×

Picard successive approximation method for solving differential equations arising in fractal heat transfer with local fractional derivative. (English) Zbl 1470.35418

Summary: The Fourier law of one-dimensional heat conduction equation in fractal media is investigated in this paper. An approximate solution to one-dimensional local fractional Volterra integral equation of the second kind, which is derived from the transformation of Fourier flux equation in discontinuous media, is considered. The Picard successive approximation method is applied to solve the temperature field based on the given Mittag-Leffler-type Fourier flux distribution in fractal media. The nondifferential approximate solutions are given to show the efficiency of the present method.

MSC:

35R11 Fractional partial differential equations
35A35 Theoretical approximation in context of PDEs
35Q79 PDEs in connection with classical thermodynamics and heat transfer
80A19 Diffusive and convective heat and mass transfer, heat flow
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Crank, J.; Nicolson, P., A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Mathematical Proceedings of the Cambridge Philosophical Society, 43, 50-67 (1947) · Zbl 0029.05901
[2] Shih, T. M., A literature survey on numerical heat transfer, Numerical Heat Transfer, 5, 4, 369-420 (1982) · doi:10.1080/10407788208913456
[3] Choi, B. C.; Churchill, S. W., A technique for obtaining approximate solutions for a class of integral equations arising in radiative heat transfer, International Journal of Heat and Fluid Flow, 6, 1, 42-48 (1985) · doi:10.1016/0142-727X(85)90030-X
[4] Povstenko, Y. Z., Fractional heat conduction equation and associated thermal stress, Journal of Thermal Stresses, 28, 1, 83-102 (2004) · doi:10.1080/014957390523741
[5] Dehghan, M., The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons & Fractals, 32, 2, 661-675 (2007) · Zbl 1139.35352 · doi:10.1016/j.chaos.2005.11.010
[6] Ioannou, Y.; Fyrillas, M. M.; Doumanidis, C., Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer, Engineering Analysis with Boundary Elements, 36, 8, 1278-1283 (2012) · Zbl 1352.65653 · doi:10.1016/j.enganabound.2012.02.006
[7] Nadeem, S.; Akbar, N. S., Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: application of adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation, 14, 11, 3844-3855 (2009) · doi:10.1016/j.cnsns.2008.09.010
[8] Wazwaz, A.-M.; Mehanna, M. S., The combined Laplace-Adomian method for handling singular integral equation of heat transfer, International Journal of Nonlinear Science, 10, 2, 248-252 (2010) · Zbl 1215.65206
[9] Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Letters A, 360, 1, 109-113 (2006) · Zbl 1236.80010 · doi:10.1016/j.physleta.2006.07.065
[10] Raftari, B.; Vajravelu, K., Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall, Communications in Nonlinear Science and Numerical Simulation, 17, 11, 4149-4162 (2012) · Zbl 1316.76111 · doi:10.1016/j.cnsns.2012.01.032
[11] Joneidi, A. A.; Ganji, D. D.; Babaelahi, M., Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, International Communications in Heat and Mass Transfer, 36, 7, 757-762 (2009) · doi:10.1016/j.icheatmasstransfer.2009.03.020
[12] Cattani, C.; Laserra, E., Spline-wavelets techniques for heat propagation, Journal of Information & Optimization Sciences, 24, 3, 485-496 (2003) · Zbl 1049.65104 · doi:10.1080/02522667.2003.10699579
[13] Simões, N.; Tadeu, A.; António, J.; Mansur, W., Transient heat conduction under nonzero initial conditions: a solution using the boundary element method in the frequency domain, Engineering Analysis with Boundary Elements, 36, 4, 562-567 (2012) · Zbl 1352.65596 · doi:10.1016/j.enganabound.2011.10.006
[14] Hristov, J., Approximate solutions to fractional sub-diffusion equations: the heat-balance integral method, The European Physical Journal, 193, 1, 229-243 (2011) · doi:10.1140/epjst/e2011-01394-2
[15] Hristov, J., Heat-balance integral to fractional (half-time) heat diffusion sub-model, Thermal Science, 14, 2, 291-316 (2010) · doi:10.2298/TSCI1002291H
[16] Ganji, D. D.; Sajjadi, H., New analytical solution for natural convection of Darcian fluid in porous media prescribed surface heat flux, Thermal Science, 15, 2, 221-227 (2011) · doi:10.2298/TSCI100424001V
[17] Yang, X. J.; Baleanu, D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 17, 2, 625-628 (2013) · doi:10.2298/TSCI121124216Y
[18] Yang, X.-J., Picard’s approximation method for solving a class of local fractional Volterra integral equations, Advances in Intelligent Transportation Systems, 1, 3, 67-70 (2012)
[19] Nigmatullin, R. R., The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi B, 133, 1, 425-430 (1986) · doi:10.1002/pssb.2221330150
[20] Davey, K.; Prosser, R., Analytical solutions for heat transfer on fractal and pre-fractal domains, Applied Mathematical Modelling, 37, 1-2, 554-569 (2013) · Zbl 1349.80013 · doi:10.1016/j.apm.2012.02.047
[21] Yang, X. J., Advanced Local Fractional Calculus and Its Applications (2012), New York, NY, USA: World Science, New York, NY, USA
[22] Zhang, Y. Z.; Yang, A. M.; Yang, X.-J., 1-D heat conduction in a fractal medium: a solution by the local fractional Fourier series method, Thermal Science, 17, 3, 953-956 (2013) · doi:10.2298/TSCI130303041Z
[23] Hu, M.-S.; Baleanu, D.; Yang, X.-J., One-phase problems for discontinuous heat transfer in fractal media, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1296.80006 · doi:10.1155/2013/358473
[24] Yang, X. J., Local Fractional Functional Analysis and Its Applications (2011), Hong Kong: Asian Academic, Hong Kong
[25] Li, M.; Zhao, W., On bandlimitedness and lag-limitedness of fractional Gaussian noise, Physica A, 392, 9, 1955-1961 (2013) · doi:10.1016/j.physa.2012.12.035
[26] Li, M., Approximating ideal filters by systems of fractional order, Computational and Mathematical Methods in Medicine, 2012 (2012) · Zbl 1233.92048 · doi:10.1155/2012/365054
[27] Li, M.; Zhao, W., On \(1 / f\) noise, Mathematical Problems in Engineering, 2012 (2012) · Zbl 1264.94060 · doi:10.1155/2012/673648
[28] Heydari, M. H.; Hooshmandasl, M. R.; Maalek Ghaini, F. M.; Li, M., Chebyshev wavelets method for solution of nonlinear fractional integro-differential equations in a large interval, Advances in Mathematical Physics, 2013 (2013) · Zbl 1291.65245 · doi:10.1155/2013/482083
[29] Ray, S. S.; Bera, R. K., Analytical solution of a fractional diffusion equation by Adomian decomposition method, Applied Mathematics and Computation, 174, 1, 329-336 (2006) · Zbl 1089.65108 · doi:10.1016/j.amc.2005.04.082
[30] Wang, Q., Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Applied Mathematics and Computation, 182, 2, 1048-1055 (2006) · Zbl 1107.65124 · doi:10.1016/j.amc.2006.05.004
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.