THERMAL SCIENCE

International Scientific Journal

Zeliha Körpinar

ON NUMERICAL SOLUTIONS FOR THE CAPUTO-FABRIZIO FRACTIONAL HEAT-LIKE EQUATION

ABSTRACT
In this article, Laplace homotopy analysis method in order to solve fractional heat-like equation with variable coefficients, are introduced. Laplace homotopy analysis method, founded on combination of homotopy methods and Laplace transform is used to supply a new analytical approximated solutions of the fractional partial differential equations in case of the Caputo-Fabrizio. The solutions obtained are compared with exact solutions of these equations. Reliability of the method is given with graphical consequens and series solutions. The results show that the method is a powerfull and efficient for solving the fractional heat-like equations with variable coefficients.
KEYWORDS
Laplace homotopy analysis method, fractional heat-like equations, Caputo-Fabrizio derivative, approximate solution
PAPER SUBMITTED: 2017-06-14
PAPER REVISED: 2017-11-15
PAPER ACCEPTED: 2017-11-21
PUBLISHED ONLINE: 2018-01-07
DOI REFERENCE: https://doi.org/10.2298/TSCI170614274K
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement, PAGES [87 - 95]
REFERENCES
[1] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006
[2] Podlubny, I., Fractional Differential Equation, Academic Press, San Diego, 1999
[3] Inc, M., The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, Journal of Mathematical Analysis and Applications, 345 (2008), 1, pp. 476-484
[4] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, 1993
[5] Baleanu, D., Diethelm, K., Scalas, E., Trujllo, J., Fractional Calculus Models and Numerical Methods, Complexity,Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012
[6] Yilmazer, R., Inc, M., Tchier, F. and Baleanu, D., Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator, Entropy, 18(2) (2016), 49.
[7] Hristov, J., Subdiffusion model with time-dependent diffusion coefficient Integral-Balance Solution and Analysis, Thermal Sciences, 21 (2017), 1, pp. 69-80
[8] Kadem, A., Baleanu, D., Homotopy perturbation method for the coupled fractional Lotka- Volterra equations, Rom. J. Phys., 56 (2011), pp.332-338
[9] Hristov J., Steady-State Heat Conduction in a Medium with Spatial Non-Singular fading memory Derivation of Caputo-Fabrizio Space-Fractional Derivative, Thermal Sciences, 21 (2017), 2, pp.827-839
[10] Kilic, B., Inc, M., The first integral method for the time fractional Kaup-Boussinesq system with time dependent coefficient, Applied Mathematics and Computation, 254 (2015), pp. 70-74
[11] Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr.Fract.Differ.Appl.,1 (2015), pp.73-85
[12] Losada, J., Nieto, J.J., Properties of the new fractional derivative without singular kernel, Progr.Fract.Differ.Appl.,1 (2015), pp. 87-92
[13] Atangana, A., Badr, S., Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), pp. 4439-4453.
[14] Atangana, A., Badr, S., Extension of the RLC electrical circuit to fractional derivative without singular kernel, Adv.Mech.Eng. 7 (2015), pp.1-6.
[15] Franc, E., Goufo, D., Application of the Caputo-Fabrizio Fractional Derivative without Singular Kernel to Korteweg-de Vries-Burgers Equation, Mathematical Modelling and Analysis, 21 (2016), 2, pp. 188-198
[16] Alkahtani, B.S.T., Atangana, A., Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016), pp.539-546.
[17] Atangana, A., On the new fractional derivative and application to nonlinear Fisher's reactiondiffusion equation, Applied Mathematics and Computation, 273 (2016), pp. 948-956.
[18] Badr, S., Atangana, A., Analysis of non-homogeneous heat model with new trend of derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016), pp. 566-571
[19] Alqahtani, R.T., Fixed-point theorem for Caputo-Fabrizio fractional Nagumo equation with nonlinear diffusion and convection, J. Nonlinear Sci. Appl., 9 (2016), pp.1991-1999.
[20] Mohamed, M.S., Gepreel, K.A., Al-Malki, F.A., Al-Humyani, M., Approximate solutions of the generalized Abel's integral equations using the extension Khan's homotopy analysis transformation method. J. Appl. Math. (2015), 357861, pp. 1-9.
[21] Gupta, V.G., Kumar, P: Approximate solutions of fractional linear and nonlinear differential equations using Laplace homotopy analysis method, Int. J. Nonlinear Sci. 19 (2015), 2, pp.113-120
[22] Dehghan, M., Manafian, J., Saadatmandi, A., The solution of the linear fractional partial differential equations using the homotopy analysis method. Z. Naturforsch. A, 65 (2010), 11, 935-949
[23] Xu, H, Cang, J, Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett. A, 372 (2008), 8, pp.1250-1255
[24] Jafari, H., Das, S., Tajadodi, H., Solving a multi-order fractional differential equation using homotopy analysis method. J. King Saud Univ. Sci., 23 (2011), 2, pp.151-155
[25] Diethelm, K., Ford, N.J., Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput., 154 (2004), pp.621-640
[26] Goufo, E.F., Doungmo, M.K.P., Mwambakana, J.N., Duplication in a model of rock fracture with fractional derivative without singular kernel., Open Math. 13 (2015), 1, pp.839-846
[27] Morales-Delgado, V.F., Gómez-Aguilar, J.F., Yépez-Martnez, H., Baleanu, D., Escobar- Jimenez, R.F., Olivares-Peregrino, V.H., Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, (2016), 164
[28] Arqub, O.A., El-Ajou, A., Solution of the fractional epidemic model by homotopy analysis method. J. King Saud Univ. Sci. 25 (2013), 1, pp.73-81
[29] Vong, S., Wang, Z., A compact difference scheme for a two dimensional fractional Klein- Gordon equation with Neumann boundary conditions. J. Comput. Phys. 274 (2014), pp.268-282
[30] Atangana, A, On the new fractional derivative and application to nonlinear Fisher's reactiondiffusion equation. Appl. Math. Comput. 273 (2016), pp.948-956
[31] Liao. S, Homotopy Analysis Method in Nonlinear Differential Equations. Springer, Berlin, 2012
[32] Atangana, A, Alkahtani, BST, Analysis of the Keller-Segel model with a fractional derivative without singular kernel. Entropy, 17 (2015), 6, pp.4439-4453
[33] Momani, S., Analytic approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl. Math. Comput. 165 (2005), pp. 459-472
[34] Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynm., 29 (2002), pp.145-155
[35] Andrezei, H., Multi-dimensional solutions of space-time-fractional diffusion equations, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458 (2002), pp. 429-450
[36] Klafter, J., Blumen, A., Shlesinger, M.F., Fractal behavior in trapping and reaction: A random walk study, J. Stat. Phys. 36 (1984), pp.561-578.
[37] Metzler, R., Klafter, J., Boundary value problems fractional diffusion equations, Physica A, 278 (2000), pp.107-125
[38] Debnath, L., Bhatta, D., Solution to few linear inhomogenous partial differential equation in fluid mechanics, Fract. Calc. Appl. Anal. 7 (2004), 1, pp. 21-36
[39] Momani, S., Odibat, Z., Analytic approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006), pp.271-279
[40] Shou, D.H., He, J.H., Beyond Adomian methods: The variational iteration method for solving heat-like and wave-like equations with variable coefficients, Phys. Lett. A 73 (2007), 1, pp. 1-5
PDF VERSION [DOWNLOAD]
On numerical solutions for the Caputo-Fabrizio fractional heat-like equation

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence