International Scientific Journal


Recently a new kind of derivatives, namely the conformable derivative is introduced which have not many drawbacks of other fractional derivatives. Two types of KdV equations with conformable derivative are investigated in this paper. Existence and uniqueness of two different equations of KdV class with conformable derivatives are investigated. It is also shown that the invariant subspace method can be extended to find the exact solutions of these equations.
PAPER REVISED: 2019-09-20
PAPER ACCEPTED: 2019-10-05
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 6, PAGES [S2107 - S2117]
  1. M. U. Ali, T. Kamran, W. Kassab, Solution of a fractional order integral equation via fixed point theorem in pseudomodular metric space, UNIVERSITY POLITEHNICA OF BUCHAREST SCIENTIFIC BULLETIN-SERIES A-APPLIED MATHEMATICS AND PHYSICS 80 (1) (2018) 71- 80.
  2. Q. Huang, R. Zhdanov, Symmetries and exact solutions of the time fractional harry-dym equation with riemann-liouville derivative, Physica A: Statistical Mechanics and its Applications 409 (2014) 110-118.
  3. J. Hu, Y. Ye, S. Shen, J. Zhang, Lie symmetry analysis of the time fractional kdv-type equation, Appl Math Comput 233 (2014) 439-444.
  4. R. Sahadevan, T. Bakkyaraj, Invariant analysis of time fractional generalized burgers and korteweg-de vries equations, J Math Anal Appl 393 (2012) 341-347.
  5. M. S. Hashemi, D. Baleanu, On the time fractional generalized fisher equation: group similarities and analytical solutions, Communications in Theoretical Physics 65 (1) (2016) 11.
  6. S. Pashayi, M. S. Hashemi, S. Shahmorad, Analytical lie group approach for solving fractional integro-differential equations, Communications in Nonlinear Science and Numerical Simulation 51 (2017) 66-77.
  7. F. Bahrami, R. Najafi, M. S. Hashemi, On the invariant solutions of space/time-fractional diffusion equations, Indian Journal of Physics (2017) 1-9.
  8. G. Wang, M. S. Hashemi, Lie symmetry analysis and soliton solutions of time-fractional k (m, n) equation, Pramana 88 (1) (2017) 7.
  9. M. S. Hashemi, F. Bahrami, R. Najafi, Lie symmetry analysis of steady-state fractional reaction-convection-diffusion equation, Optik-International Journal for Light and Electron Optics 138 (2017) 240-249.
  10. R. Najafi, F. Bahrami, M. S. Hashemi, Classical and nonclassical lie symmetry analysis to a class of nonlinear time-fractional differential equations, Nonlinear Dynamics 87 (3) (2017) 1785-1796.
  11. E. Yaşar, Y. Yıldırım, C. M. Khalique, Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional sawada-kotera-ito equation, Results in Physics 6 (2016) 322-328.
  12. R. Gazizov, A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Computers Mathematics with Applications 66 (5) (2013) 576-584.
  13. A. Ouhadan, E. El Kinani, Invariant subspace method and some exact solutions of time fractional modi ed kuramoto-sivashinsky equation, British Journal of Mathematics Computer Science 15 (4).
  14. R. Sahadevan, T. Bakkyaraj, Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations, Fractional Calculus and Applied Analysis 18 (1) (2015) 146-162.
  15. R. Sahadevan, P. Prakash, Exact solution of certain time fractional nonlinear partial differential equations, Nonlinear Dynamics 85 (1) (2016) 659-673.
  16. R. Sahadevan, P. Prakash, Exact solutions and maximal dimension of invariant subspaces of time fractional coupled nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation 42 (2017) 158-177.
  17. S. Azimi-Kavar, M. S. Hashemi, Analytical solutions of nonlinear time-space fractional schrödinger equation, Journal of Advanced Physics 6 (2) (2017) 297-302.
  18. S. Zhang, H.-Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional pdes, Physics Letters A 375 (7) (2011) 1069-1073.
  19. H. Jafari, H. Tajadodi, D. Baleanu, A. A. Al-Zahrani, Y. A. Alhamed, A. H. Zahid, Fractional sub-equation method for the fractional generalized reaction duffing model and nonlinear fractional sharma-tasso-olver equation, Central European Journal of Physics 11 (10) (2013) 1482-1486.
  20. H. Jafari, H. Tajadodi, D. Baleanu, A. A. Al-Zahrani, Y. A. Alhamed, A. H. Zahid, Exact solutions of boussinesq and kdv-mkdv equations by fractional sub-equation method, Romanian Reports in Physics 65 (4) (2013) 1119-1124.
  21. S. Sahoo, S. S. Ray, Improved fractional sub-equation method for (3+ 1)-dimensional generalized fractional kdv-zakharov-kuznetsov equations, Computers & Mathematics with Applications 70 (2) (2015) 158-166.
  22. Z. Bin, (g'/g)-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Communications in Theoretical Physics 58 (5) (2012) 623.
  23. A. Bekir, Ö. Güner, Exact solutions of nonlinear fractional differential equations by (g'/g)- expansion method, Chinese Physics B 22 (11) (2013) 110202.
  24. S. Sahoo, S. S. Ray, Solitary wave solutions for time fractional third order modified kdv equation using two reliable techniques (g'/g)-expansion method and improved (g'/g)-expansion method, Physica A: Statistical Mechanics and its Applications 448 (2016) 265-282.
  25. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Academic press, 1998.
  26. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl 1 (2) (2015) 1-13.
  27. A. Atangana, I. Koca, Chaos in a simple nonlinear system with atangana-baleanu derivatives with fractional order, Chaos, Solitons & Fractals 89 (2016) 447-454.
  28. 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.
  29. R. A. Ferreira, Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one, Journal of Difference Equations and Applications 19 (5) (2013) 712-718.
  30. C. Zhai, W. Yan, C. Yang, A sum operator method for the existence and uniqueness of positive solutions to riemann-liouville fractional differential equation boundary value problems, Communications in Nonlinear Science and Numerical Simulation 18 (4) (2013) 858-866.
  31. Y. Ding, Z. Wei, J. Xu, D. O’Regan, Extremal solutions for nonlinear fractional boundary value problems with p-laplacian, Journal of Computational and Applied Mathematics 288 (2015) 151-158.
  32. M. Morgado, N. Ford, P. Lima, Analysis and numerical methods for fractional differential equations with delay, Journal of Computational and Applied Mathematics 252 (Supplement C) (2013) 159 - 168, selected papers on Computational and Mathematical Methods in Science and Engineering (CMMSE). .
  33. C. Zhai, L. Xu, Properties of positive solutions to a class of four-point boundary value problem of caputo fractional differential equations with a parameter, Communications in Nonlinear Science and Numerical Simulation 19 (8) (2014) 2820-2827.
  34. Y. Luchko, M. Yamamoto, General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems, Fractional Calculus and Applied Analysis 19 (3) (2016) 676-695.
  35. B. Alkahtani, A. Atangana, Controlling the wave movement on the surface of shallow water with the caputo-fabrizio derivative with fractional order, Chaos, Solitons & Fractals 89 (2016) 539-546.
  36. J.-D. Djida, A. Atangana, More generalized groundwater model with space-time caputo fabrizio fractional differentiation, Numerical Methods for Partial Differential Equations 33 (5) (2017) 1616-1627.
  37. M. Eslami, H. Rezazadeh, The first integral method for wu-zhang system with conformable time-fractional derivative, Calcolo 53 (3) (2016) 475-485.
  38. K. Hosseini, P. Mayeli, R. Ansari, Modified kudryashov method for solving the conformable time-fractional klein-gordon equations with quadratic and cubic nonlinearities, Optik - International Journal for Light and Electron Optics 130 (2017) 737 - 742.
  39. Y. Lenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of burgers- type equations with conformable derivative, Waves in Random and Complex Media 27 (1) (2017) 103-116.
  40. O. Tasbozan, Y. Çenesiz, A. Kurt, New solutions for conformable fractional boussinesq and combined kdv-mkdv equations using jacobi elliptic function expansion method, The European Physical Journal Plus 131 (7) (2016) 244.
  41. H. Yépez-Martinez, J. Gómez-Aguilar, A. Atangana, First integral method for non-linear differential equations with conformable derivative, Mathematical Modelling of Natural Phenomena 13 (1) (2018) 14.
  42. S. Momani, An explicit and numerical solutions of the fractional kdv equation, Mathematics and Computers in Simulation 70 (2) (2005) 110-118.
  43. Q. Wang, Homotopy perturbation method for fractional kdv-burgers equation, Chaos, Solitons & Fractals 35 (5) (2008) 843-850.
  44. A. El-Ajou, O. A. Arqub, S. Momani, Approximate analytical solution of the nonlinear fractional kdv-burgers equation: a new iterative algorithm, Journal of Computational Physics 293 (2015) 81-95.
  45. M. S. Hashemi, S. Abbasbandy, M. Alhuthali, H. Alsulam, Conservation laws and symmetries of mkdv-kp equation, Rom. J. Phys 60 (2015) 904-917.
  46. V. Benci, D. Fortunato, Hylomorphic solitons for the benjamin-ono and the fractional kdv equations, Nonlinear Analysis: Theory, Methods & Applications 144 (2016) 41-57.
  47. T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57 - 66.
  48. V. A. Galaktionov, S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science, Chapman Hall/CRC, 2007.

© 2022 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence