A fractional model of the diffusion equation and its analytical solution using Laplace transform

Scientia Iranica

Volumn 19, Issue 4, 2012, Pages 1117-1123

  Kumar, S ^a   Yildirim, A ^b   Khan, Yasir ^c   Wei, L ^d  

^a NATIONAL INSTITUTE OF TECHNOLOGY   (India)

^b EGE UNIVERSITY   (Turkey)

^c ZHEJIANG UNIVERSITY   (China)

^d XI'AN JIAOTONG UNIVERSITY   (China)

Author keywords

Analytical solution; Diffusion equation; Fractional derivatives; Homotopy perturbation method; Laplace transform method; Mittag Leffler function

Indexed keywords

DIFFUSION; LAPLACE TRANSFORMS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

DIFFUSION EQUATIONS; FRACTIONAL DERIVATIVES; HOMOTOPY PERTURBATION METHOD (HPM); LAPLACE TRANSFORM METHOD; MITTAG-LEFFLER FUNCTIONS;

PERTURBATION TECHNIQUES;

ANALYTICAL FRAMEWORK; DIFFUSION; LAPLACE TRANSFORM; NUMERICAL MODEL; PERTURBATION;

EID: 84865264582     PISSN: 10263098     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.scient.2012.06.016     Document Type: Article

References (33)

1. K.B. Oldham, and J. Spanier The Fractional Calculus 1974 Academic Press New York
2. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 2003 John Wiley and Sons, Inc. New York
3. I. Podlubny Fractional Differential Equations 1999 Academic Press New York
4. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Application of Fractional Differential Equations 2006 Elsevier Amsterdam
5. S.S. Ray, and R.K. Bera Analytical solution of a fractional diffusion equation by Adomian decomposition method Appl. Math. Comput. 174 1 2006 329 336 (Pubitemid 43288955)
6. S. Das Analytical solution of a fractional diffusion equation by variational iteration method Comput. Math. Appl. 57 3 2009 483 487
7. J.H. He Homotopy perturbation technique Comput. Methods Appl. Mech. Eng. 178 3 1999 257 262
8. J.H. He Application of homotopy perturbation method to nonlinear wave equations Chaos Solitons Fractals 26 3 2005 695 700
9. J.H. He A coupling method of homotopy technique and perturbation technique for nonlinear problems Int. J. Non-Linear Mech. 35 2000 37 43
10. J.H. He Some asymptotic methods for strongly nonlinear equations Internat. J. Modern Phys. B 20 10 2006 1141 1199
11. J.H. He A new perturbation technique which is also valid for large parameters J. Sound Vib. 229 2000 1257 1263
12. A. Yildirim, and H. Kocak Homotopy perturbation method for solving the spacetime fractional advectiondispersion equation Adv. Water Resour 32 12 2009 1711 1716
13. A. Yildirim, and Y. Gulkanat Analytical approach to fractional ZakharovKuznetsov equations by He's homotopy perturbation method Commun. Theor. Phys. 53 2010 1005 1010
14. A. Yildirim Analytical approach to FokkerPlanck equation with space and time fractional derivatives by means of the homotopy perturbation method J. King Saud Univ., Eng. Sci. 22 2010 257 264
15. S. Kumar, Y. Khan, and A. Yildirim A mathematical modelling arising in the chemical systems and its approximate numerical solution Asia-Pac. J. Chem. Eng. 2011 10.1002/apj.647
16. N.A. Khan, N.U. Khan, M. Ayaz, and A. Mahmood Analytical method for solving the time fractional Swift Hohenberg (S-H) equation Comput. Math. Appl. 16 8 2011 2182 2185
17. N.A. Khan, M. Ayaz, N.U. Khan, and L. Jin On approximate solutions for the time-fractional reactiondiffusion equation of Fisher type Int. J. Phys. Sci. 6 10 2011 2483 2496
18. N.A. Khan, N.-U. Khan, A. Ara, and M. Jamil Approximate analytical solutions of fractional reactiondiffusion equations J. King Saud Univ., Eng. Sci. 24 2010 111 118
19. A. Golbabai, and K. Sayevand Fractional calculusa new approach to the analysis of generalized fourth-order diffusion-wave equations Comput. Math. Appl. 61 8 2011 2227 2231
20. Y. Khan, Q. Wu, N. Faraz, and A. Yildirim The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet Comput. Math. Appl. 61 11 2011 3391 3399
21. A. Ghorbani, and J. Saberi-Nadjafi He's homotopy perturbation method for calculating adomian polynomials Int. J. Nonlinear Sci. Numer. Simul. 8 2 2007 229 232
22. A. Ghorbani Beyond adomian's polynomials: He polynomials Chaos Solitons Fractals 39 3 2009 1486 1492
23. M. Khan, and M. Hussain Application of Laplace decomposition method on semi-infinite domain Numer. Algorithms 56 2 2011 211 218
24. H. Jafari, C.M. Khalique, and M. Nazari Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations Appl. Math. Lett. 24 11 2011 1799 1805
25. Y. Khan, and Q. Wu Homotopy perturbation transform method for nonlinear equations using He's polynomials Comput. Math. Appl. 61 8 2011 1963 1967
26. Y. Khan, N. Faraz, S. Kumar, and A. Yildirim A coupling method of homotopy method and Laplace transform for fractional modells U.P.B. Sci. Bull., Series A 74 1 2012 57 68
27. M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim On the coupling of the homotopy perturbation method and Laplace transformation Math. Comput. Modelling 53 9 2011 1937 1945
28. Yu. Luchko, and R. Gorenflo An operational method for solving fractional differential equations with the Caputo derivatives Acta Math. Vietnam. 24 1999 207 233
29. O.L. Moustafa On the Cauchy problem for some fractional order partial differential equations Chaos Solitons Fractals 18 1 2003 135 140
30. G. Samko, A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives: Theory and Applications 1993 Gordon and Breach Yverdon
31. F. Mainardi On the initial value problem for the fractional diffusion-wave equation S. Rionero, T. Ruggeeri, Waves and Stability in Continuous Media 1994 World Scientific Singapore 246 251
32. K. Abbaoui, and Y. Cherruault New ideas for proving convergence of decomposition methods Comput. Math. Appl. 29 1995 103 108
33. M. Dehghan Application of Adomian decomposition method for two dimensional parabolic equation subject to nonstandard boundary specification Appl. Math. Comput. 157 2 2004 549 560