On the numerical simulation and convergence study for system of non-linear fractional dynamical model of marriage

Abstract

In this article, an implementation of an efficient numerical method for solving the system of coupled non-linear fractional (Caputo sense) dynamical model of marriage (FDMM) is introduced. The proposed system describes the dynamics of love affair between couples. The method is based on the spectral collocation method using Legendre polynomials. The proposed method reduces FDMM to a system of algebraic equations, which solved using Newton iteration method. Special attention is given to study the convergence analysis and deduce an error upper bound of the resulting approximate solution. Numerical simulation is given to show the validity and the accuracy of the proposed method.

Keywords

• Coupled non-linear fractional dynamical model of marriage,Caputo fractional derivatives,Legendre collocation method

References

1. Bell, W. W., Special Functions for Scientists and Engineers, Great Britain, Butler and Tanner Ltd, Frome and London, (1968).
2. Cherif, A., and Barley, K., Stochastic nonlinear dynamics of interpersonal and romantic relationships, Applied Mathematics and Computation, 217, 6273-6281, (2011).
3. Eid, A., and Khader, M. M., Numerical studies using FDM for viscous dissipation and thermal radiation efects on the slip flow and heat transfer due to a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity, New Trends in Mathematical Sciences, 4(1), 38-50, (2016).
4. Gottman, J. M., Murray, J. D, Swanson, C. C. Tyson, R., and Swanson, K. R., The Mathematics of Marriage, Cambridge, MA: MIT Press, (2002).
5. Jafari, H. and Momani, S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Physics Letter A, 370, 388-396, (2007).
6. Khader, M. M., On the numerical solutions for the fractional diffusion equation, Communications in Nonlinear Science and Numerical Simulation, 16, 2535-2542, (2011).
7. Khader, M. M., Numerical treatment for solving the perturbed fractional PDEs using hybrid techniques, Journal of Computational Physics, 250, 565-573, (2013).
8. Khader, M. M., Numerical treatment for solving fractional Riccati differential equation, Journal of the Egyptian Mathematical Society, 21, 32-37, (2013).

9. Khader, M. M., On the numerical solution and convergence study for system of non-linear fractional diffusion equations, Canadian Journal of Physics, 92(12), 1658-1666, (2014).
10. Khader, M. M., and Hendy, A. S., The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method, International Journal of Pure and Applied Mathematics, 74(3), 287-297, (2012).
11. Khader, M. M., and Sweilam, N. H., Numerical and analytical study for integro-differential equations using spectral collocation method, New Trends in Mathematical, 3(4), 144-153, (2015).
12. Khader, M. M., and Hendy, A. S., A numerical technique for solving fractional variational problems, Mathematical Methods in Applied Sciences, 36(10), 1281-1289, (2013).
13. Khader, M. M., EL-Danaf, T. S., and Hendy, A. S., A computational matrix method for solving systems of high order fractional differential equations, Applied Mathematical Modelling, 37, 4035-4050, (2013).
14. Khader, M. M., Sweilam, N. H., and Mahdy, A. M. S., Numerical study for the fractional differential equations generated by optimization problem using Chebyshev collocation method and FDM, Applied Mathematics and Information Science, 7(5), 2011-2018, (2013).
15. Martin, M. T. C., and Bumpass, B. L., Recent trends in marital disruption, Demography, 26(1), 37-51, (1989).
16. Miller, K. S., and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, NY, USA, (1993).
17. Oldham, K. B., and Spanier, J., The Fractional Calculus, Academic Press, New York, (1974).
18. Ozalp, N., and Koca, I., A fractional order nonlinear dynamical model of interpersonal relationships, Advances in Difference Equations, 189(1), 1-7, (2012).
19. Podlubny, I., Fractional Differential Equations, Academic Press, New York, (1999).
20. Srivastava, H. M., Kilbas, A. A., and Trujillo, J. J., Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, (2006).
21. Strogatz, S. H., Nonlinear Dynamics and Caos: With Applications in to Physics, Biology, Chemistry and Engineering, Addison-Wesley, Reading, MA, (1994).
22. Sweilam, N. H., and Khader, M. M., A Chebyshev pseudo-spectral method for solving fractional integro-differential equations, ANZIAM, 51, 464-475, (2010).
23. Sweilam, N. H., Khader, M. M. and Nagy, A. M., Numerical solution of two-sided space-fractional wave equation using finite difference method, Journal of Computional and Applied Mathematics, 235, 2832-2841, (2011).
24. Sweilam, N. H., Khader, M. M., and Mahdy, A. M. S., Numerical studies for fractional-order Logistic differential equation with two different delays, Journal of Applied Mathematics, 2012, Article ID 764894, 14 pages, (2012).