A new numerical technique for local fractional diffusion equation in fractal heat transfer

Volume 9, Issue 10, pp 5621--5628 http://dx.doi.org/10.22436/jnsa.009.10.09

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Authors

Xiao-Jun Yang - School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China. - State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China. J. A. Tenreiro Machado - Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida, 4249-015 Porto, Portugal. Dumitru Baleanu - Department of Mathematics, Cankya University, Ogretmenler Cad. 14, Balgat-06530, Ankara, Turkey. - Institute of Space Sciences, Magurele-Bucharest, Romania. Feng Gao - School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China. - State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China.

Abstract

In this paper, a new numerical approach, embedding the differential transform (DT) and Laplace trans- form (LT), is firstly proposed. It is considered in the local fractional derivative operator for obtaining the non-differential solution for diffusion equation in fractal heat transfer.

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Keywords

• Numerical solution
• diusion equation
• dierential transform
• Laplace transform
• fractal heat transfer
• local fractional derivative.

MSC

•  76R50
•  26A33
•  44A10
•  28A80

References

• [1] R. Almeida, A. B. Malinowska, D. F. M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51 (2010), 12 pages


• [2] M. Alquran, K. Al-Khaled, M. Ali, A. Ta'any, The combined Laplace transform-differential transform method for solving linear non-homogeneous PDEs, J. Math. Comput. Sci., 2 (2012), 690--701


• [3] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Models and numerical methods, World Sci., 3 (2012), 10--16


• [4] D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional dynamics and control, Springer, Berlin (2012)


• [5] A. H. Bhrawy, E. H. Doha, D. Baleanu, S. S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293 (2015), 142--156


• [6] Y. Cao, W.-G. Ma, L.-C. Ma, Local fractional functional method for solving diffusion equations on Cantor sets, Abstr. Appl. Anal., 2014 (2014), 6 pages


• [7] A. Carpinteri, P. Cornetti, A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos Solitons Fractals, 13 (2002), 85--94


• [8] C. Cattani, H. M. Srivastava, X.-J. Yang, Fractional dynamics, De Gruyter Open, Berlin (2015)


• [9] K. Diethelm, N. J. Ford, A. D. Freed, Y. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Methods Appl. Mech. Engrg, 194 (2005), 743--773


• [10] A. K. Golmankhaneh, A. K. Golmankhaneh, D. Baleanu, On nonlinear fractional Klein-Gordon equation, Signal Process., 91 (2011), 446--451


• [11] R. Goren o, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, Udine, (1996), 223-276, CISM Courses and Lectures, Springer, Vienna (1997)


• [12] R. Goren o, F. Mainardi, D. Moretti, G. Pagnini, P. Paradisi, Fractional diffusion: probability distributions and random walk models, Non extensive thermodynamics and physical applications, Villasimius, (2001), Phys. A, 305 (2002), 106--112


• [13] J.-H. He, A tutorial review on fractal spacetime and fractional calculus, Internat. J. Theoret. Phys., 53 (2014), 3698--3718


• [14] J. Hristov, Approximate solutions to fractional subdiffusion equations, Eur. Phys. J. Spec. Top., 193 (2013), 229--243


• [15] H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1962--1969


• [16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North- Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (1997)


• [17] K. M. Kolwankar, A. D. Gangal, Local fractional Fokker-Planck equation, Phys. Rev. Lett., 80 (1998), 214--217


• [18] C. P. Li, F. H. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149--179


• [19] F. Liu, V. V. Anh, I. Turner, P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), 233--245


• [20] H.-Y. Liu, J.-H. He, Z.-B. Li, Fractional calculus for nanoscale flow and heat transfer, Internat. J. Numer. Methods Heat Fluid Flow, 24 (2014), 1227--1250


• [21] C.-F. Liu, S.-S. Kong, S.-J. Yuan, Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem, Therm. Sci., 17 (2013), 715--721


• [22] Y. Luchko, R. Goren o, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam., 24 (1999), 207--233


• [23] Y. F. Luchko, H. M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appl., 29 (1995), 73--85


• [24] C. I. Muresan, C. Ionescu, S. Folea, R. De Keyser, Fractional order control of unstable processes: the magnetic levitation study case, Nonlinear Dyn., 80 (2014), 1761--1772


• [25] Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21 (2008), 194--199


• [26] M. D. Ortigueira, Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering, Springer, Dordrecht (2011)


• [27] I. Podlubny, A. Chechkin, T. Skovranek, Y. Q. Chen, B. M. Vinagre Jara, Matrix approach to discrete fractional calculus. II. Partial fractional differential equations, J. Comput. Phys., 228 (2009), 3137--3153


• [28] Y. A. Rossikhin, M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev., 63 (2010), 52 pages


• [29] M. G. Sakar, F. Erdogan, A. Yıldırım, Variational iteration method for the time-fractional Fornberg-Whitham equation, Comput. Math. Appl., 63 (2012), 1382--1388


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


• [31] N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131 (2002), 517--529


• [32] S.Wei, W. Chen, Y.-C. Hon, Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations, Therm. Sci., 19 (2015), 59--67


• [33] B. J. West, P. Grigolini, R. Metzler T. F. Nonnenmacher, Fractional diffusion and Lévy stable processes, Phys. Rev. E, 55 (1997), 99--106


• [34] X.-J. Yang, Advanced local fractional calculus and its applications, World Sci., New York (2012)


• [35] 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


• [36] X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)


• [37] X.-J. Yang, J. T. Machado, J. Hristov, Nonlinear dynamics for local fractional Burgers' equation arising in fractal ow, Nonlinear Dyn., 84 (2015), 3--7


• [38] X.-J. Yang, H. M. Srivastava, An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 499--504


• [39] A.-M. Yang, Y.-Z. Zhang, Y. Long, The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar, Therm. Sci., 17 (2013), 707--713


• [40] Y.-Z. Zhang, Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform, Therm. Sci., 18 (2014), 677--681