Submission Date | May 9, 2022 |

Acceptance Date | September 8, 2022 |

Published in Issue | Year 2022 Volume: 12 Issue: 2 |

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