Document Type : Original Article
Authors
1 Nevsehir Haci Bektas Veli University, Department of Mathematics, Nevsehir, Turkey
2 Mutah University, Department of Mathematics and Statistics, Al-Karak, Jordan
3 Prairie View A&M University, Department of Mathematics, Prairie View, TX, USA
Abstract
In this study, our main goal is to study the exact traveling wave solutions of some recent nonlinear evolution equations, namely, modified generalized (3+1)-dimensional time-fractional Kadomtsev–Petviashvili (KP) and Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equations of conformable type. We employed a consistent analytical method called the generalized Riccati equation mapping method, along with a conformable derivative to extract the multiple kinks, bi-symmetry soliton, bright and dark soliton solutions, periodic solutions, and singular solutions for suggested equations. The theoretical method is based on the Riccati equation and a number of empirical solutions have been proposed that do not exist in the literature. Furthermore, as the order of the fractional derivative approaches one, the exact solutions obtained by the current method are reduced to classical solutions. The obtained results show that the present technique is effective, easy to implement, and a strong tool for solving nonlinear fractional partial differential equations, and produces a very large number of solutions.
Keywords
[1] Şenol, B., & Demiroğlu, U. (2019). Frequency frame approach on loop shaping of first order plus time delay systems using fractional order PI controller. ISA transactions, 86, 192-200.
[2] He, J.H., & Ji, F.Y. (2019). Two-scale mathematics and fractional calculus for thermodynamics. Thermal Science, 23(4), 2131-2133.
[3] Wang, G., Liu, Y., Wu, Y., & Su, X. (2020). Symmetry analysis for a seventh-order generalized KdV equation and its fractional version in fluid mechanics. Fractals, 28(3), 2050044-134.
[4] Kumar, A., Komaragiri, R., & Kumar, M. (2019). Design of efficient fractional operator for ECG signal detection in implantable cardiac pacemaker systems. International Journal of Circuit Theory and Applications, 47(9), 1459-1476.
[5] Mashayekhi, S., Hussaini, M.Y., & Oates, W. (2019). A physical interpretation of fractional viscoelasticity based on the fractal structure of media: theory and experimental validation. Journal of the Mechanics and Physics of Solids, 128, 137-150.
[6] Abdou, M.A. (2019). On the fractional order space-time nonlinear equations arising in plasma physics. Indian Journal of Physics, 93(4), 537-541.
[7] Iyiola, O.S., Oduro, B., & Akinyemi, L. (2021). Analysis and solutions of generalized Chagas vectors reinfestation model of fractional order type. Chaos, Solitons Fractals, 145, 110797.
[8] Owusu-Mensah, I., Akinyemi, L., Oduro, B., & Iyiola, O.S. (2020). A fractional order approach to modeling and simulations of the novel COVID-19. Adv. Differ. Equ., 2020(1), 1-21. https://doi.org/10.1186/s13662-020-03141-7. Epub 2020 Dec 3. PMID: 33288983; PMCID: PMC7711272
[9] Pellegrino, E., Pezza, L., & Pitolli, F. (2020). A collocation method in spline spaces for the solution of linear fractional dynamical systems. Mathematics and Computers in Simulation, 176, 266-278.
[10] Tasbozan, O., Çenesiz, Y., Kurt, A., & Baleanu, D. (2017). New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method. Open Physics, 15(1), 647-651.
[11] Rezazadeh, H. (2018). New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Optik, 167, 218-227.
[12] Vahidi, J., Zabihi, A., Rezazadeh, H., & Ansari, R. (2021). New extended direct algebraic method for the resonant nonlinear Schrödinger equation with Kerr law nonlinearity. Optik, 227, 165936.
[13] Senol, M. (2020). New analytical solutions of fractional symmetric regularized-long-wave equation. Revista Mexicana de Fisica, 66(3 May-Jun), 297-307.
[14] Rezazadeh, H., Ullah, N., Akinyemi, L., Shah, A., Mirhosseini-Alizamin, S.M., Chu, Y.M., & Ahmad, H. (2021). Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov's method. Results in Phys., 24, 104179.
[15] Osman, M.S., Korkmaz, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., & Zhou, Q. (2018). The unified method for conformable time fractional Schrödinger equation with perturbation terms. Chinese Journal of Physics, 56(5), 2500-2506.
[16] Kolebaje, O., Bonyah, E., & Mustapha, L. (2019). The first integral method for two fractional non-linear biological models. Discrete Continuous Dynamical Systems-S, 12(3), 487.
[17] Ghanbari, B., Osman, M. S., & Baleanu, D. (2019). Generalized exponential rational function method for extended Zakharov-Kuzetsov equation with conformable derivative. Modern Physics Letters A, 34(20), 1950155.
[18] Akinyemi, L., Senol, M., & Iyiola, O.S. (2021). Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method, Math. Comput. Simul., 182, 211-233. https://doi.org/10.1016/j.matcom.2020.10.017
[19] Senol, M., Akinyemi, L., Ata, A., & Iyiola, O.S. (2021). Approximate and generalized solutions of conformable type Coudrey-Dodd-Gibbon-Sawada-Kotera equation. International Journal of Modern Physics B, 35(02), 2150021.
[20] Rezazadeh, H., Inc, M., & Baleanu, D. (2020). New Solitary Wave Solutions for Variants of (3+1)- Dimensional Wazwaz-Benjamin-Bona-Mahony Equations. Frontiers in Phys., 8, 1-11.
[21] Seadawy, A.R., Ali, K.K., & Nuruddeen, R.I. (2019). A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations. Results Phys., 12, 2234-2241.
[22] Az-Zo’bi, E.A. (2019). New kink solutions for the van der Waals p‐system, Mathematical Methods in the Applied Sciences, 42(18), 6216-6226.
[23] Az-Zo’bi, E.A. (2019). Peakon and solitary wave solutions for the modified Fornberg-Whitham equation using simplest equation method. International Journal of Mathematics and Computer Sci., 14(3), 635-645.
[24] Az-Zo’bi, E.A., AlZoubi, W.A., Akinyemi, L., et al. (2021). Abundant closed-form solitons for timefractional integro–differential equation in fluid dynamics. Opt. Quant. Electron, 53, 132. https://doi.org/10.1007/s11082-021-02782-6
[25] Vahidi, J., Zekavatmand, S.M., Rezazadeh, H., Inc, M., Akinlar, M.A., & Chu, Y.M. (2021). New solitary wave solutions to the coupled Maccari’s system. Results in Physics, 21, 103801.
[26] Akinyemi, L., Senol, M., Mirzazadeh, M., & Eslami, M. (2021). Optical solitons for weakly nonlocal Schrödinger equation with parabolic law nonlinearity and external potential. Optik, 230 1-9.
[27] Leta, T.D., Liu, W., El Achab, A., & Rezazadeh, H. (2021). A Bekir Dynamical Behavior of Traveling Wave Solutions for a (2+1)-Dimensional Bogoyavlenskii Coupled System. Qualitative Theory of Dynamical Systems, 20(1), 1-22.
[28] Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S.P., & Belic, M. (2018). Optical solitons having weak non-local nonlinearity by two integration schemes, Optik, 164 380–384.
[29] Hosseini, K., Salahshour, S., & Mirzazadeh, M. (2021). Bright and dark solitons of a weakly nonlocal Schrödinger equation involving the parabolic law nonlinearity. Optik, 227, 166042.
[30] Hosseini, K., Mirzazadeh, M., & Gómez-Aguilar, J.F. (2020). Soliton solutions of the Sasa–Satsuma equation in the monomode optical fibers including the beta-derivatives. Optik, 224, 165425.
[31] Hosseini, K., Mirzazadeh, M., Vahidi, J., & Asghari, R. (2020). Optical wave structures to the Fokas– Lenells equation. Optik, 207, 164450.
[32] Hosseini, K., Mirzazadeh, M., Ilie, M., & Gómez-Aguilar, J.F. (2020). Biswas–Arshed equation with the beta time derivative: optical solitons and other solutions. Optik, 217, 164801.
[33] Hosseini, K., Mirzazadeh, M., Rabiei, F., Baskonus, H.M., & Yel, G. (2020). Dark optical solitons to the Biswas–Arshed equation with high order dispersions and absence of the self-phase modulation. Optik, 209, 164576.
[34] Akinyemi, L., Şenol, M., Rezazadeh, H., Ahmad, H., & Wang, H. (2021). Abundant optical soliton solutions for an integrable (2+1)-dimensional nonlinear conformable Schrödinger system. Results in Phys. 104177. https://doi.org/10.1016/j.rinp.2021.104177
[35] El-Tawil M.A. & Huseen S.N. (2012). The Q-homotopy analysis method (q-HAM). Int. J. Appl. Math. Mech., 8 (15), 51-75.
[36] Akinyemi L. (2019). q-Homotopy analysis method for solving the seventh-order time-fractional Lax's Korteweg–de Vries and Sawada–Kotera equations. Comp. Appl. Math., 38(4), 1-22.
[37] Akinyemi L., Iyiola O.S., & Akpan U. (2020). Iterative methods for solving fourth- and sixth order timefractional Cahn-Hillard equation. Math. Meth. Appl. Sci., 43(7), 4050–4074. https://doi.org/10.1002/mma.6173
[38] Az-Zo’bi, E.A. (2018). A reliable analytic study for higher-dimensional telegraph equation, Journal of Mathematics and Computer Science, 18, 423–429.
[39] Senol M., Iyiola O.S., Daei Kasmaei H., & Akinyemi L. (2019). Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent-Miodek system with energy-dependent Schrödinger potential. Adv. Differ. Equ., 2019, 1-21.
[40] Senol, M., Tasbozan, O., & Kurt, A. (2019). Numerical solutions of fractional Burgers' type equations with conformable derivative. Chinese Journal of Physics, 58, 75-84.
[41] Senol, M., Kurt, A., Atilgan, E., & Tasbozan, O. (2019). Numerical solutions of fractional BoussinesqWhitham-Broer-Kaup and diffusive Predator-Prey equations with conformable derivative. New Trends in Mathematical Sciences, 7(3), 286-300.
[42] Senol M. (2020). Analytical and approximate solutions of (2+1)-dimensional time-fractional BurgersKadomtsev-Petviashvili equation. Commun. Theor. Phys., 72(5), 1-11.
[43] Johnston, S.J., Jafari, H., Moshokoa, S.P., Ariyan, V.M., & Baleanu, D. (2016). Laplace homotopy perturbation method for Burgers equation with space-and time-fractional order. Open Physics, 14(1), 247-252.
[44] Akinyemi L. & Iyiola O.S. (2020). Exact and approximate solutions of time-fractional models arising from physics via Shehu transform. Math. Meth. Appl. Sci., 43(12), 7442-7464. https://doi.org/10.1002/mma.6484
[45] Akinyemi L. (2020). A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction. Comp. Appl. Math., 39, 1-34. https://doi.org/10.1007/s40314-020-01212-9
[46] Akinyemi L. & Huseen S.N., (2020). A powerful approach to study the new modified coupled Korteweg–de Vries system. Math. Comput. Simul., 177, 556-567. https://doi.org/10.1016/j.matcom.2020.05.021
[47] Akinyemi L. & Iyiola O.S. (2020). A reliable technique to study nonlinear time-fractional coupled Korteweg-de Vries equations. Adv. Differ. Equ., 2020, 1-27. https://doi.org/10.1186/s13662-020-02625-w
[48] Adomian G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Kluwer.
[49] Az-Zo’bi, E.A. (2014). An Approximate Analytic Solution for Isentropic Flow by An Inviscid Gas Equations, Archives of Mechanics, 66(3), 203-212.
[50] Az-Zo’bi, E.A. (2013). Construction of Solutions for Mixed Hyperbolic Elliptic Riemann Initial Value System of Conservation Laws, Applied Mathematical Modeling, 37, 6018-6024.
[51] Az-Zo’bi, E.A., Al Dawoud K., & Marashdeh, M.F. (2015). Numeric-analytic solutions of mixed-type systems of balance laws, Applied Mathematics and Comput., 265, 133–143.
[52] Akinyemi, L., Senol, M., & Huseen, S.N. (2021). Modified homotopy methods for generalized fractional perturbed Zakharov-Kuznetsov equation in dusty plasma. Adv. Differ. Equ. 2021(1), 1-27.
https://doi.org/10.1186/s13662-020-03208-5
https://doi.org/10.1186/s13662-020-03208-5
[53] Wazwaz A.M. & El-Tantawy S. (2016). A new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation. Nonlinear Dyn., 84, 1107–1112.
[54] Khalil R., Al Horani M., Yousef A., & Sababheh M. (2014). A new definition of fractional derivative. J. Comput. Appl. Math., 264, 65-70.
[55] Abdeljawad T. (2015). On conformable fractional calculus. J. Comput. Appl. Math., 279, 57–66.
[56] El-Ganaini S. & Al-Amr M.O. (2019). New abundant wave solutions of the conformable space-time fractional (4+1)-dimensional Fokas equation in water waves. Comput. Math. Appl., 78(6), 2094-2106. https://doi.org/10.1016/j.camwa.2019.03.050
[57] Bekir A. & Guner O. (2013). Bright and dark soliton solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation. Pramana- J. Phys., 81, 203-214.
[58] Taghizadeh N., Mirzazadeh M., & Noori S.R.M. (2012). Exact solutions of the generalized Benjamin equation and (3+1)-dimensional Gkp equation by the extended tanh method. Appl. Math. Int. J., 7(1), 175-187.
[59] Wazwaz A.M. (2005). Exact solutions of compact and noncompact structures for the KP-BBM equation. Appl. Math. Comput., 169(1), 700-712.
[60] Wazwaz A.M. (2012). Multiple-soliton solutions for a (3+1)-dimensional generalized KP equation. Commun. Nonlinear Sci. Numer. Simul., 17, 491–495.
[61] Wazwaz A.M. (2011). Multi-front waves for extended form of modified Kadomtsev-Petviashvili equations. Appl. Math. Mech., 32(7), 875–880.
)