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.
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
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.
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
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
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.
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.
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
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.
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.
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
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
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
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
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