An analysis of time-fractional heat transfer problem using two-scale approach. (English) Zbl 1480.35386

Summary: Porous media have been a significant subject of research for a long time due to their applicability in various sciences. This paper investigates the heat transfer phenomenon in the porous media. A convergent solution is obtained for a two-dimensional time-fractional equation arising in a porous soil heat transfer. He’s polynomial and He’s variational iteration method are used to accomplish the required goals. The fractional derivative used in the article is described by He’s definition. He’s fractional complex transform is used to convert the fractional differential equation into its traditional partner differential equation, which can be solved iteratively. Graphical representations of the results are provided to demonstrate the efficacy of the methods used.


35R11 Fractional partial differential equations
35A25 Other special methods applied to PDEs
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI


[1] Ain, QT; He, JH; Anjum, N.; Ali, M., The fractional complex transform: a novel approach to the time-fractional SCHRÖDINGER equation, Fractals, 28, 7, 2050141-2055578 (2020)
[2] Ain, QT; He, JH, On two-scale dimension and its applications, Therm. Sci., 23, 1707-1712 (2019)
[3] Ali, M.; Anjum, N.; Ain, QT; He, JH, Homotopy perturbation method for the attachment oscillator arising in nanotechnology, Fibers Polym., 2, 96 (2020)
[4] Anjum, N.; Ain, QT, Application of he’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation, Therm. Sci., 00, 450-450 (2019)
[5] Anjum, N.; He, JH, Laplace transform: making the variational iteration method easier, Appl. Math. Lett., 92, 134-138 (2019) · Zbl 1414.34014
[6] Anjum, N.; He, JH, Nonlinear dynamic analysis of vibratory behavior of a graphene nano/microelectromechanical system, Math. Methods Appl. Sci., 2, 968 (2020)
[7] Anjum, N.; He, JH, Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromagnetic force under zero initial conditions, Alexandria Eng. J., 3, 61 (2020)
[8] Anjum, N.; He, JH, Homotopy perturbation method for N/MEMS oscillators, Math. Methods Appl. Sci., 7, 1002 (2020)
[9] Anjum, N.; He, JH, Higher-order homotopy perturbation method for conservative nonlinear oscillators generally and microelectromechanical systems’ oscillators particularly, Int. J. Modern Phys. B., 20, 503130 (2020) · Zbl 1454.34032
[10] Anjum, N.; He, JH, Two modifications of the homotopy perturbation method for nonlinear oscillators, J. Appl. Comput. Mech., 9, 1420-1425 (2020)
[11] Cao, XQ; Guo, YN; Zhang, CZ; Hou, SC; Peng, KC, Different groups of variational principles for Whitham-Broer-Kaup equations in shallow water, J. Appl. Comput. Mech., 2, 17 (2020)
[12] Cao, XQ; Ya-Nan, G.; Shi-Cheng, H.; Cheng-Zhuo, Z.; Ke-Cheng, P., Variational principles for two kinds of coupled nonlinear equations in shallow water, Symmetry., 12, 850 (2020)
[13] Cao, XQ; Hou, SC; Guo, YN, Variational principle for (2+1)-dimensional Broer-Kaup equations with fractal derivatives, Fractals, 28, 2050107 (2020)
[14] Cao, XQ, Generalized variational principles for Boussinesq equation systems, Acta Phys. Sin., 2, 105-11 (2011)
[15] Cao, XQ; Jun-Qiang, S.; Wei-Min, Z.; Jun, Z., Variational principles for two kinds of extended Korteweg—de Vries equations, Chin. Phys. B., 20, 9, 090401 (2011)
[16] Faraz, N.; Khan, Y.; Jafari, H.; Yildirim, A.; Madani, M., Fractional variational iteration method via modified Riemann-Liouville derivative, J. King Saud Univ. Sci., 23, 4, 413-417 (2011)
[17] He, JH, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178, 3, 257-262 (1999) · Zbl 0956.70017
[18] He, JH, Variational iteration method for autonomous ordinary differential systems, App. Math. Comput., 114, 2-3, 115-123 (2000) · Zbl 1027.34009
[19] He, JH, Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 2, 4, 235-236 (1997)
[20] He, JH, Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34, 4, 699-708 (1999) · Zbl 1342.34005
[21] He, JH; Ain, QT, New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle, Therm. Sci., 24, 659-685 (2020)
[22] He, JH; Elagan, SK; Li, ZB, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376, 257-259 (2012) · Zbl 1255.26002
[23] He, JH; Li, ZB; Wang, Q., A new fractional derivative and its application to explanation of polar bear hairs, J. King Saud Univ. Sci., 28, 190-192 (2016)
[24] He, JH; Li, ZB, Converting fractional differential equations into partial differential equations, Therm. Sci., 16, 331-334 (2012)
[25] He, JH; Sun, C., A variational principle for a thin film equation, J. Math. Chem., 57, 2075-2081 (2019) · Zbl 1462.76021
[26] He, JH, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech., 35, 37-43 (2000) · Zbl 1068.74618
[27] He, JH, A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives, Int. J. Numer. Methods Heat Fluid Flow., 6, 0961-5539 (2020)
[28] He, JH, Fractal calculus and its geometrical explanation, Results Phys., 10, 272-276 (2018)
[29] He, JH, Lagrange crisis and generalized variational principle for 3D unsteady flow, Int J Numer Methods Heat Fluid Flow, 11, 0961-5539 (2019)
[30] He, JH, The simplest approach to nonlinear oscillators, Results Phys., 15, 102546 (2019)
[31] He, JH, Α review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear Sci. Numer. Simul., 1, 51-70 (2000) · Zbl 0966.65056
[32] Khan, Y., A variational approach for novel solitary solutions of FitzHugh-Nagumo equation arising in the nonlinear reaction-diffusion equation, Int. J. Numer. Methods Heat Fluid Flow, 2, 17 (2020)
[33] Khan, Y.; Faraz, N.; Yildirim, A., New soliton solutions of the generalized Zakharov equations using He’s variational approach, Appl. Math. Lett., 24, 6, 965-968 (2011) · Zbl 1211.35071
[34] Khan, Y., A new necessary condition of soliton solutions for Kawahara equation arising in physics, Optik, 155, 273-275 (2018)
[35] Khan, Y., Fractal modification of complex Ginzburg-Landau model arising in the oscillating phenomena, Results Phys., 18, 103324 (2020)
[36] Khan, Y.; Wu, Q., Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Comput. Math. Appl., 61, 8, 1963-1967 (2011) · Zbl 1219.65119
[37] Khan, Y.; Faraz, N.; Yildirim, A.; Wu, Q., Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science, Comput. Math. Appl., 62, 5, 2273-2278 (2011) · Zbl 1231.35288
[38] Khan, Y.; Wu, Q.; Faraz, N.; Yildirim, A.; Madani, M., A new fractional analytical approach via a modified Riemann-Liouville derivative, Appl. Math. Lett., 25, 10, 1340-1346 (2012) · Zbl 1251.65101
[39] Li, XX; Tian, D.; He, CH; He, JH, A fractal modification of the surface coverage model for an electrochemical arsenic sensor, Electro Chemica Acta., 296, 491-493 (2019)
[40] Li, ZB; He, JH, Fractional complex transform for fractional differential equations, Math. Comput. Appl., 15, 970-973 (2010) · Zbl 1215.35164
[41] Li, ZB; Zhu, WH; He, JH, Exact solutions of time-fractional heat conduction equation by the fractional complex transform, Therm. Sci., 16, 335-338 (2012)
[42] Liu, FJ; Li, ZB; Zhang, S.; Liu, HY, He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy, Therm. Sci., 19, 1155-1159 (2015)
[43] Liu, HY; Li, Z.; Yao, Y., A fractional nonlinear system for release oscillation of silver ions from hollow fibers, J. Low Freq. Noise Vib. Active Control., 38, 88-92 (2019)
[44] Ren, ZF; Yao, SW; He, JH, He’s multiple scales method for nonlinear vibrations, J. Low Freq. Noise Vib. Active Control., 38, 1708-1712 (2019)
[45] Tian, D.; Ain, QT; Anjum, N., Fractal N/MEMS: From pull-in instability to pull-in stability, Fractals., 2, 19 (2020)
[46] Wang, KJ, A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge, Eur. Phys. J. plus, 135, 871 (2020)
[47] Wang, KJ, Variational principle and approximate solution for the generalized burgers-huxley equation with fractal derivative, Fractals., 11, 13 (2021)
[48] Wang, KJ, G-D Wang variational principle and approximate solution for the fractal generalized benjamin-bona-mahony-burgers equation in fluid mechanics, Fractals., 10, 97 (2021)
[49] Wang, KL; He, CH, A remark on Wang’s fractal variational principle, Fractals, 27, 1950134-1950276 (2019) · Zbl 1434.35267
[50] Wang, KL; Liu, SY, He’s fractional derivative and its application for fractional Fornberg-Whitham equation, Therm. Sci., 21, 2049-2055 (2017)
[51] Wang, Y.; An, JY, Amplitude-frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion, J. Low Freq. Noise Vib. Active Control., 38, 1008-1012 (2019)
[52] Wang, Y.; Deng, Q., Fractal derivative model for tsunami traveling, Fractals, 27, 1950017 (2019)
[53] Wang, YAN; An, JY; Wang, X., A variational formulation for anisotropic wave traveling in a porous medium, Fractals, 27, 1950047 (2019)
[54] Xu, Y.; Yang, T.; Fuller, CR; Sun, Y.; Liu, Z., A theoretical analysis on the active structural acoustical control of a vibration isolation system with a coupled plate-shell foundation, Int. J. Mech. Sci., 170, 105334 (2020)
[55] Yang, XJ; Baleanu, D.; Khan, Y.; Mohyud-Din, ST, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Rom. J. Phys., 59, 1-2, 36-48 (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.