An Integral Collocation Approach Based on Legendre Polynomials for Solving Riccati, Logistic and Delay Differential Equations

M. M. Khader, A. M. S. Mahdy, M. M. Shehata

Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, KSA; Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt , Riyadh, KSA.
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

DOI: 10.4236/am.2014.515228                

Abstract

In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equations with variable coefficients. The properties of the Legendre polynomials are used to reduce the proposed problems to the solution of non-linear system of algebraic equations using Newton iteration method. We give numerical results to satisfy the accuracy and the applicability of the proposed schemes.

Keywords

Integral Collocation Formulation, Spectral Method, Riccati, Logistic and Delay Differential Equations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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
[2]	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
[3]	Sweilam, N.H., Khader, M.M. 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, 2013-2020.
[4]	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
[5]	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
[6]	Sweilam, N.H., Khader, M.M. and Adel, M. (2012) On the Stability Analysis of Weighted Average Finite Difference Methods for Fractional Wave Equation. Fractional Differential Calculus, 2, 17-75. http://dx.doi.org/10.7153/fdc-02-02
[7]	Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Order Integro-Differential Equations. ANZIAM, 51, 464-475. http://dx.doi.org/10.1017/S1446181110000830
[8]	Sweilam, N.H., Khader, M.M. and Al-Bar, R.F. (2008) Homotopy Perturbation Method for Linear and Nonlinear System of Fractional Integro-Differential Equations. International Journal of Computational Mathematics and Numerical Simulation, 1, 73-87.
[9]	Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) On the Numerical Solution for the Linear Fractional Klein-Gordon Equation Using Legendre Pseudospectral Method. International Journal of Mathematics and Computer Applications Research, 2, 1-10.
[10]	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
[11]	Khader, M.M., Mahdy, A.M.S. and Shehata, M.M. (2014) Approximate Analytical Solution to the Time-Fractional Biological Population Model Equation. Jokull, 64, 378-394.
[12]	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.
[13]	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, Article ID: 764894.
[14]	Reid, W.T. (1972) Riccati Differential Equations Mathematics in Science and Engineering. Academic Press, New York.
[15]	Lasiecka, I. and Triggiani, R. (1991) Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lecture Notes in Control and Information Sciences, Springer, Berlin.
[16]	Bahnasawi, A.A., El-Tawil, M.A. and Abdel-Naby, A. (2004) Solving Riccati Differential Equation Using ADM. Applied Mathematics and Computation, 157, 503-514. http://dx.doi.org/10.1016/j.amc.2003.08.049
[17]	Tan, Y. and Abbasbandy, S. (2008) Homotopy Analysis Method for Quadratic Riccati Differential Equation. Communications in Nonlinear Science and Numerical Simulation, 13, 539-546. http://dx.doi.org/10.1016/j.cnsns.2006.06.006
[18]	Cushing, J.M. (1998) An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics. http://dx.doi.org/10.1137/1.9781611970005
[19]	Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1996) An Introduction to Dynamical Systems. Springer, Berlin.
[20]	Ausloos, M. The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications XVI. 411, 2006.
[21]	Pastijn, H. (2006) Chaotic Growth with the Logistic Model of P.-F. Verhulst, Understanding Complex Systems. The Logistic Map and the Route to Chaos. 3-11.
[22]	Suansook, Y. and Paithoonwattanakij, K. (2009) Dynamic of Logistic Model at Fractional Order. IEEE International Symposium on Industrial Electronics.
[23]	Fridman, E., Fridman, L. and Shustin, E. (2000) Steady Modes in Relay Control Systems with Time Delay and Periodic Disturbances. Journal of Dynamic Systems, Measurement, and Control, 122, 732-737. http://dx.doi.org/10.1115/1.1320443
[24]	Davis, C.L. (2002) Modification of the Optimal Velocity Traffic Model to Include Delay due to Driver Reaction Time. Physica A, 319, 557-567. http://dx.doi.org/10.1016/S0378-4371(02)01457-7
[25]	Kuang, Y. (1993) Delay Differential Equations with Applications in Population Biology. Academic Press, Boston, San Diego, New York.
[26]	Epstein, I. and Luo, Y. (1991) Differential Delay Equations in Chemical Kinetics: Nonlinear Models: The Cross-Shaped Phase Diagram and the Originator. Journal of Chemical Physics, 95, 244-254. http://dx.doi.org/10.1063/1.461481
[27]	Mai-Duy, N., See, H. and Tran-Cong, T. (2009) A Spectral Collocation Technique Based on Integrated Chebyshev Polynomials for Biharmonic Problems in Irregular Domains. Applied Mathematical Modelling, 33, 284-299. http://dx.doi.org/10.1016/j.apm.2007.11.002
[28]	Bhrawy, A.H. and Alofi, A.S. (2013) The Operational Matrix of Fractional Integration for Shifted Chebyshev Polynomials. Applied Mathematics Letters, 26, 26-31. http://dx.doi.org/10.1016/j.aml.2012.01.027
[29]	Khader, M.M., Sweilam, N.H. and Mahdy, A.M.S. (2011) An Efficient Numerical Method for Solving the Fractional Diffusion Equation. Journal of Applied Mathematics and Bioinformatics, 1, 1-12.
[30]	Bell, W.W. (1968) Special Functions for Scientists and Engineers. Great Britain, Butler and Tanner Ltd, Frome and London.
[31]	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