Research Article
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Year 2025, Volume: 11 Issue: 2, 197 - 209, 30.06.2025

Sevil Çulha Ünal

https://doi.org/10.28979/jarnas.1684271

Abstract

References

  • E. Yusufoğlu, New solitonary solutions for the MBBM equations using exp-function method, Physics Letters A 372 (2008) 442-446.
  • A-M. Wazwaz, Partial differential equations and solitary waves theory, Nonlinear Physical Science, Springer, 2009.
  • E. M. E. Zayed, S. Al-Joudi, Applications of an extended (G^'/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Mathematical Problems in Engineering (2010) 768573.
  • K. Khan, M. A. Akbar, S. M. R. Islam, Exact solutions for (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations, SpringerPlus 3 (1) (2014).
  • A. Yokus, A. Sulaiman, M. T. Gulluoglu, H. Bulut, Stability analysis, numerical and exact solutions of the (1+1)-dimensional NDMBBM equation, ITM Web of Conferences 22 (2018) 01064.
  • X-W. Yan, S-F. Tian, M-J. Dong, X-B. Wang, T-T. Zhang, Nonlocal symmetries, conservation laws and interaction solutions of the generalised dispersive modified Benjamin-Bona-Mahony equation, Zeitschrift für Naturforschung A 73 (5) (2018) 399-405.
  • F. Taşcan, A. Akbulut, Construction of exact solutions to partial differential equations with CRE method, Communications in Advanced Mathematical Sciences 2 (2) (2019) 105-113.
  • T. Alotaibi, A. Althobaiti, Exact solutions of the nonlinear modified Benjamin-Bona-Mahony equation by an analytical method, Fractal and Fractional 6 (399) (2022).
  • M. Shakeel, A. Ullah, E. R. El-Zahar, N. A. Shah, J. D. Chung, Generalized exp-function method to find closed form solutions of nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves, Mathematics 10 (7) (2022).
  • B. Kopçasız, E. Yaşar, μ-symmetries and μ-conservation laws for the nonlinear dispersive modified Benjamin-Bona-Mahony equation, Journal of Mathematical Sciences and Modelling 6 (3) (2023) 87-96.
  • H. M. Baskonus, H. Bulut, Analytical studies on the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves, Waves in Random and Complex Media 25 (4) (2015).
  • S-W. Yao, K. U. Tariq, M. Inc, R. N. Tufail, Modulation instability analysis and soliton solutions of the modified BBM model arising in dispersive medium, Results in Physics 46 (2023) 106274.
  • I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
  • B. Ahmad, J. J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstract and Applied Analysis (2009) 494720.
  • Y. Wang, S. Liang, Q. Wang, Existence results for fractional differential equations with integral and multi-point boundary conditions, Boundary Value Problems (2018) https://doi.org/10.1186/s13661-017-0924-4.
  • M. Şenol, A. Ata, Approximate solution of time-fractional KdV equations by residual power series method, Journal of Balıkesir University Institute of Science and Technology 20 (1) (2018) 430-439.
  • G. Akram, M. Sadaf, M. Abbas, I. Zainab, S. R. Gillani, Efficient techniques for traveling wave solutions of time-fractional Zakharov-Kuznetsov equation, Mathematics and Computers in Simulation 193 (2022) 607-622.
  • M. M. A. Qurashi, Z. Korpinar, D. Baleanu, M. Inc, A new iterative algorithm on the time-fractional Fisher equation: residual power series method, Advances in Mechanical Engineering 9 (9) (2017).
  • Z. Körpınar, The residual power series method for solving fractional Klein-Gordon equation, Sakarya University Journal of Science 21 (3) (2017) 285-293.
  • F. Batool, G. Akram, Application of extended Fan sub-equation method to (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution, Optical and Quantum Electronics 49 (375) (2017).
  • Md. Asaduzzaman, F. Jesmin, Construction of the closed form wave solutions for TFSMCH and (1+1) dimensional TFDMBBM equations via the EMSE technique, Fractal and Fractional 9 (72) (2025).
  • N. Sribua-Iam and S. Chinviriyasit, New analytical wave solutions of fractional order DMBBM and Bateman-Burgers equations, Frontiers in Applied Mathematics and Statistics (2025) https://doi.org/10.3389/fams.2025.1568834.
  • H. U. Rehman, A. Amer, A. Amer, New solitary wave solutions of generalized fractional Tzitzѐica-type evolution equations using Sardar sub-equation method, Optical and Quantum Electronics 55 (13) (2023).
  • H-D. Guo, T-C. Xia, B-B. Hu, High-order lumps, high-order breathers and hybrid solutions for an extended (3+1)-dimensional Jimbo-Miwa equation in fluid dynamics, Nonlinear Dynamics 100 (1) (2020) 601-614.
  • H-D. Guo, T-C. Xia, B-B. Hu, Dynamics of abundant solutions to the (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama equation, Applied Mathematics Letters 105 (2020) 106301.
  • N. Ullah, M. I. Asjad, J. Awrejcewicz, T. Muhammad, D. Baleanu, On soliton solutions of fractional-order nonlinear model appears in physical sciences, AIMS Mathematics 7 (5) (2022) 7421-7440.
  • M. I. Asjad, N. Ullah, H. Rehman, D. Baleanu, Optical solitons for conformable space-time fractional nonlinear model, Journal of Mathematics and Computer Science 27 (2022) 28-41.
  • A. K. Alsharidi, A. Bekir, Discovery of new exact wave solutions to the M-fractional complex three coupled Maccari’s system by Sardar sub-equation scheme, Symmetry 15 (2023).
  • M. Raheel, A. Zafar, A. Bekir, K. U. Tariq, Exact wave solutions and obliqueness of truncated M-fractional Heisenberg ferromagnetic spin chain model through two analytical techniques, Waves in Random and Complex Media (2023) https://doi.org/10.1080/17455030.2023.2173550.
  • T. Rasool, R. Hussain, H. Rezazadeh, A. Ali, U. Demirbilek, Novel soliton structures of truncated M-fractional (4+1)-dim Fokas wave model, Nonlinear Engineering 12 (2023).
  • M. M. A. Khater, Multi-vector with nonlocal and non-singular kernel ultrashort optical solitons pulses waves in birefringent fibers, Chaos, Solitons and Fractals 167 (2023).
  • C. Pleumpreedaporn, S. Pleumpreedaporn, E. J. Moore, S. Sirisubtawee, S. Sungnul, Novel exact traveling wave solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with Atangana’s space and time beta-derivatives via the Sardar subequation method, Thai Journal of Mathematics 22 (1) (2024).
  • A. A. Altwaty, J. J. Amhalhil, A. E. Sakori, N. F. Meriki, The Sardar sub-equation technique for obtaining some optical solitons of cubic nonlinear Schrödinger equation involving beta derivatives with Kerr law nonlinearity, The Scientific Journal of University of Benghazi 37 (1) (2024) 28-36.
  • M. Awadalla, A. Taishiyeva, R. Myrzakulov, J. Alahmadi, A. A. Zaagan, A. Bekir, Exact analytical soliton solutions of the M-fractional Akbota equation, Scientific Reports 14 (2024) 13360.
  • D. Chou, H. U. Rehman, A. Amer, A. Amer, New solitary wave solutions of generalized fractional Tzitzeica‑type evolution equations using Sardar sub‑equation method, Optical and Quantum Electronics 55 (2023) 1148.
  • Md. N. Hossain, M. M. Miah, M. Alosaimi, F. Alsharif, M. Kanan, Exploring novel soliton solutions to the time-fractional coupled Drinfel'd-Sokolov-Wilson equation in industrial engineering using two efficient techniques, Fractal and Fractional 8 (2024) 352.
  • H. Qawaqneh, A. Altalbe, A. Bekir, K. U. Tariq, Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential, AIMS Mathematics 9 (9) (2024).
  • A. Said, A. Bakkar, H. Khan, C. Cattani, F. Tchier, Construction of mechanically preserved optical travelling wave solution for fractional generalized reaction Duffing model, Journal of Applied and Computational Mechanics (2025) https://doi.org/10.22055/jacm.2025.48049.4941.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014) 65-70.
  • T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
  • H. Rezazadeh, M. Inc, D. Baleanu, New solitary wave solutions for variants of (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations, Frontiers in Physics 8 (2020).
  • H. Miyakawa, S. Takeuchi, Applications of a duality between generalized trigonometric and hyperbolic functions, Journal of Mathematical Analysis and Applications 502 (1) (2021).
  • E. Neuman, Inequalities for the generalized trigonometric, hyperbolic and Jacobian elliptic functions, Journal of Mathematical Inequalities 9 (3) (2015) 709-726.
  • H. Kobayashi, S. Takeuchi, Applications of generalized trigonometric functions with two parameters, Communications on Pure and Applied Analysis 18 (3) (2019) 1509-1521.
  • S. Takeuchi, Applications of generalized trigonometric functions with two parameters II, Differential Equations and Applications 11 (4) (2019) 563-575.
  • S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, Journal of Mathematical Analysis and Applications 444 (2) (2016) 1000-1014.
  • O. Büttner, M. Bauer, A. Rueff, S. O. Demokritov, B. Hillebrands, A. N. Slavin, M. P. Kostylev, B. A. Kalinikos, Space- and time-resolved Brillouin light scattering from nonlinear spin-wave packets, Ultrasonics 38 (2000) 443-449.
  • M. Mossa Al-Sawalha, S. Mukhtar, A. S. Alshehry, M. Alqudah, M. S. Aldhabani, Kink soliton phenomena of fractional conformable Kairat equations, AIMS Mathematics, 10 (2) (2025) 2808-2828.

Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation

Year 2025, Volume: 11 Issue: 2, 197 - 209, 30.06.2025

Sevil Çulha Ünal

https://doi.org/10.28979/jarnas.1684271

Abstract

In mathematical physics, the nonlinear dispersive modified Benjamin-Bono-Mahony equation is an essential nonlinear evolution equation. This paper employs the Sardar sub-equation approach to derive the analytic solutions of the space-time fractional nonlinear dispersive modified Benjamin-Bona-Mahony equation with a conformable derivative. These solutions, obtained using the proposed approach, are presented in specialized generalized hyperbolic and trigonometric forms. The soliton solutions are also obtained using the presented approach. Additionally, some of these solutions are illustrated in both two- and three-dimensional graphs for visualization. Finally, the suggested approach is novel, easy, and productive for obtaining analytic solutions and is applicable to numerous nonlinear fractional partial differential equations.

Keywords

Sardar sub-equation approach fractional NDMBBM equation soliton solutions special generalized trigonometric and hyperbolic functions

References

  • E. Yusufoğlu, New solitonary solutions for the MBBM equations using exp-function method, Physics Letters A 372 (2008) 442-446.
  • A-M. Wazwaz, Partial differential equations and solitary waves theory, Nonlinear Physical Science, Springer, 2009.
  • E. M. E. Zayed, S. Al-Joudi, Applications of an extended (G^'/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Mathematical Problems in Engineering (2010) 768573.
  • K. Khan, M. A. Akbar, S. M. R. Islam, Exact solutions for (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations, SpringerPlus 3 (1) (2014).
  • A. Yokus, A. Sulaiman, M. T. Gulluoglu, H. Bulut, Stability analysis, numerical and exact solutions of the (1+1)-dimensional NDMBBM equation, ITM Web of Conferences 22 (2018) 01064.
  • X-W. Yan, S-F. Tian, M-J. Dong, X-B. Wang, T-T. Zhang, Nonlocal symmetries, conservation laws and interaction solutions of the generalised dispersive modified Benjamin-Bona-Mahony equation, Zeitschrift für Naturforschung A 73 (5) (2018) 399-405.
  • F. Taşcan, A. Akbulut, Construction of exact solutions to partial differential equations with CRE method, Communications in Advanced Mathematical Sciences 2 (2) (2019) 105-113.
  • T. Alotaibi, A. Althobaiti, Exact solutions of the nonlinear modified Benjamin-Bona-Mahony equation by an analytical method, Fractal and Fractional 6 (399) (2022).
  • M. Shakeel, A. Ullah, E. R. El-Zahar, N. A. Shah, J. D. Chung, Generalized exp-function method to find closed form solutions of nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves, Mathematics 10 (7) (2022).
  • B. Kopçasız, E. Yaşar, μ-symmetries and μ-conservation laws for the nonlinear dispersive modified Benjamin-Bona-Mahony equation, Journal of Mathematical Sciences and Modelling 6 (3) (2023) 87-96.
  • H. M. Baskonus, H. Bulut, Analytical studies on the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves, Waves in Random and Complex Media 25 (4) (2015).
  • S-W. Yao, K. U. Tariq, M. Inc, R. N. Tufail, Modulation instability analysis and soliton solutions of the modified BBM model arising in dispersive medium, Results in Physics 46 (2023) 106274.
  • I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
  • B. Ahmad, J. J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstract and Applied Analysis (2009) 494720.
  • Y. Wang, S. Liang, Q. Wang, Existence results for fractional differential equations with integral and multi-point boundary conditions, Boundary Value Problems (2018) https://doi.org/10.1186/s13661-017-0924-4.
  • M. Şenol, A. Ata, Approximate solution of time-fractional KdV equations by residual power series method, Journal of Balıkesir University Institute of Science and Technology 20 (1) (2018) 430-439.
  • G. Akram, M. Sadaf, M. Abbas, I. Zainab, S. R. Gillani, Efficient techniques for traveling wave solutions of time-fractional Zakharov-Kuznetsov equation, Mathematics and Computers in Simulation 193 (2022) 607-622.
  • M. M. A. Qurashi, Z. Korpinar, D. Baleanu, M. Inc, A new iterative algorithm on the time-fractional Fisher equation: residual power series method, Advances in Mechanical Engineering 9 (9) (2017).
  • Z. Körpınar, The residual power series method for solving fractional Klein-Gordon equation, Sakarya University Journal of Science 21 (3) (2017) 285-293.
  • F. Batool, G. Akram, Application of extended Fan sub-equation method to (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution, Optical and Quantum Electronics 49 (375) (2017).
  • Md. Asaduzzaman, F. Jesmin, Construction of the closed form wave solutions for TFSMCH and (1+1) dimensional TFDMBBM equations via the EMSE technique, Fractal and Fractional 9 (72) (2025).
  • N. Sribua-Iam and S. Chinviriyasit, New analytical wave solutions of fractional order DMBBM and Bateman-Burgers equations, Frontiers in Applied Mathematics and Statistics (2025) https://doi.org/10.3389/fams.2025.1568834.
  • H. U. Rehman, A. Amer, A. Amer, New solitary wave solutions of generalized fractional Tzitzѐica-type evolution equations using Sardar sub-equation method, Optical and Quantum Electronics 55 (13) (2023).
  • H-D. Guo, T-C. Xia, B-B. Hu, High-order lumps, high-order breathers and hybrid solutions for an extended (3+1)-dimensional Jimbo-Miwa equation in fluid dynamics, Nonlinear Dynamics 100 (1) (2020) 601-614.
  • H-D. Guo, T-C. Xia, B-B. Hu, Dynamics of abundant solutions to the (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama equation, Applied Mathematics Letters 105 (2020) 106301.
  • N. Ullah, M. I. Asjad, J. Awrejcewicz, T. Muhammad, D. Baleanu, On soliton solutions of fractional-order nonlinear model appears in physical sciences, AIMS Mathematics 7 (5) (2022) 7421-7440.
  • M. I. Asjad, N. Ullah, H. Rehman, D. Baleanu, Optical solitons for conformable space-time fractional nonlinear model, Journal of Mathematics and Computer Science 27 (2022) 28-41.
  • A. K. Alsharidi, A. Bekir, Discovery of new exact wave solutions to the M-fractional complex three coupled Maccari’s system by Sardar sub-equation scheme, Symmetry 15 (2023).
  • M. Raheel, A. Zafar, A. Bekir, K. U. Tariq, Exact wave solutions and obliqueness of truncated M-fractional Heisenberg ferromagnetic spin chain model through two analytical techniques, Waves in Random and Complex Media (2023) https://doi.org/10.1080/17455030.2023.2173550.
  • T. Rasool, R. Hussain, H. Rezazadeh, A. Ali, U. Demirbilek, Novel soliton structures of truncated M-fractional (4+1)-dim Fokas wave model, Nonlinear Engineering 12 (2023).
  • M. M. A. Khater, Multi-vector with nonlocal and non-singular kernel ultrashort optical solitons pulses waves in birefringent fibers, Chaos, Solitons and Fractals 167 (2023).
  • C. Pleumpreedaporn, S. Pleumpreedaporn, E. J. Moore, S. Sirisubtawee, S. Sungnul, Novel exact traveling wave solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with Atangana’s space and time beta-derivatives via the Sardar subequation method, Thai Journal of Mathematics 22 (1) (2024).
  • A. A. Altwaty, J. J. Amhalhil, A. E. Sakori, N. F. Meriki, The Sardar sub-equation technique for obtaining some optical solitons of cubic nonlinear Schrödinger equation involving beta derivatives with Kerr law nonlinearity, The Scientific Journal of University of Benghazi 37 (1) (2024) 28-36.
  • M. Awadalla, A. Taishiyeva, R. Myrzakulov, J. Alahmadi, A. A. Zaagan, A. Bekir, Exact analytical soliton solutions of the M-fractional Akbota equation, Scientific Reports 14 (2024) 13360.
  • D. Chou, H. U. Rehman, A. Amer, A. Amer, New solitary wave solutions of generalized fractional Tzitzeica‑type evolution equations using Sardar sub‑equation method, Optical and Quantum Electronics 55 (2023) 1148.
  • Md. N. Hossain, M. M. Miah, M. Alosaimi, F. Alsharif, M. Kanan, Exploring novel soliton solutions to the time-fractional coupled Drinfel'd-Sokolov-Wilson equation in industrial engineering using two efficient techniques, Fractal and Fractional 8 (2024) 352.
  • H. Qawaqneh, A. Altalbe, A. Bekir, K. U. Tariq, Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential, AIMS Mathematics 9 (9) (2024).
  • A. Said, A. Bakkar, H. Khan, C. Cattani, F. Tchier, Construction of mechanically preserved optical travelling wave solution for fractional generalized reaction Duffing model, Journal of Applied and Computational Mechanics (2025) https://doi.org/10.22055/jacm.2025.48049.4941.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014) 65-70.
  • T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
  • H. Rezazadeh, M. Inc, D. Baleanu, New solitary wave solutions for variants of (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations, Frontiers in Physics 8 (2020).
  • H. Miyakawa, S. Takeuchi, Applications of a duality between generalized trigonometric and hyperbolic functions, Journal of Mathematical Analysis and Applications 502 (1) (2021).
  • E. Neuman, Inequalities for the generalized trigonometric, hyperbolic and Jacobian elliptic functions, Journal of Mathematical Inequalities 9 (3) (2015) 709-726.
  • H. Kobayashi, S. Takeuchi, Applications of generalized trigonometric functions with two parameters, Communications on Pure and Applied Analysis 18 (3) (2019) 1509-1521.
  • S. Takeuchi, Applications of generalized trigonometric functions with two parameters II, Differential Equations and Applications 11 (4) (2019) 563-575.
  • S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, Journal of Mathematical Analysis and Applications 444 (2) (2016) 1000-1014.
  • O. Büttner, M. Bauer, A. Rueff, S. O. Demokritov, B. Hillebrands, A. N. Slavin, M. P. Kostylev, B. A. Kalinikos, Space- and time-resolved Brillouin light scattering from nonlinear spin-wave packets, Ultrasonics 38 (2000) 443-449.
  • M. Mossa Al-Sawalha, S. Mukhtar, A. S. Alshehry, M. Alqudah, M. S. Aldhabani, Kink soliton phenomena of fractional conformable Kairat equations, AIMS Mathematics, 10 (2) (2025) 2808-2828.
There are 48 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Research Article
Authors

Sevil Çulha Ünal 0000-0001-7447-9219

Early Pub Date June 30, 2025
Publication Date June 30, 2025
Submission Date April 25, 2025
Acceptance Date June 29, 2025
Published in Issue Year 2025 Volume: 11 Issue: 2

Cite

APA Çulha Ünal, S. (2025). Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Advanced Research in Natural and Applied Sciences, 11(2), 197-209. https://doi.org/10.28979/jarnas.1684271
AMA Çulha Ünal S. Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. JARNAS. June 2025;11(2):197-209. doi:10.28979/jarnas.1684271
Chicago Çulha Ünal, Sevil. “Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”. Journal of Advanced Research in Natural and Applied Sciences 11, no. 2 (June 2025): 197-209. https://doi.org/10.28979/jarnas.1684271.
EndNote Çulha Ünal S (June 1, 2025) Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Advanced Research in Natural and Applied Sciences 11 2 197–209.
IEEE S. Çulha Ünal, “Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”, JARNAS, vol. 11, no. 2, pp. 197–209, 2025, doi: 10.28979/jarnas.1684271.
ISNAD Çulha Ünal, Sevil. “Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”. Journal of Advanced Research in Natural and Applied Sciences 11/2 (June 2025), 197-209. https://doi.org/10.28979/jarnas.1684271.
JAMA Çulha Ünal S. Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. JARNAS. 2025;11:197–209.
MLA Çulha Ünal, Sevil. “Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”. Journal of Advanced Research in Natural and Applied Sciences, vol. 11, no. 2, 2025, pp. 197-09, doi:10.28979/jarnas.1684271.
Vancouver Çulha Ünal S. Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. JARNAS. 2025;11(2):197-209.
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