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Sains Malaysiana 47(11)(2018): 28992905 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 Received: 21 February
2018/Accepted: 2 July 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 REFERENCES
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Gordon and Breach Science Publishers. *Corresponding author; email: ishak_h@ukm.edu.my
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