Reduced differential transform method for solving time and space local fractional partial differential equations

Volume 10, Issue 10, pp 5230--5238 http://dx.doi.org/10.22436/jnsa.010.10.09

Publication Date: October 19, 2017 Submission Date: April 01, 2017

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Authors

Omer Acan - Department of Mathematics, Art and Science Faculty, Siirt University, Siirt, Turkey. Maysaa Mohamed Al Qurashi - Department of Mathematics, Faculty of Art and Science, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia. Dumitru Baleanu - Department of Mathematics and Computer Sciences, Faculty of Art and Science, Cankaya University, Ankara, Turkey. - Institute of Space Sciences, Magurele-Bucharest, Romania.

Abstract

We apply the new local fractional reduced differential transform method to obtain the solutions of some linear and nonlinear partial differential equations on Cantor set. The reported results are compared with the related solutions presented in the literature and the graphs are plotted to show their behaviors. The results prove that the presented method is faster and easy to apply.

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Keywords

• Approximate solution
• local fractional derivative
• partial differential equations
• reduced differential transform method

MSC

•  35R11
•  35A22
•  65R10

References

• [1] O. Acan , N-dimensional and higher order partial differential equation for reduced differential transform method, Science Institute, Selcuk Univ., Turkey (2016)


• [2] O. Acan, The existence and uniqueness of periodic solutions for a kind of forced rayleigh equation, Gazi Univ. J. Sci., 29 (2016), 645–650.
  • Google Scholar


• [3] O. Acan, O. Firat, Y. Keskin, The use of conformable variational iteration method, conformable reduced differential transform method and conformable homotopy analysis method for solving different types of nonlinear partial differential equations, Proceedings of the 3rd International Conference on Recent Advances in Pure and Applied Mathematics, Bodrum, Turkey (2016)
  • Google Scholar


• [4] O. Acan, O. Firat, Y. Keskin, G. Oturanc, Solution of conformable fractional partial differential equations by reduced differential transform method, Selcuk J. Appl. Math., (2016), in press.
  • Google Scholar


• [5] O. Acan, O. Firat, Y. Keskin, G. Oturanc, Conformable variational iteration method, New Trends Math. Sci., 5 (2017), 172–178.
  • View Article
  • Google Scholar


• [6] O. Acan, Y. Keskin , Approximate solution of Kuramoto-Sivashinsky equation using reduced differential transform method , AIP Conference Proceedings, 1648 (2015), 470003.
  • View Article
  • Google Scholar


• [7] O. Acan, Y. Keskin, Reduced differential transform method for (2+1) dimensional type of the Zakharov-Kuznetsov ZK(n,n) equations, AIP Conference Proceedings, 1648 (2015), 370015.
  • View Article
  • Google Scholar


• [8] O. Acan, Y. Keskin, A comparative study of numerical methods for solving (n + 1) dimensional and third-order partial differential equations, J. Comput. Theor. Nanosci., 13 (2016), 8800–8807.
  • View Article
  • Google Scholar


• [9] O. Acan, Y. Keskin , A new technique of Laplace Pade reduced differential transform method for (1 + 3) dimensional wave equations, New Trends Math. Sci., 5 (2017), 164–171.
  • View Article
  • Google Scholar


• [10] M. O. Al-Amr, New applications of reduced differential transform method, Alexandria Eng. J., 53 (2014), 243–247.
  • View Article
  • Google Scholar


• [11] D. Baleanu, A. H. Bhrawy, R. A. Van Gorder, New trends on fractional and functional differential equations, Abstr. Appl. Anal., 2015 (2015), 2 pages.
  • Google Scholar


• [12] D. Baleanu, K. Sayevand, Performance evaluation of matched asymptotic expansions for fractional differential equations with multi-order, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 59 (2016), 3–12.
  • MathSciNet
  • Google Scholar


• [13] D. Baleanu, J. A. Tenreiro Machado, C. Cattani, M. C. Baleanu, X.-J. Yang, Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators, Abstr. Appl. Anal., 2014 (2014), 6 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] N. Baykus, M. Sezer , Solution of high-order linear Fredholm integro-differential equations with piecewise intervals, Numer. Methods Partial Differential Equations, 27 (2011), 1327–1339.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional derivatives, Z. Angew. Math. Mech., 75 (1995), 623–635.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] Z.-J. Cui, Z.-S. Mao, S.-J. Yang, P.-N. Yu, Approximate analytical solutions of fractional perturbed diffusion equation by reduced differential transform method and the homotopy perturbation method, Math. Probl. Eng., 2013 (2013), 7 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] A. K. Golmankhaneh, X.-J. Yang, D. Baleanu, Einstein field equations within local fractional calculus, Rom. J. Phys., 60 (2015), 22–31.
  • Google Scholar


• [18] Z.-H. Guo, O. Acan, S. Kumar , Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems, Therm. Sci., 20 (2016), S739–S742.
  • View Article
  • Google Scholar


• [19] P. K. Gupta, Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method, Comput. Math. Appl., 61 (2011), 2829–2842.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] J.-H. He, A tutorial review on fractal spacetime and fractional calculus, Internat. J. Theoret. Phys., 53 (2014), 3698–3718.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] M. A. E. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., 2014 (2014), 7 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan, F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Adv. Mech. Eng., 8 (2016), 1–6.
  • View Article
  • Google Scholar


• [23] Y. Keskin , Partial differential equations by the reduced differential transform method, Science Institute, Selcuk Univ., Turkey (2010)


• [24] Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 741–750.
  • View Article
  • Google Scholar


• [25] Y. Keskin, G. Oturanc , Reduced differential transform method for generalized KdV equations, Math. Comput. Appl., 15 (2010), 382–393.
  • View Article
  • MathSciNet
  • Google Scholar


• [26] Y. Keskin, G. Oturanc, The reduced differential transform method: A new approach to factional partial differential equations, Nonlinear Sci. Lett. A., 1 (2010), 207–217.
  • Google Scholar


• [27] E. Kurul, N. B. Savasaneril, Solution of the two-dimensional heat equation for a rectangular plate, New Trends Math. Sci., 3 (2015), 76–82.
  • View Article
  • MathSciNet
  • Google Scholar


• [28] M. Ma, D. Baleanu, Y. Gasimov, X.-J. Yang, New results for multidimensional diffusion equations in fractal dimensional space , Rom. J. Phys., 61 (2016), 784–794.
  • Google Scholar


• [29] F. Mainardi , Fractional relaxation-oscillation and fractional diffusion-wave phenomena , Chaos Solitons Fractals, 7 (1996), 1461–1477.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
  • Google Scholar


• [31] A. Razminia, D. Baleanu, Fractional order models of industrial pneumatic controllers, Abstr. Appl. Anal., 2014 (2014), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [32] S. G. Samko, A. A. Kilbas, O. I. Marichev , Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolski˘ı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
  • Google Scholar


• [33] A. Saravanan, N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation, J. Egyptian Math. Soc., 21 (2013), 259–265.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] M. Z. Sarikaya, H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc., 145 (2017), 1527–1538.
  • View Article
  • MathSciNet
  • Google Scholar


• [35] W. R. Schneider, W. Wyss , Fractional diffusion and wave equations , J. Math. Phys., 30 (1989), 134–144.
  • View Article
  • Google Scholar


• [36] B. K. Singh, Mahendra , A numerical computation of a system of linear and nonlinear time dependent partial differential equations using reduced differential transform method, Int. J. Differ. Equ., 2016 (2016), 8 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] X.-J. Yang, Local fractional functional analysis and its applications, Asian Academic Publisher, Hong Kong (2011)
  • Google Scholar


• [38] X.-J. Yang, Advanced local fractional calculus and its applications, World Science, New York (2012)
  • Google Scholar


• [39] X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets , Appl. Math. Lett., 47 (2015), 54–60.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] X.-J. Yang, F. Gao, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 73 (2017), 203–210.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] X.-J. Yang, L.-Q. Hua , Variational iteration transform method for fractional differential equations with local fractional derivative, Abstr. Appl. Anal., 2014 (2014), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [42] X.-J. Yang, H. M. Srivastava, C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (2015), 752–761.
  • Google Scholar


• [43] X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani , On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 5 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [44] X.-J. Yang, J. A. Tenreiro Machado, C. Cattani, F. Gao, On a fractal LC-electric circuit modeled by local fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 47 (2017), 200–206.
  • View Article
  • Google Scholar


• [45] X.-J. Yang, J. A. Tenreiro Machado, J. J. Nieto , A new family of the local fractional PDEs , Fund. Inform., 151 (2017), 63–75.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [46] X.-J. Yang, J. A. Tenreiro Machado, H. M. Srivastava , A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Appl. Math. Comput., 274 (2016), 143–151.
  • View Article
  • MathSciNet
  • Google Scholar


• [47] Y. Zhang, C. Cattani, X.-J. Yang, Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains, Entropy, 17 (2015), 6753–6764.
  • View Article
  • MathSciNet
  • Google Scholar


• [48] Y. Zhang, X.-J. Yang, An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics, Appl. Math. Model., 40 (2016), 1793–1799.
  • View Article
  • MathSciNet
  • Google Scholar