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Solution for fractional forced KdV equation using fractional natural decomposition method

1 Department of Mathematics, Karnatak University, Dharwad-580003, Karnataka, India
2 Department of Mathematics, Faculty of Science, Davangere University, Shivagangothri, Davangere-577007, Karnataka, India
3 Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India

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

The fractional natural decomposition method (FNDM) is employed in the present investigation to find the solution for fractional forced Korteweg-de Vries (FF-KdV) equation. Three distinct cases are chosen for each equation to validate and illustrate the effectiveness of the future technique. The behaviour for different values of Froude number (Fr) has been presented to assure the proficiency and reliability and of the considered method. Moreover, we captured the behaviour of the FNDM solution for distinct arbitrary order. The obtained results elucidate that, the considered method is very effective and easy to employ while analyse the behaviour of nonlinear fractional differential equations arising in connected areas of science and technology.
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1. J. Liouville, Memoire surquelques questions de geometrieet de mecanique, et sur un nouveau genre de calcul pour resoudreces questions, J. Ec. Polytech., 13 (1832), 1-69.

2. G. F. B. Riemann, Auffassung der Integration und Differentiation, Gesammelte Mathematische Werke, Leipzig, 1896.

3. M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, 1969.

4. K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, Wiley, New York, 1993.

5. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

6. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.

7. D. Baleanu, Z. B. Guvenc, J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer Dordrecht Heidelberg, London and New York, 2010.

8. N. H. Sweilam, M. M. A. Hasan, D. Baleanu, New studies for general fractional financial models of awareness and trial advertising decisions, Chaos, Solitons Fractals, 104 (2017), 772-784.    

9. D. Baleanu, G. C. Wu, S. D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos, Solitons Fractals, 102 (2017), 99-105.    

10. P. Veeresha, D. G. Prakasha, H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29 (2019), 013119.

11. W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus, A powerful approach for fractional Drinfeld-Sokolov-Wilson equation with Mittag-Leffler law, Alexandria Eng. J., (2019). DOI: 10.1016/j.aej.2019.11.002.

12. E. F. D. Goufo, Application of the Caputo-Fabrizio Fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Math. Model. Numer. Anal., 21 (2016), 188-198.    

13. P. Veeresha, D. G. Prakasha, H. M. Baskonus, Novel simulations to the time-fractional Fisher's equation, Math. Sci., 13 (2019), 33-42.    

14. A. Prakash, P. Veeresha, D. G. Prakasha, A homotopy technique for fractional order multidimensional telegraph equation via Laplace transform, Eur. Phys. J. Plus, 134 (2019), 19.

15. B. Li, Y. Chen, H. Zhang, Explicit exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order, Chaos, Solitons Fractals, 15 (2003), 647-654.    

16. E. F. D. Goufo, H.M. Tenkam, M. Khumalo, A behavioral analysis of KdVB equation under the law of Mittag-Leffler function, Chaos, Solitons Fractals, 125 (2019), 139-145.    

17. P. Veeresha, D. G. Prakasha, H. M. Baskonus, Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method, Math. Sci., 13 (2019), 115-128.    

18. A. Prakash, P. Veeresha, D. G. Prakasha, A homotopy technique for fractional order multidimensional telegraph equation via Laplace transform, Eur. Phys. J. Plus, 134 (2019), 1-18.    

19. P. Dubard, P. Gaillard, C. Kleina, On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation, Eur. Phys. J. Special Topics, 185 (2010), 247-258.    

20. P. Veeresha, D. G. Prakasha, M. A. Qurashi, A reliable technique for fractional modified Boussinesq and approximate long wave equations, Differ Equ, 2019 (2019), 253.

21. E. F. D. Goufo, A. Atangana, Dynamics of traveling waves of variable order hyperbolic Liouville equation, Regulation and control, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 645-662.

22. P. Veeresha, D. G. Prakasha, D. Kumar, An efficient technique for nonlinear time-fractional KleinFock-Gordon equation, Appl. Math. Comput., 364 (2020), 124637.

23. D. G. Prakasha, P. Veeresha, J. Singh, Fractional approach for equation describing the water transport in unsaturated porous media with Mittag-Leffler kernel, Front. Phys., 7 (2019), 193.

24. E. F. D. Goufo, I. T. Toudjeu, Around chaotic disturbance and irregularity for higher order traveling waves, J. Math., 2018 (2018), 1-11.

25. P. Veeresha, D. G. Prakasha, A novel technique for (2 + 1)-dimensional time-fractional coupled Burgers equations, Math. Comput. Simulation, 116 (2019), 324-345.

26. E. F. D. Goufo, A. Atangana, Extension of fragmentation process in a kinetic-diffusive-wave system, Thermal Science, 19 (2015), 13-23.    

27. P. Veeresha, D. G. Prakasha, Solution for fractional Zakharov-Kuznetsov equations by using two reliable techniques, Chinese J. Phys., 60 (2019), 313-330.    

28. K. B. Liaskos, A. A. Pantelous, I. A. Kougioumtzoglou, Implicit analytic solutions for the linear stochastic partial differential beam equation with fractional derivative terms, Systems Control Lett., 121 (2018), 38-49.    

29. A. Jeffrey, M. N. B. Mohamad, Exact solutions to the KdV-Burgers' equation, Wave Motion, 14 (1991), 369-375.    

30. E. F. D. Goufo, S. Kumar, Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities, Math. Probl. Eng., 2017 (2017), 1-9.

31. W. Zhang, Q. Chang, B. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order, Chaos, Solitons Fractals, 13 (2002), 311-319.    

32. F. Dias, J. M. Vanden-Broeck, Generalized critical free-surface flows, J. Eng. Math., 42 (2002), 291-301.    

33. S. S. Shen, On the accuracy of the stationary forced Korteweg-De Vries equation as a model equation for flows over a bump, Q. Appl. Math., 53 (1995), 701-719.    

34. R. Camassa, T. Wu, Stability of forced solitary waves, Philos. Trans. R. Soc. Lond. A, 337 (1991), 429-466.    

35. N. J. Zabuski, M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.

36. D. G. Crighton, Applications of KdV, Acta Appl. Math., 15 (1995), 39-67.

37. W. Hereman, Shallow Water Waves and Solitary Waves, In: R. Meyers, Ed., Mathematics of Complexity and Dynamical Systems, Springer, New York, USA, 2012.

38. T. Y. T. Wu, Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech., 184 (1987), 75-99.    

39. V. D. David, Z. A. Aziz, F. Salah, Analytical approximate solution for the forced Korteweg-de Vries (FKdV) on critical flow over a hole using homotopy analysis method, Journal Teknologi (Sciences & Engineering), 78 (2016), 107-112.

40. G. Adomian, A new approach to nonlinear partial differential equations, J. Math. Anal. Appl., 102 (1984), 420-434.    

41. M. S. Rawashdeh, H. Al-Jammal, New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM, Advances in Difference Equations, 235 (2016), 1-19.

42. M. S. Rawashdeh, H. Al-Jammal, Numerical solutions for system of nonlinear fractional ordinary differential equations using the FNDM, Mediterr. J. Math., 13 (2016), 4661-4677.    

43. M. S. Rawashdeh, The fractional natural decomposition method: theories and applications, Math. Meth. Appl. Sci., 40 (2017), 2362-2376.    

44. D. G. Prakasha, P. Veeresha, M. S. Rawashdeh, Numerical solution for (2+1)-dimensional timefractional coupled Burger equations using fractional natural decomposition method, Math. Meth. Appl. Sci., 42 (2019), 3409-3427.    

45. D. G. Prakasha, P. Veeresha, H. M. Baskonus, Two novel computational techniques for fractional Gardner and Cahn-Hilliard equations, Comp. and Math. Methods, 1 (2019), 1-19.

46. Z. J. Xiao, G. B. Ling, Analytic solutions to forced KdV equation, Commun. Theor. Phys. 52 (2009), 279-283.    

47. P. A. Milewski, The forced Korteweg-De Vries equation as a model for waves generated by topography, Cubo, 6 (2014), 33-51.

48. V. D. David, F. Salah, M. Nazari, Approximate analytical solution for the forced Korteweg-de Vries equation, J. Appl. Math., 2013 (2013), 1-9.

49. S. Lee, Dynamics of trapped solitary waves for the forced KdV equation, Symmetry, 10 (2018), 1-13.

50. K. G. Tay, W. K. Tiong, Y. Y. Choy, Method of lines and pseudospectral solutions of the forced Korteweg-De Vries equation with variable coefficients arises in elastic tube, International Journal of Pure and Applied Mathematics, 116 (2017), 985-999.

51. G. M. Mittag-Leffler, Sur la nouvelle fonction Eα(x), C. R. Acad. Sci. Paris, 137 (1903), 554-558.

52. Z. H. Khan, W. A. Khan, N-Transform - Properties and Applications, NUST J. Engg. Sci., 1 (2008), 127-133.

53. D. Loonker, P. K. Banerji, Solution of fractional ordinary differential equations by natural transform, Int. J. Math. Eng. Sci., 12 (2013), 1-7.

© 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)

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