
AIMS Mathematics, 2020, 5(4): 30353055. doi: 10.3934/math.2020197.
Research article Special Issues
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 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, AlBalqa 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
Received: , Accepted: , Published:
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): 30353055. doi: 10.3934/math.2020197
References:
 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 MittagLeffler Laws, Int. J. Heat Mass Transfer, 138 (2019), 12221227.
 2. D. Kumar, J. Singh, D. Baleanu, On the analysis of vibration equation involving a fractional derivative with MittagLeffler law, Mathematical Methods in the Applied Sciences, 2019.
 3. S. Bhatter, A. Mathur, D. Kumar, et al. A new analysis of fractional DrinfeldSokolovWilson 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), 20172025.
 5. D. Kumar, J. Singh, D. Baleanu, et al. On the local fractional wave equation in fractal strings, Math. Methods Appl. Sci., 42 (2019), 15881595.
 6. A. Tassaddiq, I. Khan, K. S. Nisar, Heat transfer analysis in sodium alginate based nanofluid using MoS2 nanoparticles: AtanganaBaleanu fractional model, Chaos, Solitons Fractals, 130 (2020), 109445.
 7. A. Shaikh, K. S. Nisar, Transmission dynamics of fractional order Typhoid fever model using CaputoFabrizio operator, Chaos, Solitons Fractals, 128 (2019), 355365.
 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), 4045.
 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 AtanganaBaleanu derivative for fractional integrodifferential equations, Chaos, Solitons Fractals, 125 (2019), 194200
 11. A. Akgül, Reproducing kernel method for fractional derivative with nonlocal and nonsingular kernel, In: Fractional Derivatives with MittagLeffler Kernel, Springer International Publishing, 2019, 112.
 12. A. Akgül, A novel method for a fractional derivative with nonlocal and nonsingular kernel, Chaos, Solitons Fractals, 114 (2018), 478482.
 13. A. Akgül, A new method for approximate solutions of fractional order boundary value problems, Neural, Parallel Sci. Comput., 22 (2014), 223237.
 14. A. Akgül, E. K. Akgül, A Novel method for solutions of fourthorder 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 timefractional Robertson equations with 1D diffusion, Chaos, Solitons Fractals, 94 (2017), 17.
 17. A. Kurt, O. Tasbozan, D. Baleanu, New solutions for conformable fractional NizhnikNovikovVeselov 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 BurgersKortewegde Vries equation, Bull. Math. Sci. Appl, 17 (2016), 1723.
 19. A. Kurt, H. Rezazadeh, M. Senol, et al. Two effective approaches for solving fractional generalized HirotaSatsuma coupled KdV system arising in interaction of long waves, J. Ocean Eng. Sci., 4 (2019), 2432.
 20. M. Senol, O. Tasbozan, A. Kurt, Numerical solutions of fractional Burgers' Type equations with conformable derivative, Chin. J. Phys., 58 (2019), 7584.
 21. M. Senol, A. Kurt, E. Atilgan, et al. Numerical solutions of fractional BoussinesqWhithamBroerKaup and diffusive PredatorPrey equations with conformable derivative. New Trends Math. Sci., 7 (2019), 286300.
 22. A. Kurt, O. Tasbozan, Approximate analytical solutions to conformable modified Burgers equation using homotopy analysis method, Ann. Math. Silesianae, 33 (2019), 159167.
 23. A. Akgül, Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a PowellEyring nonNewtonian fluid, J. Taibah Univ. Sci., 13 (2019), 858863.
 24. A. Akgül, On the solution of higherorder difference equations, Math. Methods Appl. Sci., 40 (2016), 61656171.
 25. A. Akgül, A novel method for the solution of blasius equation in semiinfinite domains An International, J Optim. Control: Theor. Appl., 7 (2017), 225233.
 26. B. Boutarfa, A. Akgül, M. Inc, New approach for the FornbergWhitham type equations, J. Comput. Appl. Math., 312 (2017), 1326.
 27. J. N. Shin, J. H. Tang, M. S. Wu, Solution of shallowwater equations using leastsquares finiteelement method, Acta Mech. Sin., 24 (2008), 523532.
 28. T. Ozer, Symmetry group analysis of Benney system and an application for shallowwater equations. Mech. Res. Commun., 32 (2005), 24154.
 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 shallowwater equation in oceans, Z. Naturfors, 68 (2013), 17.
 32. S. Kumar, A. Kumar, I. K. Argyros, A new analysis for the KellerSegel model of fractional order, Numer. Alg., 75 (2017), 213228.
 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), 95106.
 34. A. S. Arife, S. K. Vanani, F. Soleymani, The Laplace homotopy analysis method for solving a general fractional diffusion equation arising in nanohydrodynamics, J. Comput. Theory Nanosci, 10 (2014), 14.
 35. O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 3152.
 36. A. ElAjou, 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), 119133 37. O. A. Arqub, Z. AboHammour, R. AlBadarneh, et al. A reliable analytical method for solving higherorder 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), 111.
 39. A. ElAjou, O. A. Arqub, S. Momani, Approximate analytical solution of the nonlinear fractional KdVBurgers equation: A new iterative algorithm, J. Comput. Phys., 293 (2015), 8195.
 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), 782789.
 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), 593600.
 43. S. Abbasbandy, F. S. Zakaria, Soliton solutions for the fifthorder KdV equation with the homotopy analysis method, Nonlinear Dyn., 51 (2008), 8387.
 44. S. Abbasbandy, Homotopy analysis method for the Kawahara equation, Nonlinear Anal. B, 11 (2010), 307312.
 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), 9911000.
 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 homotopyanalysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 20032016.
Reader Comments
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *