Approximate Solutions for A Fractional Shallow Water Flow Model

Year 2023, Volume: 5 Issue: 2, 65 - 75, 20.11.2023

Hira Tarıq , Hadi Rezazadeh , Mehmet Şenol , Orkun Taşbozan , Ali Kurt

https://doi.org/10.54286/ikjm.1256664

Abstract

This paper presents the solutions of fractional Drinfeld-Sokolov-Wilson (DSW) equations
that occur in shallow water flow models using the residual power series method.
The fractional derivatives and integrals are considered in the conformable sense. In
addition, surface plots of the solutions are given. The solutions and results show that
the present method is very efficient and effective due to the lack of a need for complex
calculations and that the method also has a wide range of practicability in the resolution
of partial differential fractional equations.

Keywords

Conformable Calculus, Residual Power Series Method, Drinfeld-Sokolov-Wilson Equation, Shallow water waves

References

• Kulish, V. V., Lage, J. L. (2002). Application of fractional calculus to fluid mechanics. J. Fluids Eng., 124(3), 803-806.
• Cruz Duarte, J.M., Rosales Garcia, J., Correa Cely, C.R., Garcia Perez, A., Avina Cervantes, J.G. (2018). A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications. Communications in Nonlinear Science and Numerical Simulation, 61, 138-148.
• He, J.H., Ji, F.Y. (2019). Two-scale mathematics and fractional calculus for thermodynamics. Thermal Science, 23(4), 2131-2133.
• Landman, K. A., Pettet, G. J., Newgreen, D. F. (2003). Mathematical models of cell colonization of uniformly growing domains. Bulletin of mathematical biology, 65(2), 235-262.
• Tarasov, V. E. (2019). On history of mathematical economics: Application of fractional calculus. Mathematics, 7(6), 509.
• Ionescu, C., Kelly, J.F. (2017). Fractional calculus for respiratory mechanics: Power law impedance, viscoelasticity, and tissue heterogeneity. Chaos, Solitons & Fractals, 102, 433-440.
• Tufenkci, S., Senol, B., Alagoz, B. B. (2019, September). Disturbance Rejection Fractional Order PID Controller Design in v-domain by Particle SwarmOptimization. In 2019 International Artificial Intelligence and Data Processing Symposium(IDAP) (pp. 1-6). IEEE.
• Yavuz, M., Ozdemir, N., Okur, Y. Y. (2016, July). Generalized differential transformmethod for fractional partial differential equation from finance. In Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia (pp. 778-785).
• Atangana, A., Owolabi, K.M. (2018). New numerical approach for fractional differential equations. MathematicalModelling of Natural Phenomena, 13(1), 3.
• Kurt, A., Tasbozan, O., Baleanu, D. (2017). New solutions for conformable fractional Nizhnik-Novikov- Veselov system via G’/G G’/G expansion method and homotopy analysis methods. Optical and Quantum Electronics, 49, 1-16.
• Akinyemi, L. (2019). q-Homotopy analysis method for solving the seventh-order time-fractional Laxâs Korteweg-de Vries and Sawada-Kotera equations. Computational and AppliedMathematics, 38(4), 191.
• Akinyemi, L., Iyiola, O. S. (2020). Exact and approximate solutions of time-fractional models arising from physics via Shehu transform.MathematicalMethods in the Applied Sciences, 43(12), 7442-7464.
• Senol, M., Atpinar, S., Zararsiz, Z., Salahshour, S., Ahmadian, A. (2019). Approximate solution of timefractional fuzzy partial differential equations. Computational and AppliedMathematics, 38(1), 1-18.
• Hammouch, Z., Mekkaoui, T. (2012). Adomian decomposition method for solving a time-fractional Burger-Huxley’s equation. Nonlinear Studies, 19(3).
• Rezazadeh, H., Vahidi, J., Zafar, A., Bekir, A. (2019). The Functional VariableMethod to Find New Exact Solutions of theNonlinear Evolution Equations withDual-Power-LawNonlinearity. International Journal of Nonlinear Sciences and Numerical Simulation. DOI: https://doi.org/10.1515/ijnsns-2019-0064.
• Korkmaz, A., Hepson, O. E., Hosseini, K., Rezazadeh, H., Eslami, M. (2018). Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class. Journal of King Saud University-Science. 32 (1), 567-574.
• Park, C., Khater, M.M., Abdel-Aty, A.H., Attia, R.A., Rezazadeh, H., Zidan, A.M., Mohamed, A.B. (2020). Dynamical analysis of the nonlinear complex fractional emerging telecommunication model with higher order dispersive cubic-quintic. Alexandria Engineering Journal, 59(3), 1425-1433.
• Raza, N., Afzal, U., Butt, A. R., Rezazadeh, H. (2019). Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities. Optical and Quantum Electronics, 51(4), 107.
• Mirhosseini-Alizamini, S. M., Rezazadeh, H., Srinivasa, K., Bekir, A. (2020). New closed form solutions of the new coupled Konno-Oono equation using the new extended direct algebraic method. Pramana, 94, 1-12.
• 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.
• Arqub, O. A. (2013). Series solution of fuzzy differential equations under strongly generalized differentiability. Journal of Advanced Research in AppliedMathematics, 5(1), 31-52.
• Arqub, O. A., El-Ajou, A., Bataineh, A. S., Hashim, I. (2013, January). A representation of the exact solution of generalized Lane-Emden equations using a new analytical method. In Abstract and Applied Analysis (Vol. 2013). Hindawi.
• El-Ajou, A., Arqub, O. A., Zhour, Z. A., Momani, S. (2013). New results on fractional power series: theories and applications. Entropy, 15(12), 5305-5323.
• Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
• Senol, M., Tasbozan, O., Kurt, A. (2019). Numerical solutions of fractional Burgers’ type equations with conformable derivative. Chinese Journal of Physics, 58, 75-84.
• Tariq, H., Akram, G. (2017). New traveling wave exact and approximate solutions for the nonlinear Cahn-Allen equation: evolution of a nonconserved quantity. Nonlinear Dynamics, 88, 581-594.
• Tariq, H., Sadaf,M., Akram, G., Rezazadeh, H., Baili, J., Lv, Y.P., Ahmad, H. (2021). Computational study for the conformable nonlinear Schrödinger equation with cubic-quintic-septic nonlinearities. Results in Physics, 30, 104839.
• Tariq, H., Günerhan, H., Rezazadeh, H., Adel, W. (2021). A numerical approach for the nonlinear temporal conformable fractional foamdrainage equation. Asian-European Journal ofMathematics, 14(06), 2150089.
• Tariq, H., Akram, G. (2017). Residual power series method for solving time-space-fractional Benney- Lin equation arising in falling film problems. Journal of Applied Mathematics and Computing, 55(1), 683-708.
• Jaradat, H. M., Al-Shara, S., Khan, Q. J., Alquran, M., Al-Khaled, K. (2016). Analytical solution of timefractional Drinfeld-Sokolov-Wilson system using residual power series method. IAENG International Journal of AppliedMathematics, 46(1), 64-70.