The fractional-order differential model of the pollution for a system of lakes
محورهای موضوعی : Numerical Analysisbijan hasani lichae 1 , mehran nemati 2
1 - Department of Mathematics, Fouman Branch, Islamic Azad University, Fouman, Iran
2 - Department of Mathematics, roudbar Branch, Islamic Azad University, roudbar, Iran
کلید واژه: Numerical solution, Laplace Adomian decomposition method, System of fractional-order differential equations of the pollution, Caputo fractional derivative,
چکیده مقاله :
Pollution produced by human is a serious danger to the planet Earth in our time. In recent decades, a lot of efforts have been made to monitor and control the pollution to save the environment. In this paper, the fractional-order differential model of the pollution for a system of lakes has been introduced. There are three components; the amount of the pollution in lake 1, , the amount of the pollution in lake 2, , and the amount of the pollution in lake 3, , at any time . The aim of this work is to get numerical solution of the proposed fractional-order model by Laplace Adomian decomposition method (LADM). The numerical solution has been obtained in a series form. The solution has been compared with the solutions of some other numerical approaches. The results illustrate the ability and accuracy of the present method. The Caputo form has been applied for fractional derivatives. All of computations have been done in Maple.
Pollution produced by human is a serious danger to the planet Earth in our time. In recent decades, a lot of efforts have been made to monitor and control the pollution to save the environment. In this paper, the fractional-order differential model of the pollution for a system of lakes has been introduced. There are three components; the amount of the pollution in lake 1, , the amount of the pollution in lake 2, , and the amount of the pollution in lake 3, , at any time . The aim of this work is to get numerical solution of the proposed fractional-order model by Laplace Adomian decomposition method (LADM). The numerical solution has been obtained in a series form. The solution has been compared with the solutions of some other numerical approaches. The results illustrate the ability and accuracy of the present method. The Caputo form has been applied for fractional derivatives. All of computations have been done in Maple.
1. Culshaw, R.V. and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells. Mathematical biosciences, 2000. 165(1): p. 27-39.
2. Arafa, A., S. Rida, and M. Khalil, A fractional-order model of HIV infection with drug therapy effect. Journal of the Egyptian Mathematical Society, 2014. 22(3): p. 538-543.
3. Hindmarsh, J.L. and R. Rose, A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B, 1984. 221(1222): p. 87-102.
4. Chen, T., H.L. He, and G.M. Church, Modeling gene expression with differential equations, in Biocomputing'99. 1999, World Scientific. p. 29-40.
5. Lichae, B.H., J. Biazar, and Z. Ayati, The Fractional Differential Model of HIV-1 Infection of CD4+ T-Cells with Description of the Effect of Antiviral Drug Treatment. Computational and mathematical methods in medicine, 2019. 2019.
6. Lichae, B.H., J. Biazar, and Z. Ayati, A class of Runge–Kutta methods for nonlinear Volterra integral equations of the second kind with singular kernels. Advances in Difference Equations, 2018. 2018(1): p. 349.
7. Hasani Lichae, B., J. Biazar, and Z. Ayati, Asymptotic decomposition method for fractional order Riccati differential equation. Computational Methods for Differential Equations, 2021. 9(1): p. 63-78.
8. Hoggard, J., Lake Pollution Modeling, Virginia Tech, 2007.
9. Biazar, J., L. Farrokhi, and M.R. Islam, Modeling the pollution of a system of lakes. Applied Mathematics and Computation, 2006. 178(2): p. 423-430.
10. Biazar, J., M. Shahbala, and H. Ebrahimi, VIM for solving the pollution problem of a system of lakes. Journal of Control Science and Engineering, 2010. 2010: p. 8.
11. Merdan, M., A new application of modified differential transformation method for modelling the pollution of a system of lakes. Selçuk Journal of Applied Mathematics, 2010. 11(2): p. 27-40.
12. Yüzbaşı, Ş., N. Şahin, and M. Sezer, A collocation approach to solving the model of pollution for a system of lakes. Mathematical and Computer Modelling, 2012. 55(3-4): p. 330-341.
13. Khalid, M., et al., Solving polluted lakes system by using perturbation-iteration method. International Journal of Computer Applications, 2015. 114(4).
14. Tatom, F.B., The relationship between fractional calculus and fractals. Fractals, 1995. 3(01): p. 217-229.
15. Heymans, N. and J.-C. Bauwens, Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheologica Acta, 1994. 33(3): p. 210-219.
16. Giona, M. and H.E. Roman, Fractional diffusion equation on fractals: One-dimensional case and asymptotic behaviour. Journal of physics A: Mathematical and general, 1992. 25(8): p. 2093.
17. Arafa, A., S. Rida, and M. Khalil, The effect of anti-viral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model. Applied Mathematical Modelling, 2013. 37(4): p. 2189-2196.
18. Haq, F., et al., Numerical solution of fractional order smoking model via laplace Adomian decomposition method. Alexandria Engineering Journal, 2017.
19. Ertürk, V.S., Z.M. Odibat, and S. Momani, An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of T-cells. Computers & Mathematics with applications, 2011. 62(3): p. 996-1002.
20. Diethelm, K., The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. 2010: Springer.
21. Garrappa, R., Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics, 2018. 6(2): p. 16.
22. Marinca, V. and N. Herisanu, Optimal Homotopy Asymptotic Method, in The Optimal Homotopy Asymptotic Method. 2015, Springer. p. 9-22.
23. Marinca, V. and N. Herişanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. International Communications in Heat and Mass Transfer, 2008. 35(6): p. 710-715.
24. Marinca, V. and N. Herişanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method. Journal of Sound and Vibration, 2010. 329(9): p. 1450-1459.
25. Marinca, V., N. Herişanu, and I. Nemeş, Optimal homotopy asymptotic method with application to thin film flow. Central European Journal of Physics, 2008. 6(3): p. 648.
26. Diethelm, K., N.J. Ford, and A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 2002. 29(1-4): p. 3-22.
27. Ghoreishi, M., A.M. Ismail, and A. Alomari, Application of the homotopy analysis method for solving a model for HIV infection of CD4+ T-cells. Mathematical and Computer Modelling, 2011. 54(11): p. 3007-3015.
28. Liao, S.J., The proposed homotopy analysis technique for the solution of nonlinear problems, 1992, Ph. D. Thesis, Shanghai Jiao Tong University.
29. Ongun, M.Y., The Laplace adomian decomposition method for solving a model for HIV infection of CD4+ T cells. Mathematical and Computer Modelling, 2011. 53(5): p. 597-603.
30. Merdan, M. and T. Khan, Homotopy perturbation method for solving viral dynamical model. CÜ Fen-Edebiyat Fakültesi, Fen Bilimleri Dergisi, 2010. 31: p. 65-77.
31. Odibat, Z. and S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos, Solitons & Fractals, 2008. 36(1): p. 167-174.
32. Hong, J., C. Huang, and X. Wang, Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths. arXiv preprint arXiv:1704.04144, 2017.
33. Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo, Preface, 2006, Elsevier.
34. Aguirre, J. and D. Tully, Lake pollution model, 1999, May.
35. Biazar, J., et al., An alternate algorithm for computing Adomian polynomials in special cases. Applied Mathematics and Computation, 2003. 138(2-3): p. 523-529.
36. Ayati, Z. and J. Biazar, On the convergence of Homotopy perturbation method. Journal of the Egyptian Mathematical Society, 2015. 23(2): p. 424-428.
