On convergence of homotopy analysis method and its application to fractional integro-differential equations

Quaestiones Mathematicae

Volumn 36, Issue 1, 2013, Pages 93-105

  Abbasbandy, S ^a   Hashemi, M S ^b   Hashim, I ^c  

^a IMAM KHOMEINI INTERNATIONAL UNIVERSITY   (Iran)

^b UNIVERSITY OF BONAB   (Iran)

^c UNIVERSITI KEBANGSAAN MALAYSIA   (Malaysia)

Author keywords

convergence control parameter; Fractional integro differential equation; homotopy analysis method

Indexed keywords

EID: 84876037987     PISSN: 16073606     EISSN: 1727933X     Source Type: Journal    
DOI: 10.2989/16073606.2013.780336     Document Type: Article

References (37)

1. Abbasbandy, S. 2007. The application of homotopy analysis method to solve a generalized HirotaSatsuma coupled KdV equation. Phys. Lett. A, 361: 478-483.
2. Abbasbandy, S. 2010. Homotopy analysis method for the Kawahara equation. Nonlinear Anal. RWA, 11: 307-312.
3. Abbasbandy, S., Babolian, E. and Ashtiani, M. 2009. Numerical solution of the generalized Zakharov equation by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul., 14: 4114-4121.
4. Abbasbandy, S. and Hayat, T. 2009. Solution of the MHD Falkner-Skan flow by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul., 14: 3591-3598.
5. Abbasbandy, S., Hayat, T., Mahomed, F.M. and Ellahi, R. 2009. On comparison of exact and series solutions for thin film flow of a third grade fluid. Int. J. Numer. Method Fluids, 61(9): 987-994.
6. Abdulaziz, O., Bataineh, A.S. and Hashim, I. 2010. On convergence of homotopy analysis method and its modification for fractional modified KdV equations. J. Appl. Math. Comput., 33: 61-81.
7. Abel, N. 1881. Solution de quelques problèmes à laide dintégrales définites, 16-18. Norway: Christiania Grondahl.
8. Arikoglu, A. and Ozkol, I. 2007. Solution of fractional differential equations by using differential transform method. Chaos, Solitons & Fractals, 34: 1473-1481.
9. Arikoglu, A. and Ozkol, I. 2009. Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons & Fractals, 40: 521-529.
10. Bataineh, A.S., Alomari, A.K., Noorani, M.S.M., Hashim, I. and Nazar, R. 2009. Series Solutions of Systems of Nonlinear Fractional Differential Equations. Acta. Appl. Math., 105: 189-198.
11. Chen, Q., Suki, B. and An, K.N. 2004. Dynamic mechanical properties of agarose gels modeled by a fractional derivative model. J. Biomech. Eng., 126: 666-71.
12. Craiem, D. and Armentano, R. 2007. A fractional derivative model to describe arterial viscoelasticity. Biorheology, 44: 251-263.
13. Ellahi, R., Ariel, P.D., Hayat, T. and Asghar, S. 2010. Effect of heat transfer on a third grade fluid in a flat Channel. Int. J. Numer. Methods Fluids, 63: 847-859.
14. Folland, G.B. 1999. Real analysis: Modern techniques and their applications, 2nd Ed Wiley, New York
15. Ganjiani, M. 2010. Solution of nonlinear fractional differential equations using homotopy analysis method. Appl. Math. Model., 34: 1634-1641.
16. Hosseinnia, S.H., Ranjbar, A. and Momani, S. 2008. Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Comput. Math. Appl., 56: 3138-3149.
17. Kiss, M.Z., Varghese, T. and Hall, T.J. 2004. Viscoelastic characterization of in vitro canine tissue. Phys. Med. Biol., 49: 4207-4218.
18. Klatt, D., Hamhaber, U., Asbach, P., Braun, J. and Sack, I. 2007. Noninvasive assessment of the rheological behavior of human organs using multifrequency Mr elastography: a study of brain and liver viscoelasticity. Phys. Med. Biol., 52: 7281-7294.
19. Kurulay, M. and Bayram, M. 2010. Approximate analytical solution for the fractional modified KdV by differential transform method. Commun. Nonlinear Sci. Numer. Simul., 15: 1777-1782.
20. Lacroix, S. 1819. Traité du calcul differentiel et du calcul intégral, Paris: Courcier.
21. Leibniz, G. 1962. Leibnizsche Mathematische Schriften, Hildesheim: Georg Olm Verlag.
22. Liao S.J. The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992.
23. Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
24. The Homotopy Analysis Method in Nonlinear Differential Equations, Higher Education Press and Springer, Beijing and Heidelberg, 2012.
25. Malik, M.Y., Hussain, A., Nadeem, S. and Hayat, T. 2009. Flow of a third grade fluid between coaxial cylinders with variable viscosity. Z. Naturforsch., 64(a): 588-596.
26. Momani, S. and Noor, M.A. 2006. Numerical methods for fourth order fractional integro-differential equations. Appl. Math. Comput., 182: 754-760.
27. Momani, S. and Qaralleh, A. 2006. An Efficient Method for Solving Systems of Fractional Integro-Differential Equations. Comput. Math. Appl., 52: 459-470.
28. Nadeem, S. 2009. Thin film flow of a third grade fluid with variable viscosity. Z. Naturforsch., 64(a): 553-558.
29. Nadeem, S., Abbasbandy, S. and Hussain, M. 2009. Series solutions of boundary layer flow of a micropolar fluid near the stagnation point towards a shrinking sheet. Z. Naturforsch., 64(a): 575-582.
30. Nadeem, S. and Ali, M. 2009. Analytical solutions for pipe flow of a fourth grade fluid with Reynold and Vogel's models of viscosities. Commun. Nonlinear Sci. Numer. Simul., 14: 2073-2090.
31. Nadeem, S. and Awais, M. 2008. Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity. Phys. Lett. A, 372: 4965-4972.
32. Podlubuy, I. 1999. Fractional Differential Equations, New York: Academic Press.
33. Rawashdeh, E.A. 2006. Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput., 176: 1-6.
34. Rostam, K.S. and Hassan, M.S. 2010. Solving a System of Linear Fredholm Fractional Integro-differential Equations Using Homotopy Perturbation Method. Aust. J. Basic Appl. Sci., 4: 633-638.
35. Rudin, W. 1976. Principles of mathematical Analysis, 3rd Ed, New York: McGraw-Hill.
36. Sinkus, R., Siegmann, K., Xydeas, T., Tanter, M., Claussen, C. and Fink, M. 2007. MR elastography of breast lesions: understanding the solid/liquid duality can improve the specificity of contrast-enhanced Mr mammography. Magn. Reson. Med., 58: 1135-1144.
37. Zurigat, M., Momani, S. and Alawneh, A. 2009. Homotopy analysis method for systems of fractional integro-differential equations. Neur. Parallel Sci. Comput., 17: 169-186.