Sains Malaysiana 47(11)(2018): 2899–2905

http://dx.doi.org/10.17576/jsm-2018-4711-33

 

Homotopy Decomposition Method for Solving Higher-Order Time-Fractional Diffusion Equation via Modified Beta Derivative

(Kaedah Penguraian Homotopi bagi Menyelesaikan Persamaan Resapan Pecahan-Masa Peringkat Tinggi menerusi Terbitan Terubah Suai Beta)

 

SALAH ABUASAD1 & ISHAK HASHIM2*

 

1Faculty of Sciences, King Faisal University, 31982 Hofuf, Al-Hasa, Saudi Arabia

 

2School of Mathematical Sciences, Faculty Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

 

Diserahkan: 21 Februari 2018/Diterima: 2 Julai 2018

 

ABSTRACT

In this paper, the homotopy decomposition method with a modified definition of beta fractional derivative is adopted to find approximate solutions of higher-dimensional time-fractional diffusion equations. To apply this method, we find the modified beta integral for both sides of a fractional differential equation first, then using homotopy decomposition method we can obtain the solution of the integral equation in a series form. We compare the solutions obtained by the proposed method with the exact solutions obtained using fractional variational homotopy perturbation iteration method via modified Riemann-Liouville derivative. The comparison shows that the results are in a good agreement.

 

Keywords: Beta derivative; fractional differential equation; fractional diffusion equation; homotopy decomposition method

 

ABSTRAK

Dalam kertas ini, kaedah penguraian homotopi dengan takrif terbitan pecahan beta terubah suai diadaptasi untuk mencari penyelesaian penghampiran bagi persamaan resapan pecahan-masa peringkat tinggi. Untuk menggunakan kaedah ini, kami dapatkan dahulu kamiran beta terubah suai bagi kedua-dua belah persamaan terbitan pecahan itu, kemudian dengan menggunakan kaedah penguraian homotopi kami boleh dapatkan penyelesaian bagi persamaan kamiran itu dalam bentuk siri. Kami bandingkan penyelesaian yang diperoleh dengan penyelesaian tepat yang diperoleh menerusi kaedah usikan berlelar homotopi ubahan dengan terbitan Riemann-Liouville. Perbandingan menunjukkan kedua-dua kaedah memberikan penyelesaian yang sangat hampir.

 

Kata kunci: Kaedah penguraian homotopi; persamaan resapan pecahan; persamaan terbitan pecahan; terbitan beta

RUJUKAN

Abdullah, F.A. 2013. Simulations of Hirschsprung’s Disease using fractional differential equations. Sains Malaysiana 42(5): 661-666.

Atangana, A. 2015. Derivative with a New Parameter: Theory, Methods and Applications. London: Elsevier.

Atangana, A. & Belhaouari, S. 2013. Solving partial differential equation with space- and time-fractional derivatives via homotopy decomposition method. Mathematical Problems in Engineering 2013: Article ID. 318590.

Atangana, A. & Secer, A. 2013. The time-fractional coupled- Korteweg-de Vries equations. Abstract and Applied Analysis 2013: Article ID. 947986.

Atangana, A. & Botha, J. 2012. Analytical solution of the groundwater flow equation obtained via homotopy decomposition method. Journal of Earth Science & Climatic Change 3: 1000115.

Çetinkaya, A. & Kıymaz, O. 2013. The solution of the time-fractional diffusion equation by the generalized differential transform method. Mathematical and Computer Modelling 57: 2349-2354.

Das, S. 2009a. Analytical solution of a fractional diffusion equation by variational iteration method. Computers & Mathematics with Applications 57: 483-487.

Das, S. 2009b. A note on fractional diffusion equations. Chaos, Solitons & Fractals 42: 2074-2079.

Gorenflo, R., Mainardi, F. Moretti, D. & Paradisi, P. 2002. Time fractional diffusion: A discrete random walk approach. Nonlinear Dynamics 29: 129-143.

Guo, S.M.L. & Li, Y. 2013. Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation. Applied Mathematics and Computation 219: 5909-5917.

He, J.H. 1999. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 178: 257- 262.

Huang, F. & Liu, F. 2005. The time fractional diffusion equation and the advection-dispersion equation. The ANZIAM Journal 46: 317-330.

Jumarie, G. 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications 51: 1367-1376.

Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J. 2006. Theory and Applications of Fractional Differential Equations. Volume 204. New York: Elsevier.

Li, X., Xu, M. & Jiang, X. 2009. Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition. Applied Mathematics and Computation 208: 434-439.

Lin, Y. & Xu, C. 2007. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics 225: 1533-1552.

Miller, K.S. & Ross, B. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley-Interscience.

Nigmatullin, R. 1986. The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi (b) 133: 425-430.

Nigmatullin, R. 1984. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience. Physica Status Solidi (b) 123: 739-745.

Oldham, K. & Spanier, J. 1974. The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Volume 111. New York: Academic Press.

Podlubny, I. 1999. Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. San Diego, CA: Academic Press.

Ray, S.S. & Bera, R. 2006. Analytical solution of a fractional diffusion equation by Adomian decomposition method. Applied Mathematics and Computation 174: 329-336.

Samko, S.G., Kilbas, A.A. & Marichev, O.I. 1993. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers.

 

*Pengarang untuk surat-menyurat; email: ishak_h@ukm.edu.my

 

 

 

 

 

 

 

 

 

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