A coupling method involving the Sumudu transform and the variational iteration method for a class of local fractional diffusion equations

Gao, Feng¹ ; Srivastava, H. M.² ; Gao, Ya-Nan¹ ; Yang, Xiao-Jun¹

¹ School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, P. R. China;State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, P. R. China
² Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada;Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, P. R. China

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 11, p. 5830-5835.

Résumé

In this article, a coupling of the variational iteration method with the Sumudu transform via the local fractional calculus operator is proposed for the first time. As a testing example, the exact solution for the local fractional diffusion equation in fractal one-dimensional space is obtained. The method provided an accurate and efficient technique for solving the local fractional PDEs.

Détail

DOI : 10.22436/jnsa.009.11.11

Classification : 35R11, 26A33, 35A15
Keywords: Diffusion equation, exact solution, variational iteration method, Sumudu transform, local fractional calculus.

Bibliographie

[1] Blasiak, S. Time-fractional heat transfer equations in modeling of the non-contacting face seals, Int. J. Heat Mass Transf., Volume 100 (2016), pp. 79-88

[2] Chen, L.; Basu, B.; McCabe, D. Fractional order models for system identification of thermal dynamics of buildings, Energy and Buildings, Volume 133 (2016), pp. 381-388

[3] Damor, R. S.; Kumar, S.; Shukla, A. K. Solution of fractional bioheat equation in terms of Fox's H-function, SpringerPlus, Volume 5 (2016), pp. 1-10

[4] Ezzat, M. A. State space approach to thermoelectric fluid with fractional order heat transfer, Heat Mass Transf., Volume 48 (2012), pp. 71-82

[5] Ezzat, M. A.; El-Bary, A. A. Effects of variable thermal conductivity and fractional order of heat transfer on a perfect conducting infinitely long hollow cylinder, Int. J. Thermal Sci., Volume 108 (2016), pp. 62-69

[6] Hristov, J. Heat-balance integral to fractional (half-time) heat diffusion sub-model, Thermal Sci., Volume 14 (2010), pp. 291-316

[7] Huang, F.; Liu, F. The space-time fractional diffusion equation with Caputo derivatives, J. Appl. Math. Comput., Volume 19 (2005), pp. 179-190

[8] Jesus, I. S.; Machado, J. A. T. Fractional control of heat diffusion systems, Nonlinear Dyn., Volume 54 (2008), pp. 263-282

[9] Kulish, V. V.; Large, J. L. Fractional-diffusion solutions for transient local temperature and heat flux, ASME J. Heat Transf., Volume 122 (2000), pp. 372-376

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

[11] Liu, L.; Zheng, L.; Liu, F.; Zhang, X. An improved heat conduction model with Riesz fractional Cattaneo-Christov flux, Int. J. Heat Mass Transf., Volume 103 (2016), pp. 1191-1197

[12] Ma, M.; Baleanu, D.; Gasimov, Y. S.; Yang, X.-J. New results for multidimensional diffusion equations in fractal dimensional space, Rom. J. Phys., Volume 61 (2016), pp. 784-794

[13] Povstenko, Y. Z. Fractional heat conduction in infinite one-dimensional composite medium, J. Thermal Stresses, Volume 36 (2013), pp. 351-363

[14] Priya, G. S.; Prakash, P.; Nieto, J. J.; Kayar, Z. Higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions, Numer. Heat Trans., Part B: Fund.: Int. J. Comput. Methodol., Volume 63 (2013), pp. 540-559

[15] Simpson, R.; Jaques, A.; Núñez, H.; Ramirez, C.; Almonacid, A. Fractional calculus as a mathematical tool to improve the modeling of mass transfer phenomena in food processing, Food Eng. Rev., Volume 5 (2013), pp. 45-55

[16] Singh, J.; Kumar, D.; Nieto, J. J. A reliable algorithm for a local fractional Tricomi equation arising in fractal transonic flow, Entropy, Volume 18 (2016), pp. 1-8

[17] Srivastava, H. M.; Golmankhaneh, A. K.; Baleanu, D.; Yang, X.-J. Local fractional Sumudu transform with application to IVPs on Cantor sets, Abstr. Appl. Anal., Volume 2014 (2014), pp. 1-7

[18] Suzuki, A.; Niibori, Y.; Fomin, S. A.; Chugunov, V. A.; Hashida, T. Analysis of water injection in fractured reservoirs using a fractional-derivative-based mass and heat transfer model, Math. Geosci., Volume 47 (2015), pp. 31-49

[19] Yang, X.-J. Advanced local fractional calculus and its applications, World Science Publisher, New York, 2012

[20] Yang, X.-J.; Baleanu, D. Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Sci.,, Volume 17 (2013), pp. 625-628

[21] Yang, X.-J.; Baleanu, D.; Srivastava, H. M. Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., Volume 47 (2015), pp. 54-60

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

[23] Yang, X.-J.; Srivastava, H. M. 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., Volume 29 (2015), pp. 499-504