Geodesic
Approximate solution for system of fractional non-linear dynamical marriage model using Bernstein polynomials
Khader, Mohamed M.1 ; Alqahtani, Rubayyi T.2
1 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh: 11566, Saudi Arabia;Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
2 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh: 11566, Saudi Arabia
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 3, p. 865-873.

Voir la notice de l'article provenant de la source International Scientific Research Publications

This paper is devoted to present the approximate solutions with helping of an efficient numerical method for the nonlinear coupled system of dynamical marriage model in the fractional of Riemann-Liouville sense (FDMM). The proposed system describes the dynamics of love affair between a couple. The proposed method is dependent on the use of useful properties of the operational matrices of Bernstein polynomials. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FDMM to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical simulation is given to show the validity and the accuracy of the proposed algorithm. We introduce a comparison with the obtained solution using Runge-Kutta method.
DOI : 10.22436/jnsa.010.03.02
Classification : 41A04, 65N20
Keywords: Fractional dynamical model of marriage, Riemann-Liouville fractional derivatives, operational matrix, Bernstein polynomials.

Khader, Mohamed M. 1 ; Alqahtani, Rubayyi T. 2

1 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh: 11566, Saudi Arabia;Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
2 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh: 11566, Saudi Arabia 
@article{JNSA_2017_10_3_a1,
     author = {Khader, Mohamed M. and Alqahtani, Rubayyi T.},
     title = {Approximate solution for system of fractional non-linear dynamical marriage model using {Bernstein} polynomials},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {865-873},
     publisher = {mathdoc},
     volume = {10},
     number = {3},
     year = {2017},
     doi = {10.22436/jnsa.010.03.02},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.03.02/}
}
TY  - JOUR
AU  - Khader, Mohamed M.
AU  - Alqahtani, Rubayyi T.
TI  - Approximate solution for system of fractional non-linear dynamical marriage model using Bernstein polynomials
JO  - Journal of nonlinear sciences and its applications
PY  - 2017
SP  - 865
EP  - 873
VL  - 10
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.03.02/
DO  - 10.22436/jnsa.010.03.02
LA  - en
ID  - JNSA_2017_10_3_a1
ER  - 
%0 Journal Article
%A Khader, Mohamed M.
%A Alqahtani, Rubayyi T.
%T Approximate solution for system of fractional non-linear dynamical marriage model using Bernstein polynomials
%J Journal of nonlinear sciences and its applications
%D 2017
%P 865-873
%V 10
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.03.02/
%R 10.22436/jnsa.010.03.02
%G en
%F JNSA_2017_10_3_a1
Khader, Mohamed M.; Alqahtani, Rubayyi T. Approximate solution for system of fractional non-linear dynamical marriage model using Bernstein polynomials. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 3, p. 865-873. doi : 10.22436/jnsa.010.03.02. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.03.02/

[1] Alipour, M.; Rostamy, D. Bernstein polynomials for solving Abel’s integral equation, J. Math. Comput. Sci., Volume 3 (2011), pp. 403-412

[2] Alipour, M.; Rostamy, D.; Baleanu, D. Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices, J. Vib. Control, Volume 19 (2013), pp. 2523-2540 | Zbl | DOI

[3] Barley, K.; Cherif, A. Stochastic nonlinear dynamics of interpersonal and romantic relationships , Appl. Math. Comput., Volume 217 (2011), pp. 6273-6281 | DOI

[4] Cheney, E. W. Introduction to approximation theory, Reprint of the second (1982) edition, AMS Chelsea Publishing, Providence, RI, 1998

[5] Gottman, J. M.; Murray, J. D.; Swanson, C. C.; Tyson, R.; Swanson, K. R. The mathematics of marriage: dynamic nonlinear models, A Bradford Book, MIT Press, Cambridge, MA, 2002 | Zbl

[6] Khader, M. M. On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., Volume 16 (2011), pp. 2535-2542 | DOI

[7] Khader, M. M. Numerical treatment for solving fractional Riccati differential equation, J. Egyptian Math. Soc., Volume 21 (2013), pp. 32-37 | Zbl | DOI

[8] Khader, M. M. Numerical treatment for solving the perturbed fractional PDEs using hybrid techniques, J. Comput. Phys., Volume 250 (2013), pp. 565-573 | Zbl | DOI

[9] Khader, M. M.; Adel, M. H. Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula, Adv. Difference Equ., Volume 2016 (2016 ), pp. 1-10 | DOI

[10] Khader, M. M.; Danaf, T. S. EL; Hendy, A. S. A computational matrix method for solving systems of high order fractional differential equations, Appl. Math. Model., Volume 37 (2013), pp. 4035-4050 | DOI

[11] Khader, M. M.; Hendy, A. S. A numerical technique for solving fractional variational problems, Math. Methods Appl. Sci., Volume 36 (2013), pp. 1281-1289 | DOI

[12] Khader, M. M.; Sweilam, N. H. Singularly perturbed BVP to estimation of diaphragm deflection in MEMS capacitive microphone: an application of ADM, Appl. Math. Comput., Volume 281 (2016), pp. 214-222 | DOI

[13] Khader, M. M.; Sweilam, N. H.; Mahdy, A. M. S. Numerical study for the fractional differential equations generated by optimization problem using Chebyshev collocation method and FDM, Appl. Math. Inf. Sci., Volume 7 (2013), pp. 2011-2018 | DOI

[14] Kreyszig, E. Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978

[15] Martin, T. C.; Bumpass, L. L. Recent trends in marital disruption, Demography, Volume 26 (1989), pp. 37-51 | DOI

[16] Oldham, K. B.; Spanier, J. The fractional calculus, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974

[17] Ozalp, N.; Koca, I. A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Difference Equ., Volume 2012 (2012 ), pp. 1-7 | Zbl | DOI

[18] Podlubny, I. Fractional differential equations , An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999

[19] Rostamy, D.; Alipour, M.; Jafari, H.; Baleanu, D. Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis, Rom. Rep. Phys., Volume 65 (2013), pp. 334-349

[20] Rostamy, D.; Karimi, K. Bernstein polynomials for solving fractional heat- and wave-like equations, Fract. Calc. Appl. Anal., Volume 15 (2012), pp. 556-571 | DOI | Zbl

[21] Strogatz, S. H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Addison- Wesley, Reading, MA, 1994

[22] Sweilam, N. H.; Khader, M. M. A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations, ANZIAM J., Volume 51 (2010), pp. 464-475 | DOI | Zbl

[23] Sweilam, N. H.; Khader, M. M.; Mahdy, A. M. S. Numerical studies for solving fractional-order Logistic equation, Int. J. Pure Appl. Math., Volume 78 (2012), pp. 1199-1210 | DOI | Zbl

[24] Sweilam, N. H.; Khader, M. M.; Nagy, A. M. Numerical solution of two-sided space-fractional wave equation using finite difference method, J. Comput. Appl. Math., Volume 235 (2011), pp. 2832-2841 | Zbl | DOI

Cité par Sources :