An efficient computational method for local fractional transport equation occurring in fractal porous media
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An efficient computational method for local fractional transport equation occurring in fractal porous media Jagdev Singh1 · Devendra Kumar2 · Sunil Kumar3 Received: 25 May 2019 / Revised: 1 April 2020 / Accepted: 9 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The present article deals with the local fractional linear transport equations (LFLTE) in fractal porous media. LFLTE play a key role in different scientific problems such as aeronomy, superconductor, semiconductors, turbulence, gas mixture, plasma and biology. A numerical scheme namely q-local fractional homotopy analysis transform method (q-LFHATM) is applied to get the solution of LFLTE. The results obtained by using of q-LFHATM show that the proposed scheme is very suitable and easy to perform with high accuracy. Keywords Local fractional transport equations · Fractal porous media · q-Local fractional homotopy analysis transform method · Local fractional laplace transform Mathematics Subject Classification 26A33 · 35R11 · 35Q99
1 Introduction The local fractional transport equations play a key role in different scientific problems such as superconductor (Betbeder-Matibet and Nozieres 1969), aeronomy (Schunk 1975) semiconductors (Blotekjaer 1970), turbulence (Daly and Harlow 1970), gas mixture (Tanenbaum 1965), plasma (Mikhailovskii and Tsypin 1984) and biology (Perthame 2006). Zaslavsky (2002) discussed the anomalous transport of fractional dynamics, Tarasov (2006) studied transport equations for fractional-order systems. Uchaikin and Sibatov ( 2008) discussed the Communicated by Jorge X. Velasco. * Devendra Kumar [email protected] Jagdev Singh [email protected] Sunil Kumar [email protected] 1
Department of Mathematics, JECRC University, Jaipur, Rajasthan 303905, India
2
Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan 302004, India
3
Department of Mathematics, National Institute of Technology, Jamshedpur 831014, India
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utilization of this equation in the disordered semiconductors, Lutz (2001) investigated the transport equations with arbitrary order derivative, Kadem et al. (2010) discussed spectral technique to solve the transport equation with fractional order, Meng Li et al. (2014) studied the approximate solutions of linear transport equation and many others. In these years local fractional calculus has attained more attraction and interest of the researchers and mathematicians because of its broad utilization such as Yang (2012a) studied heat transfer in discontinues media, Singh et al. (2019) demonstrated local fractional wave equation in fractal strings, Rayneau-Kirkhope et al. (2012) analyzed ultra-light fractals structures, Povestenko (2004) discussed heat conduction equation with arbitrary order derivative, Shih (1982) investigated on the numerical heat transfer, Wang et al. (2012) studied heat conduction associated with non-integer order derivative, Yang and Baleanu (2013) explained fractal hea
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