Non-probabilistic solutions of imprecisely defined fractional-orderdiffusion equations

doi:10.1088/1674-1056/23/12/120202

Abstract The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0< α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.

Keywords: double parametric form of fuzzy number fuzzy fractional diffusion equation ADM

Received: 13 April 2014
Revised: 15 July 2014
Accepted manuscript online:

PACS:	02.60.Cb	(Numerical simulation; solution of equations)
	02.60.-x	(Numerical approximation and analysis)
	02.90.+p	(Other topics in mathematical methods in physics)
	07.05.Mh	(Neural networks, fuzzy logic, artificial intelligence)

Corresponding Authors: M. Syed Ali, R. Saravanakumar
E-mail: sne_chak@yahoo.com;smitatapaswini@gmail.com

Cite this article:

S. Chakraverty, Smita Tapaswini Non-probabilistic solutions of imprecisely defined fractional-orderdiffusion equations 2014 Chin. Phys. B 23 120202

[1]	Samko S G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives —— Theory and Applications (Langhorne PA: Gordon and Breach Science Publishers)
[2]	Miller K S and Ross B 1993 An Introduction to the Fractional Calculus and Fractional Differential Equations (New York: John Wiley and Sons)
[3]	Oldham K B and Spanier J 1974 The Fractional Calculus (New York: Academic Press)
[4]	Kiryakova V S 1993 Generalized Fractional Calculus and Applications (England: Chapman and Hall/CRC)
[5]	Podlubny I 1999 Fractional Differential Equations (New York: Academic Press)
[6]	Petras I 2011 Fractional-Order Nonlinear Systems Modeling, Analysis and Simulation (London, New York: Higher Education Press)
[7]	Wu G C and Baleanu D 2013 Adv. Difference Equ. 2013 1
[8]	Hashim I, Abdulaziz O and Momani S 2009 Commun. Nonlinear Sci. Numer. Simul. 14 674
[9]	Mohyud-Din S T, Yildirim A and Yuluklu E 2012 Int. J. Numer. Methods Heat Fluid Flow 22 928
[10]	Odibat Z and Momani S 2008 Appl. Math. Lett. 21 194
[11]	Secer A, Akinlar M A and Cevikel A 2012 Adv. Difference Equ. 202 1
[12]	Yuste S B and Quintana-Murillo J 2012 Comput. Phys. Comm. 183 2594
[13]	Allahviranloo T, Abbasbandy S and Rouhparvar H 2011 Appl. Soft Comput. 11 2186
[14]	Hosseinnia S H, Ranjbar A and Momani S 2008 Comput. Math. Appl. 56 3138
[15]	Molliq R Y, Noorani M S M and Hashim I 2009 Nonlinear Anal.: Real World Appl. 10 1854
[16]	El-Sayed A M A, El-Kalla I L and Ziada E A A 2010 Appl. Numer. Math. 60 788
[17]	Meerschaert M M and Tadjeran C 2006 Appl. Numer. Math. 56 80
[18]	Lynch V E, Carreras B A, del-Castillo-Negrete D, Ferreira-Mejias K M and Hicks H R 2003 J. Comput. Phys. 192 406
[19]	Chakraverty S and Behera D 2013 Alexandria Eng. J. 52 557
[20]	Behera D and Chakraverty S 2013 Central Eur. J. Phys. 11 792
[21]	Sun H G, Sheng H, Cheng Y Q, Chen W and Yu Z B 2013 Chin. Phys. Lett. 30 046601
[22]	Merdan M 2013 Iranian J. Sci. Technol. 83
[23]	Yuste S B and Quintana-Murillo J 2012 Comput. Phys. Comm. 183 2594
[24]	Gorenflo R, Luchko Y and Mainardi F 2000 J. Comput. Appl. Math. 118 175
[25]	Mainardi F and Pagnini G 2003 Appl. Math. Comput. 141 51
[26]	Mainardi F 1996 Chaos, Solitons and Fractals 7 1461
[27]	Odibat Z M 2006 Appl. Math. Comput. 179 92
[28]	Cetinkaya A and Kiymaz O 2013 Math. Comput. Model. 57 2349
[29]	Safari M and Danesh M 2011 Adv. Pure Math. 1 345
[30]	Godal M A, Salah A, Khan M and Batool S I 2012 Neural Comput. & Appl.
[31]	Lin Y and Xu C 2007 J. Comput. Phys. 225 1533
[32]	Garg M and Manohar P 2010 Fract. Calc. Appl. Anal. 13 191
[33]	Momani S 2006 J. Phys. Sci. 10 30
[34]	Li X, Xu M and Jiang X 2009 Appl. Math. Comput. 208 434
[35]	Costa F S and Oliveira E C 2012 J. Math. Phys. 53 123520
[36]	Lenzi E K, Mendes R S, Fa K S, Moraes L S, da Silva L R and Lucena L S 2005 J. Math. Phys. 46 083506
[37]	Zou L, Zu D Y, Wang Z and Zong Z 2013 Chin. Phys. Lett. 30 020204
[38]	Lü Z Q, Wang Y S and Song Y Z 2013 Chin. Phys. Lett. 30 030201
[39]	Wand C and Leng Y X 2013 Chin. Phys. Lett. 30 044208
[40]	Cai J X, Qin Z L and Bai C Z 2013 Chin. Phys. Lett. 30 070202
[41]	Zhang H, Wu J J, Zhang D X, Zhang R and He Z 2013 Acta Phys. Sin. 62 210202 (in Chinese)
[42]	Wang X X, Fan D, Huang J K and Huang Y 2013 Acta Phys. Sin. 62 228101 (in Chinese)
[43]	Liu L Z, Zhang J Q, Xu G X, Liang L S and Wang M S 2014 Acta Phys. Sin. 63 010501 (in Chinese)
[44]	Liu H S, Liu D D, Jiang C H, Wang L S, Jiang Y G, Sun P and Ji Y Q 2014 Acta Phys. Sin. 63 017801 (in Chinese)
[45]	Hu G L, Ni Z P and Wang Q L 2014 Acta Phys. Sin. 63 018301 (in Chinese)
[46]	Wang W H, Zhao G Z and Liang X X 2013 Chin. Phys. B 22 120205
[47]	Li C, Zhuang Y Q, Zhang L and Jin G 2014 Chin. Phys. B 23 018501
[48]	Fu Z, Liu K X and Luo N 2014 Chin. Phys. B 23 020202
[49]	Hua W and Liu S X 2014 Chin. Phys. B 23 020309
[50]	Jin C J, Wang W and Jiang R 2014 Chin. Phys. B 23 024501
[51]	Tapaswini S and Chakraverty 2012 Int. J. Fuzzy Inf. Eng. 4 293
[52]	Tapaswini S and Chakraverty 2013 Int. J. Comput. Appl. 64 5
[53]	Tapaswini S and Chakraverty S 2014 Int. J. Artificial Intell. Soft Comput. 4 58
[54]	Fard O S, Hadi Z, Ghal-Eh N and Borzabadi A H 2009 J. Adv. Res. Sci. Comput. 1 22
[55]	Mikaeilvand N and Khakrangin S 2012 Neural Comput & Appl. 21 S307
[56]	Allahviranloo T, Abbasband S Ahmady N and Ahmady E 2009 Inf. Sci. 179 945
[57]	Prakash P and Kalaiselvi V 2012 Fuzzy Inf. Eng. 4 445
[58]	Jafari H, Saeidy M and Baleanu D 2012 Central Eur. J. Phys. 10 76
[59]	Khastan A, Nieto J J and Rodriguez-Lopez R 2011 Fuzzy Sets Syst. 177 20
[60]	Chalco-Cano Y and Roman-Flores H 2008 Chaos, Solitons and Fractals 38 112
[61]	Xu J, Liao Z and Nieto J J 2010 J. Math. Anal. Appl. 368 54
[62]	Guchhait P, Maiti M K and Maiti M 2013 Eng. Appl. Artificial Intell. 26 766
[63]	Agarwal R P, Lakshmikantham V and Nieto J J 2010 Nonlinear Anal.: Theory, Methods & Appl. 72 2859
[64]	Arshad S and Lupulescu V 2011 Electron. J. Differ. Equ. 2011 1
[65]	Arshad S and Lupulescu V 2011 Nonlinear Anal. 74 3685
[66]	Jeong J U 2013 Int. Math. Forum 5 3221
[67]	Wang H and Liu Y 2011 Int. Math. Forum 6 2535
[68]	Allahviranloo T, Salahshour S and Abbasbandy S 2012 Soft Comput. 16 297
[69]	Khodadadi E and Çelik E 2013 Fixed Point Theory Appl. 2013
[70]	Mohammed O H, Fadhel F S and Abdul-Khaleq F A 2011 J. Basrah Researches (Sciences) 37 158
[71]	Salahshour S, Allahviranloo T and Abbasbandy S 2012 Commun. Nonlinear Sci. Numer. Simul. 17 1372
[72]	Salah A, Khan M and Gondal M A 2013 Neural Comput & Appl. 23 269
[73]	Ahmadian A, Suleiman M, Salahshour S and Baleanu D 2013 Adv. Differ. Equ. 2013 1
[74]	Behera D and Chakraverty S 2013 Annals Fuzzy Math. Inform. 7 401
[75]	Adomian G 1984 J. Math. Anal. Appl. 102 420

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Non-probabilistic solutions of imprecisely defined fractional-orderdiffusion equations

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