On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials

R. F. Al-Bar

Department of Mathematics, Faculty of Science, Umm Al-Qura University, Makkah, Saudi Arabia.

DOI: 10.4236/am.2015.612184                

Abstract

In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.

Keywords

Fractional Logistic Equation, Riemann-Liouville Fractional Derivatives, Riemann-Liouville Fractional Integral, Operational Matrix, Bernstein Polynomials

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.
[2]	Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.
[3]	Burden, R.L. and Faires, J.D. (1993) Numerical Analysis. PWS, Boston.
[4]	Sweilam, N.H., Khader, M.M. and Nagy, A.M. (2011) Numerical Solution of Two-Sided Space-Fractional Wave Equation Using Finite Difference Method. Journal of Computional and Applied Mathematics, 235, 2832-2841. http://dx.doi.org/10.1016/j.cam.2010.12.002
[5]	Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Solving Fractional-Order Logistic Equation. International Journal of Pure and Applied Mathematics, 78, 1199-1210.
[6]	Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Mathematics, 2012, 1-14. http://dx.doi.org/10.1155/2012/764894
[7]	Sweilam, N.H., Khader, M.M. and Al-Bar, R.F. (2007) Numerical Studies for a Multi-Order Fractional Differential Equation. Physics Letters A, 371, 26-33. http://dx.doi.org/10.1016/j.physleta.2007.06.016
[8]	Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 16, 2535-2542. http://dx.doi.org/10.1016/j.cnsns.2010.09.007
[9]	Khader, M.M. (2013) Numerical Treatment for Solving the Perturbed Fractional PDEs Using Hybrid Techniques. Journal of Computational Physics, 250, 565-573. http://dx.doi.org/10.1016/j.jcp.2013.05.032
[10]	Khader, M.M. and Hendy, A.S. (2013) A Numerical Technique for Solving Fractional Variational Problems. Mathematical Methods in Applied Sciences, 36, 1281-1289. http://dx.doi.org/10.1002/mma.2681
[11]	Khader, M.M. and Babatin, M.M. (2013) On Approximate Solutions for Fractional Logistic Differential Equation. Mathematical Problems in Engineering, 2013, Article ID: 391901. http://dx.doi.org/10.1155/2013/391901
[12]	Khader, M.M., EL Danaf, T.S. and Hendy, A.S. (2013) A Computational Matrix Method for Solving Systems of High Order Fractional Differential Equations. Applied Mathematical Modelling, 37, 4035-4050. http://dx.doi.org/10.1016/j.apm.2012.08.009
[13]	Khader, M.M., Sweilam, N.H. and Mahdy, A.M.S. (2013) Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM. Applied Mathematics and Information Science, 7, 2011-2018. http://dx.doi.org/10.12785/amis/070541
[14]	Cushing, J.M. (1998) An Introduction to Structured Population Dynamics. Society for Industrial and Applied Mathematics, Philadelphia. http://dx.doi.org/10.1137/1.9781611970005
[15]	Pastijn, H. (2006) Chaotic Growth with the Logistic Model of P.-F. Verhulst. In: Ausloos, M. and Dirickx, M., Eds., The Logistic Map and the Route to Chaos, Understanding Complex Systems, Springer, Berlin, 3-11. http://dx.doi.org/10.1007/3-540-32023-7_1
[16]	Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1996) Chaos: An Introduction to Dynamical Systems. Springer, New York.
[17]	Ausloos, M. (2006) The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications XVI. 411 p.
[18]	Suansook, Y. and Paithoonwattanakij, K. (2009) Dynamic of Logistic Model at Fractional Order. IEEE International Symposium on Industrial Electronics, Seoul, 5-8 July 2009, 718-723. http://dx.doi.org/10.1109/isie.2009.5219765
[19]	El-Sayed, A.M.A., El-Mesiry, A.E.M. and El-Saka, H.A.A. (2007) On the Fractional-Order Logistic Equation. Applied Mathematics Letters, 20, 817-823. http://dx.doi.org/10.1016/j.aml.2006.08.013
[20]	El-Sayed, A.M.A., Gaafar, F.M. and Hashem, H.H. (2004) On the Maximal and Minimal Solutions of Arbitrary Orders Nonlinear Functional Integral and Differential Equations. Mathematical Sciences Research Journal, 8, 336-348.
[21]	Khader, M.M. and Hendy, A.S. (2012) The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudospectral Method. International Journal of Pure and Applied Mathematics, 74, 287-297.
[22]	Cheney, E.W. (1982) Introduction to Approximation Theory. 2nd Edition, AMS Chelsea Publishing, Providence.
[23]	Alipour, M., Rostamy, D. and Baleanu, D. (2013) Solving Multidimensional FOCPs with Inequality Constraint by BPs Operational Matrices. Journal of Vibration and Control, 19, 2523-2540. http://dx.doi.org/10.1177/1077546312458308
[24]	Kreyszig, E. (1978) Introduction Functional Analysis with Applications. John Wiley & Sons, New York.
[25]	Rostamy, D. and Karimi, K. (2012) Bernstein Polynomials for Solving Fractional Heat- and Wave-Like Equations. Fractional Calculus and Applied Analysis, 15, 556-571. http://dx.doi.org/10.2478/s13540-012-0039-7
[26]	Rostamy, D., Alipour, M., Jafari, H. and Baleanu, D. (2013) Solving Multi-Term Orders Fractional Differential Equations by Operational Matrices of BPs with Convergence Analysis. Romanian Reports in Physics, 65, 334-349.
[27]	Alipour, M. and Rostamy, D. (2011) Bernstein Polynomials for Solving Abel’s Integral Equation. The Journal of Mathematics and Computer Science, 3, 403-412.