AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION

Li-Guo Yuan; Zeeshan Alam

doi:10.11948/2016009

2016 Volume 6 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2016 > 6(1): 103-118

Next Article Previous Article

Li-Guo Yuan, Zeeshan Alam. AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2016, 6(1): 103-118. doi: 10.11948/2016009

Citation:

Li-Guo Yuan, Zeeshan Alam. AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2016, 6(1): 103-118. doi: 10.11948/2016009

AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION

Li-Guo Yuan¹,
Zeeshan Alam²

1 Department of Applied Mathematics, South China Agricultural University, Guangzhou 510642, China;
2 School of Mathematics, South China University of Technology, Guangzhou 510641, China

Fund Project:

Abstract

Abstract

This paper describes a new problem-solving mentality of finding optimal parameters in optimal homotopy analysis method (optimal HAM). We use particle swarm optimization (PSO) to minimize the exact square residual error in optimal HAM. All optimal convergence-control parameters can be found concurrently. This method can deal with optimal HAM which has finite convergence-control parameters. Two nonlinear fractional-order differential equations are given to illustrate the proposed algorithm. The comparison reveals that optimal HAM combined with PSO is effective and reliable. Meanwhile, we give a sufficient condition for convergence of the optimal HAM for solving fractional-order equation, and try to put forward a new calculation method for the residual error.
- Optimal homotopy analysis method /
- particle swarm optimization /
- fractional-order differential equation /
- Caputo derivative /
- residual error
MSC: 34A08;74H10;65K10

FullText(HTML)

References

[1]	O. Abdulaziz, A. Bataineh and I. Hashim, On convergence of homotopy analysis method and its modification for fractional modified KdV equations, J. Appl. Math. Comput., 33(2010), 61-81. Google Scholar
[2]	A. Alomari, M. Noorani, R. Nazar and C. Li, Homotopy analysis method for solving fractional Lorenz system, Commun. Nonlinear Sci., 15(2010), 1864-1872. Google Scholar
[3]	S. Abbas, M. Banerjee and S. Momani, Dynamical analysis of fractional-order modified logistic model, Comput. Math. Appl., 62(2011), 1098-1104. Google Scholar
[4]	K. Diethelm, The Analysis of Fractional Differential Equations, Springer Press, New York, 2010. Google Scholar
[5]	K. Diethelm, N. Ford and A. Freed, A predictor-corrector approach for the numerical solution of fractional differential equation, Nonlinear Dynam., 29(2002), 3-22. Google Scholar
[6]	T. Fan and X. You, Optimal homotopy analysis method for nonlinear differential equations in the boundary layer, Numer. Algorithms, 62(2013), 337-354. Google Scholar
[7]	J. Fan and B. Yang, The mean value theorem for multiple integral, Math. Pract. Theory, 37(2007), 197-200. (in Chinese) Google Scholar
[8]	I. Hashim, O. Abdulaziz and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci., 14(2009), 674-684. Google Scholar
[9]	J. Izadian, M. MohammadzadeAttar and M. Jalili, Numerical solution of deformation equations in homotopy analysis method, Appl. Math. Sci., 6(2012), 357-367. Google Scholar
[10]	H. Jafari, K. Sayevand, H. Tajadodi and D. Baleanu, Homotopy analysis method for solving Abel differential equation of fractional order, Cent. Eur. J. Phys., 11(2013), 1523-1527. Google Scholar
[11]	H. Jafari, H. Tajadodi and D. Baleanu, A modified variational iteration method for solving fractional riccati differential equation by adomian polynomials, Fract. Calc. Appl. Anal., 16(2013), 109-122. Google Scholar
[12]	A. Kilbas, H.Srivastava and J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier Science, Netherlands, 2006. Google Scholar
[13]	J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings IEEE International Conference on Neural Networks, 4(1995), 1942-1948. Google Scholar
[14]	J. Kennedy, R. Eberhart and Y. Shi, Swarm Intelligence, CA:Morgan Kaufman, San Francisco, 2001. Google Scholar
[15]	S. Liao, Beyond perturbation:introduction to the homotopy analysis method, Chapman&Hall/CRC Press, Boca Raton, 2003. Google Scholar
[16]	S. Liao, Homotopy analysis method in nonlinear differential equations, Higher Education Press, Beijing, 2012. Google Scholar
[17]	S. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci., 15(2010), 2003-2016. Google Scholar
[18]	S. Liao, An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Nonlin. Mech., 34(1999), 759-778. Google Scholar
[19]	Y. Liu, Z. Li and Y. Zhang, Homotopy perturbatioin method to fractional biological population equation, Fract. Diff. Calc., 1(2011), 117-124. Google Scholar
[20]	V. Marinca, N. Herisanu and I. Nemes, Optimal homotopy asymptotic method with application to thin film flow, Cent. Eur. J. Phys., 6(2008), 648-653. Google Scholar
[21]	V. Marinca and N. Herisanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Int. Commun. Heat Mass, 35(2008), 710-715. Google Scholar
[22]	V. Marinca and N. Herisanu, An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl. Math. Lett., 22(2009), 245-251. Google Scholar
[23]	M. Mohamed, Application of optimal HAM for solving the fractional order logistic equation, Appl. Comput. Math., 3(2014), 27-31. Google Scholar
[24]	Z. Niu and C. Wang, A one-step optimal homotopy analysis method for nonlinear differential equations, Commun. Nonlinear Sci., 15(2008), 2026-2036. Google Scholar
[25]	Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Comput. Math. Appl., 61(2011), 2330-2341. Google Scholar
[26]	Z. Odibat and S. Momani, Modified homotopy perturbation method:application to quadratic Riccati differential equation of fractional order, Chaos, Soliton Fract., 36(2008), 167-174. Google Scholar
[27]	N. Shaheed and R.Said, Approximate solutions of nonlinear partial differential equations by modified-homotopy analysis method, J. Appl. Math., (2013). DOI:10.1155/2013/569674. Google Scholar
[28]	N. Saberi and S. Shateyi, Application of Optimal HAM for finding feedback control of optimal control problems, Math. Probl. Eng., (2013). DOI:10.1155/2013/914741. Google Scholar
[29]	N. Sweilam, M. Khader and A. Mahdy, Numerical studies for solving fractionalorder logistic equation, Int. J. Pure Appl. Math., 78(2012), 1199-1210. Google Scholar
[30]	V. Uchaikin, Fractional Derivatives for Physicits and Engineers, Higher Educatioin Press, Beijin, 2013. Google Scholar
[31]	Y. Wu and K. Cheung, Homotopy solution for nonlinear differential equations in wave propagation problems, Wave Motion, 46(2009), 1-14. Google Scholar
[32]	K. Yabushita, M. Yamashita and K. Tsuboi, An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, J. Phys. A:Math. Theor., 40(2007), 8403-8416. Google Scholar
[33]	L. Yuan, Q. Yang and C. Zeng, Chaos detection and parameter identification in fractional-order chaotic systems with delay, Nonlinear Dynam., 73(2013), 439-448. Google Scholar
[34]	X. Zhou, C. Yang, W. Gui and T. Dong, A particle swarm optimization algorithm with variable random functions and mutation, Acta Automatica Sinica, 40(2014), 1339-1347. Google Scholar