AIMS Mathematics, 2020, 5(4): 3035-3055. doi: 10.3934/math.2020197.

Research article Special Issues

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A comparison study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow

1 Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India
2 Department of Mathematics, Balarampur College Purulia- 723143, West Bengal, India
3 Department of Mathematics, Faculty of science, Al-Balqa Applied University, Salt 19117, Jordan
4 Systems Engineering Department, King Fahd University of Petroleum and Minerals, 31261 Dhahran, Saudi Arabia
5 Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia

Special Issues: 2nd International Conference on Mathematical Modeling, Applied Analysis and Computation (ICMMAAC-19), August 8–10, 2019, JECRC University, Jaipur, India

In this study, a comparison between the modified homotopy analysis transform method (MHATM) and residual power series method (RPSM) have been given for solving time-fractional coupled shallow water equations (SWEs). The time-fractional coupled SWEs are a system of PDEs that describe the flow below a pressure surface in a fluid is considered. Rigorous convergence analysis and error estimated have been exhibited for both the featured methods. The results obtained by MHATM and RPSM are then compared with well-known exact solutions. To show the effectiveness and advantage of the featured techniques the numerical simulation of coupled SWEs has been represented graphically with tabulated data. However, the results indicate that MHATM provides more accurate value than RPSM for solving fractional coupled SWEs.
  Article Metrics

Keywords fractional shallow water equation; fractional power series; RPSM; HATM; homotopy polynomials; optimal value

Citation: Sunil Kumar,Amit Kumar ,Zaid Odibat, Mujahed Aldhaifallah, Kottakkaran Sooppy Nisar. A comparison study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow. AIMS Mathematics, 2020, 5(4): 3035-3055. doi: 10.3934/math.2020197


  • 1. D. Kumar, J. Singh, K. Tanwar, et al. A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Int. J. Heat Mass Transfer, 138 (2019), 1222-1227.    
  • 2. D. Kumar, J. Singh, D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences, 2019.
  • 3. S. Bhatter, A. Mathur, D. Kumar, et al. A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A, 537 (2020), 122578.
  • 4. J. Singh, A. Kilicman, D. Kumar, et al. Numerical study for fractional model of nonlinear predatorprey biological population dynamical system, Therm. Sci., 23 (2019), 2017-2025.    
  • 5. D. Kumar, J. Singh, D. Baleanu, et al. On the local fractional wave equation in fractal strings, Math. Methods Appl. Sci., 42 (2019), 1588-1595.    
  • 6. A. Tassaddiq, I. Khan, K. S. Nisar, Heat transfer analysis in sodium alginate based nanofluid using MoS2 nanoparticles: Atangana-Baleanu fractional model, Chaos, Solitons Fractals, 130 (2020), 109445.
  • 7. A. Shaikh, K. S. Nisar, Transmission dynamics of fractional order Typhoid fever model using Caputo-Fabrizio operator, Chaos, Solitons Fractals, 128 (2019), 355-365.    
  • 8. A. A. Abro, I. Khan, K. S. Nisar, Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit, Chaos, Solitons Fractals, 129 (2019), 40-45.    
  • 9. K. Jothimani, K. Kaliraj, Z. Hammouch, et al. New results on controllability in the framework of fractional integrodifferential equations with nondense domain, Eur. Phys. J. Plus, 134 (2019), 441.
  • 10. C. Ravichandran, K. Logeswari, F. Jarad, New results on existence in the framework of Atangana-Baleanu derivative for fractional integro-differential equations, Chaos, Solitons Fractals, 125 (2019), 194-200    
  • 11. A. Akgül, Reproducing kernel method for fractional derivative with non-local and non-singular kernel, In: Fractional Derivatives with Mittag-Leffler Kernel, Springer International Publishing, 2019, 1-12.
  • 12. A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos, Solitons Fractals, 114 (2018), 478-482.    
  • 13. A. Akgül, A new method for approximate solutions of fractional order boundary value problems, Neural, Parallel Sci. Comput., 22 (2014), 223-237.
  • 14. A. Akgül, E. K. Akgül, A Novel method for solutions of fourth-order fractional boundary value problems, Fractal Fractional, 3 (2019), 33.
  • 15. E. K. Akgül, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos: Interdiscip. J. Nonlinear Sci., 29 (2019), 023108.
  • 16. O. S. Iyiola, O. Tasbozan, A. Kurt, et al. On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion, Chaos, Solitons Fractals, 94 (2017), 1-7.    
  • 17. A. Kurt, O. Tasbozan, D. Baleanu, New solutions for conformable fractional Nizhnik-Novikov-Veselov system via G''/G expansion method and homotopy analysis methods, Opt. Quantum Electron., 49 (2017), 333.
  • 18. A. Kurt, O. Tasbozan, Y. Cenesiz, Homotopy analysis method for conformable Burgers-Kortewegde Vries equation, Bull. Math. Sci. Appl, 17 (2016), 17-23.
  • 19. A. Kurt, H. Rezazadeh, M. Senol, et al. Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves, J. Ocean Eng. Sci., 4 (2019), 24-32.    
  • 20. M. Senol, O. Tasbozan, A. Kurt, Numerical solutions of fractional Burgers' Type equations with conformable derivative, Chin. J. Phys., 58 (2019), 75-84.    
  • 21. M. Senol, A. Kurt, E. Atilgan, et al. Numerical solutions of fractional Boussinesq-Whitham-BroerKaup and diffusive Predator-Prey equations with conformable derivative. New Trends Math. Sci., 7 (2019), 286-300.
  • 22. A. Kurt, O. Tasbozan, Approximate analytical solutions to conformable modified Burgers equation using homotopy analysis method, Ann. Math. Silesianae, 33 (2019), 159-167.    
  • 23. A. Akgül, Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a Powell-Eyring non-Newtonian fluid, J. Taibah Univ. Sci., 13 (2019), 858-863.    
  • 24. A. Akgül, On the solution of higher-order difference equations, Math. Methods Appl. Sci., 40 (2016), 6165-6171.
  • 25. A. Akgül, A novel method for the solution of blasius equation in semi-infinite domains An International, J Optim. Control: Theor. Appl., 7 (2017), 225-233.
  • 26. B. Boutarfa, A. Akgül, M. Inc, New approach for the Fornberg-Whitham type equations, J. Comput. Appl. Math., 312 (2017), 13-26.    
  • 27. J. N. Shin, J. H. Tang, M. S. Wu, Solution of shallow-water equations using least-squares finiteelement method, Acta Mech. Sin., 24 (2008), 523-532.    
  • 28. T. Ozer, Symmetry group analysis of Benney system and an application for shallow-water equations. Mech. Res. Commun., 32 (2005), 241-54.
  • 29. I. Podlubny, Fractional differential equations, New York, USA, Academic Press, 1999.
  • 30. K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York, Wiley, 1993.
  • 31. S. Kumar, A numerical study for solution of time fractional nonlinear shallow-water equation in oceans, Z. Naturfors, 68 (2013), 1-7.    
  • 32. S. Kumar, A. Kumar, I. K. Argyros, A new analysis for the Keller-Segel model of fractional order, Numer. Alg., 75 (2017), 213-228.    
  • 33. A. Kumar, S. Kumar, Modified analytical approach for fractional discrete KdV equations arising in particle vibrations, Proc. Natl. Acad. Sci. India Section A: Phys. Sci. J., 88 (2018), 95-106.    
  • 34. A. S. Arife, S. K. Vanani, F. Soleymani, The Laplace homotopy analysis method for solving a general fractional diffusion equation arising in nano-hydrodynamics, J. Comput. Theory Nanosci, 10 (2014), 1-4.    
  • 35. O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31-52.    
  • 36. A. El-Ajou, O. A. Arqub, S. Momani, et al. A novel expansion iterative method for solving linear partial differential equation of fractional order, Appl. Math. Comput., 257 (2015), 119-133 37. O. A. Arqub, Z. Abo-Hammour, R. Al-Badarneh, et al. A reliable analytical method for solving higher-order initial value problems, Discrete Dyn. Natl. Soc., 2013 (2013), 673829.
  • 38. J. J. Yao, A. Kumar, S. Kumar, A fractional model to describe the Brownian motion of particles and its analytical solution., Adv. Mech. Eng., 7 (2015), 1-11.
  • 39. A. El-Ajou, O. A. Arqub, S. Momani, Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm, J. Comput. Phys., 293 (2015), 81-95.    
  • 40. S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D thesis, Shanghai Jiao Tong University, 1992.
  • 41. Z. Odibat, A study on the convergence of homotopy analysis method, Appl. Math. Comput., 217 (2010), 782-789.
  • 42. Z. Odibat, S. Momani, H. Xu, A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. Math. Modell., 34 (2010), 593-600.    
  • 43. S. Abbasbandy, F. S. Zakaria, Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dyn., 51 (2008), 83-87.
  • 44. S. Abbasbandy, Homotopy analysis method for the Kawahara equation, Nonlinear Anal. B, 11 (2010), 307-312.    
  • 45. Z. Odibat, A. S. Bataineh, An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials, Math. Meth. Appl. Sci., 38 (2015), 991-1000.    
  • 46. K. B. Oldham, J. Spanier, The Fractional Calculus: Integrations and Differentiations of Arbitrary Order, Academic Press, New York, 1974.
  • 47. S. J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2003-2016.    


This article has been cited by

  • 1. N. Valliammal, C. Ravichandran, Kottakkaran Sooppy Nisar, Solutions to fractional neutral delay differential nonlocal systems, Chaos, Solitons & Fractals, 2020, 138, 109912, 10.1016/j.chaos.2020.109912

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved