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

Author(s)
R. F. Al-Bar

Affiliation(s)

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

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

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.

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

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

Cite this paper

Al-Bar, R. (2015) On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials.*Applied Mathematics*, **6**, 2096-2103. doi: 10.4236/am.2015.612184.

Al-Bar, R. (2015) On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials.

References

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[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.

[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.