Hybridization of Genetic Algorithm with Homotopy Analysis Method for Solving Fractional Partial Differential Equations

Authors: Ahmed Entesar¹ & Omar Saber² & Waleed Al-Hayani^3
1&2&3Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Iraq

Abstract:  In this work, the Homotopy Analysis Method (HAM) is applied to solve fractional Partial Differential Equations (PDEs). The solution of HAM has improved the results by using Genetic Algorithm (GA). The hybrid method (proposed) is applied for types of problems where analytical solutions approximate are obtained. Numerical experiments are also presented.

Keywords: Homotopy Analysis Method (HAM), Genetic Algorithm (GA), Heat Like Equations, Wave Like Equations, Fractional Calculus

Download the PDF Document from here.

doi: 10.23918/eajse.v4i2p19

References

Agrawal, O. (2002). Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynam, 29, 145-155.

Al-Hayani, W. (2017). Daftardar-Jafari method for fractional heat-like and wave-like equations with variable coefficients. Applied Mathematics, 8, 215-228.

Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent. Part II. J. Roy. Astral. Soc. 13, 529-539.

Cheng,W.,  Shi, H.,  Xin, X.,  Li, D. (2011). An elitism strategy based genetic algorithm for streaming pattern discovery in wireless sensor networks. Commun. Lett. IEEE, 15, 419–421.

Dass, M. V., Ali, M. R., & Ali, M. M. (2014). Image Retrieval Using Interactive Genetic Algorithm, IEEE, International Conference on Computational Science and Computational Intelligence.

Duld, A. (1997). Lecture on algebraic topology. Berlin: Springer, Berlin.

Fujita, Y. (1990). Cauchy problems of fractional order and stable processes. Japan J. Appl. Math. 7 (3), 459-476.

Gorenflo, R., & Mainardi, F. (1997). Fractional calculus: Integral and differential equations of fractional order. In A. Carpinteri, F. Mainardi (Eds.), Fractals and fractional calculus. New York.

Hilfer, R. (1995). Foundations of fractional dynamics. Fractals, 3 (3), 549-556.

Hilfer, R. (2000). Fractional diffusion based on Riemann-Liouville fractional derivative. J. Phys. Chem. 104, 3914-3917.

Kilbas, A., Srivastava, H., & Trujillo, J. (2006). Theory and applications of fractional differential equations. Elsevier, Holland.

Klafter, J., Blumen, A., & Shlesinger, M. (1984). Fractal behavior in trapping and reaction: A random walk study. J. Stat. Phys, 36, 561-578.

Li, X., & Yin, M. (2014). Parameter estimation for chaotic systems by hybrid differential evolution algorithm and artificial bee colony algorithm. Nonlinear Dynamics, 77(1-2) 61-71.

Liao, S. J. (1992).  The proposed homotopy analysis technique for the solution of nonlinear problems, Doctoral dissertation, Shanghai Jiao Tong University.

Liao, S. J. (1997). An approximate solution technique not depending on small parameters Part 2: An application in fluid mechanics. Int. J. Nonlinear Mech. 32(5), 815-822.

Liao, S. J. (1999).  An explicit, totally analytic approximation of Blasius’ viscous flow problems. International Journal of Nonlinear Mechanics, 34 (4), 759-778.

Luchko, A.Y., & Groreflo, Y. (1998). The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98, fachbreichmathematik und informatik, FreicUniversitat Berlin.

Mainardi, F. (1997). Fractional calculus: Some basic problems in continuum and statistical mechanics. In A. Carpinteri, F. Mainardi (Eds.), Fractal and fractional calculus in continuum mechanics. Springer-Verlag, New York.

Metzler, R., & Klafter, J. (2000). Boundary value problems fractional diffusion equations. Physica A, 278, 107-125.

Miller, K., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. New York: John Wiley and Sons, Inc.

Molliq, Y.,  Noorani, M., & Hashim, I. (2009). Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal.: Real World Appl. 10, 1854-1869.

Momani, S. (2005). Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl. Math. Comput. 165, 459-472.

Oldham, K., & Spanier, J. (1974). The fractional calculus. New York:  Academic Press.

Podlubny, I. (1999). Fractional differential equations. New York: Academic Press.

Zang,W. K.,  Sun, M. H.,  Jiang, Z. N. (2016). A DNA genetic algorithm inspired by biological membrane structure. J. Comput. Theor. Nanosci, 13, 3763–3772.