Semi-analytical ‎M‎ethod to Solve the Non-linear System of Equations to Model of Evolution for Smoking Habit in ‎Spain

Subject Areas : International Journal of Industrial Mathematics

S. Noeiaghdam ¹ , K. Kamal Ali ²

1 - Baikal School of BRICS, Irkutsk National Research Technical University, Irkutsk, Russian Federation.
2 - Department of Mathematics, Faculty of Science, University of Zakho, Iraq.

Received: 2019-10-16 Accepted : 2020-09-23 Published : 2020-08-01

Keywords:

References:

[1] S. Abbasbandy, E. Shivanian, K. Vajravelu, Mathematical properties of h-curve in the frame work of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 4268-4275.
[2] S. Abbasbandy, E. Magyari, E. Shivanian, The homotopy analysis method for multiple
solutions of nonlinear boundary value problems, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 3530-3536.
[3] S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions of nonlinear boundary
value problems: novel application of homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3830-3846.
[4] S. Abbasbandy, E. Shivanian, Predictor homotopy analysis method and its application
tions in Nonlinear Science and Numerical Simulation 16 (2011) 2456-2468.
[5] S. Abbasbandy, E. Shivanian, A new analytical technique to solve Fredholm's integral equations, Numerical algorithms 56 (2011) 27-43.
[6] L. Ahmad Soltani, E. Shivanian, R. Ezzati, Convection-radiation heat transfer in solar heat exchangers lled with a porous medium: Exact and shooting homotopy analysis solution, Applied Thermal Engineering 103 (2016) 537-542.
[7] T. Allahviranloo, Z. Gouyandeh, A. Armand, Numerical solutions for fractional dierential equations by Tau-Collocation method, Applied Mathematics and Computation 271 (2015) 979-990.
[8] J. Biazar, M. A. Asadi, F. Salehi, Rational Homotopy Perturbation Method for solving sti systems of ordinary dierential equations, Applied Mathematical Modelling 39 (2015) 1291-1299.
[9] L. Bougoa, R. C. Rach, Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial dierential equations by the Adomian decomposition method, Applied Mathematics and Computation 225 (2013) 50-61.
[10] M. A. Fariborzi Araghi, S. Noeiaghdam, A novel technique based on the homotopy analysis
method to solve the rst kind Cauchy integral equations arising in the theory of airfoils, Journal of Interpolation and Approximation in Scientic Computing 2016 (2016) 1-13.
[11] M. A. Fariborzi Araghi, S. Noeiaghdam, Homotopy analysis transform method for solving generalized Abel's fuzzy integral equations of the rst kind, IEEE (2016). http://dx.doi.org/10.1109/CFIS.2015.7391645/
[12] M.A. Fariborzi Araghi, S. Noeiaghdam, Homotopy regularization method to solve the singular Volterra integral equations of the rst kind, Jordan Journal of Mathematics and Statistics (JJMS) 11 (2018) 1-12.
[13] M.A. Fariborzi Araghi, S. Noeighdam, Dynamical control of computations using the Gauss-Laguerre integration rule by applying the CADNA library, Advances and Applications in Mathematical Sciences 16 (2016) 1-18.
[14] M.A. Fariborzi Araghi, S. Noeiaghdam, A Valid Scheme to Evaluate Fuzzy Denite Integrals
by Applying the CADNA Library, International Journal of Fuzzy System Applications 6 (2017) 1-20. http://dx.doi.org/10.4018/IJFSA.2017100101/

[15] Sh. Gao, F. Zhang, Y. He, The eects of migratory bird population in a nonautonomous eco-epidemiological model, Applied Mathematical Modelling 37 (2013) 3903-3916.
[16] F. Geng, A modied variational iteration method for solving Riccati dierential equations,
Computers & Mathematics with Applications 60 (2010) 1868-1872.
[17] F. Geng, Y. Lin, M. Cui, A piecewise variational iteration method for Riccati dierential equations Open archive, Computers & Mathematics with Applications 58 (2009) 2518-2522.
[18] M. Ghoreishi, A.I.B.Md. Ismail, A.K. Alomari, Application of the homotopy analysis method for solving a model for HIV infection of CD4+ T-cells, Mathematical and Computer Modelling 54 (2011) 3007-3015.
[19] F. Guerrero, F.J. Santonja, R.J. Villanueva, Solving a model for the evolution of smoking
habit in Spain with homotopy analysis method, Nonlinear Analysis: Real World Applications 14 (2013) 549-558.
[20] Y. Hu, Y. Luo, Z. Lu, Analytical solution of the linear fractional dierential equation by Adomian decomposition method, Journal of Computational and Applied Mathematics 215 (2008) 220-229.
[21] K. Kamal Ali, H. Dutta, R. Yilmazer, S. Noeiaghdam, On the new wave behaviors of the Gilson-Pickering equation, Front. Phys. 8 (2012) 54-67.
[22] M. Khan, M.A. Gondal, I. Hussain, S. Karimi Vanani, A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi innite domain, Mathematical and Computer Modelling 55 (2012) 1143-1150.
[23] S. Kumar, D. Kumar, Fractional modelling for BBM-Burger equation by using new homotopy
analysis transform method, Journal of the Association of Arab Universities for Basic and Applied Sciences 16 (2014) 16-20.
[24] J. D. Lambert, Numerical Methods for Ordinary Dierential Systems: The Initial Value Problem, John Wiley and Sons, Chichester, UK, 1991.
[25] S. J. Liao, The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, (1992) (in English).
[26] S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, (2003).
[27] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499-513.
[28] S.J. Liao, Homotopy analysis method in nonlinear dierential equations, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2012.
[29] K. Mallory, R. A. Van Gorder, Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation, Numerical Algorithms 64 (2013) 633-657.
[30] M. Mirzazadeh, Z. Ayati, New homotopy perturbation method for system of Burgers equations, Alexandria Engineering Journal 55 (2016) 1619-1624.
[31] P. M. Ngina, R. Waema Mbogo, L. S. Luboobi, Mathematical Modelling of In-Vivo Dynamics of HIV Subject to the In uence of the CD8+ T-Cells, Applied Mathematics 8 (2017) 1153-1179.
[32] S. Noeiaghdam, A novel technique to solve the modied epidemiological model of computer viruses, SeMA Journal (2018). https://doi.org/10.1007/s40324-018-0163-3.
[33] S. Noeiaghdam, Numerical approximation of modied non-linear SIR model of computer
viruses, Contemporary Mathematics 1 (2019). http://dx.doi.org/10.37256/cm. 11201959.34-48/
[34] S. Noeiaghdam, E. Khoshrouye Ghiasi, Solving a non-linear model of HIV infection for CD4+T cells by combining Laplace transformation and Homotopy analysis method, arXiv:1809.06232.
[35] S. Noeiaghdam, E. Khoshrouye Ghiasi, An ecient method to solve the mathematical model of HIV infection for CD8+ T-cells. International Journal of Mathematical Modelling & Computations 9 (2019) 267-281.
[36] S. Noeiaghdam, M.A. Fariborzi Araghi, S. Abbasbandy, Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the  stochastic arithmetic, Numer Algor 81 (2019) 237-267. http://dx.doi.org/10. 1007/s11075-018-0546-7/
[37] S. Noeiaghdam, M.A. Fariborzi Araghi, Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method, Journal of Fuzzy Set Valued Analysis, 2017 (2017) 62-85. http://dx.doi.org/ 10.5899/2017/jfsva-00383/
[38] S. Noeiaghdam, D. Sidorov, I. Muftahov, A.V. Zhukov, Control of Accuracy on Taylor Collocation Method for Load Leveling Problem, The Bulletin of Irkutsk State University. Series Mathematics 30 (2019) 59-72. http://dx.doi.org/10.26516/1997-7670.2019.30.59/
[39] S. Noeiaghdam, D. Sidorov, V. Sizikov, N. Sidorov, Control of accuracy on Taylorcollocation
method to solve the weakly regular Volterra integral equations of the rst kind by using the CESTAC method, Applied and Computational Mathematics 19 (2020) 87-105.
[40] S. Noeiaghdam, M. Suleman, H. Budak, Solving a modied nonlinear epidemiological
model of computer viruses by homotopy analysis method, Mathematical Sciences 6 (2018) 1-12.
[41] S. Noeiaghdam, E. Zarei, H. Barzegar Kelishami, Homotopy analysis transform method for solving Abel's integral equations of the rst kind, Ain Shams Eng. J. 7 (2016) 483-495.
[42] M. Y. Ongun, The Laplace Adomian Decomposition Method for solving a model for HIV
infection of CD4+T cells, Mathematical and Computer Modelling 53 (2011) 597-603.
[43] Y. Ozturk, M. Gulsu, Numerical solution of a modied epidemiological model for computer
viruses, Applied Mathematical Modelling 39 (2015) 7600-7610.
[44] L. Pezza, F. Pitolli, A multiscale collocation method for fractional dierential problems, Mathematics and Computers in Simulation 147 (2018) 210-219.
[45] J.R. C. Piqueira, V. O. Araujo, A modied epidemiological model for computer viruses,
Applied Mathematics and Computation 213 (2009) 355-360.
[46] E. Shivanian, S. Abbasbandy, Predictor homotopy analysis method: two points second
order boundary value problems, Nonlinear Analysis: Real World Applications 15 (2014) 89-99.
[47] W. Sikander, U. Khan, N. Ahmed, S.T. Mohyud-Din, Optimal solutions for a bio mathematical model for the evolution of smoking habit, Results in Physics 7 (2017) 510-517.
[48] J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for
computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation 316 (2018) 504-515.
[49] Y. Tan, S. Abbasbandy, Homotopy analysis method for quadratic Riccati dierential equation, Communications in Nonlinear Science and Numerical Simulation 13 (2008) 539-546.
[50] H. Thabet, S. Kendre, Analytical solutions for conformable space-time fractional partial
dierential equations via fractional dierential transform, Chaos, Solitons & Fractals 109 (2018) 238-245.
[51] E.  Unal, A. Gokdogan, Solution of conformable fractional ordinary dierential equations via dierential transform method, Optik-International Journal for Light and Electron Optics 128 (2017) 264-273.
[52] R. A. Van Gorder, The variational iteration method is a special case of the homotopy analysis method, Applied Mathematics Letters 45 (2015) 81-85.
[53] H. Vazquez-Leal, F. Guerrero, Application of series method with Pade and Laplace-Pade resummation methods to solve a model for the evolution of smoking habit in Spain, Comp. Appl. Math. 33 (2014) 181-192.
[54] H. Xu, S.J. Liao, X.C. You, Analysis of nonlinear fractional partial dierential equations with the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1152-1156.
[55] L. Yakob, Endectocide-treated cattle for malaria control: A coupled  ntomologicalepidemiological model, Parasite Epidemiology and Control 1 (2016) 2-9.
[56] S. Yuzbasi, A numerical approach to solve the model for HIV infection of CD4+T cells, Applied Mathematical Modelling 36 (2012) 5876-5890.