HE’S VARIATIONAL ITERATION METHOD FOR SOLVING MODELLING THE POLLUTION OF A SYSTEM OF LAKES

Öz

In this papers,He’s variational iteration method(VIM) is implemented to for solving analytically systems of nonlinear ordinary differential equations such as modelling the pollution of a system of lakes. The proposed scheme is based on variational iteration method (VIM), Laplace transform and Padé approximants. The results to get the variational iteration method (VIM) are applied Padé approximants. Our proposed approach showed results to analytical solutions of nonlinear ordinary differential equation systems. The results are compared with the results obtained by MATLAB ode15s and the variational iteration method (VIM) are applied Padé approximants. Some plots are presented to show the reliability and simplicity of the methods.

Anahtar Kelimeler

• Padé approximants,Variational iteration method,Modelling the pollution of a system of lakes

GÖLLER SİSTEMİNİN KİRLİLİK MODELİNİN ÇÖZÜMÜ İÇİN HE’NİN VARYASYONEL İTERASYON YÖNTEMİNİN UYGULANMASI

Öz

Bu çalışmada göller sisteminin kirlilik modeli gibi nonlineer adi diferensiyel denklem sisteminin analitik çözümü için He’nin varyasyonel iterasyon yöntemi uygulandı. Önerilen yaklaşım varyasyonel iterasyon yöntemi, Laplace dönüşümü ve Padé yaklaşımlarını baz almaktadır. Varyasyonel iterasyondan elde edilen sonuçlara Padé yaklaşımları uygulanmıştır. Önerdiğimiz yaklaşım ile nonlinear adi diferensiyel denklem sisteminin analitik çözümleri gösterildi. Matlab ode15s den elde edilen sonuçlar ile VIM’e Padé yaklaşımı uygulandıktan sonra elde edilen sonuçlar karşılaştırıldı. Yöntemlerin güvenilirliği ve basitliğini göstermek için bazı grafikler sunuldu

Anahtar Kelimeler

• Padé yaklaşımı,Varyasyonel iterasyon yöntemi,Göller sistemi için kirlilik modeli

Kaynakça

1. [1] Biazar, J., Farrokhi,L. and Islam, M.R., “Modeling the pollution of a system of lakes”, Applied Mathematics and Computation., 178: 423–430 (2006).
2. [2] İnternet: J. Hoggard, Lake Pollution Modeling, Virginia Tech. Available from: http://www.math.vt.deu/pepole/hoggard/links/new/main.html (2008).
3. [3] Robertson, H.H.,” The solution of a set of reaction rate equations, in: J. Wals (Ed.), Numerical Analysis”, An Introduction, Academic Press, London, (1966).
4. [4] Giordano, F.R. and Weir,M.D., “Differential Equations: A Modern Approach”, Addison Wesley Publishing Company, (1991).
5. [5] Simmons, G.F.,”Differential Equations with Applications and Historical Notes”, McGraw-Hill (1972).
6. [6] He, J.H., “Homotopy perturbation technique, Comput Methods”, Appl. Mech. Engrg., 178: 257–62 (1999).
7. [7] He, J.H., “A coupling method of a homotopy technique and a erturbation technique for non-linear problems”, Int. J Non-linear. Mech., 35: 37–43 (2000).
8. [8] He, J.H., “Approximate analytical solution for seepage flow with fractional derivatives in porous media”, Comput. Methods Appl. Mech. Eng., 167:57–68 (1998).

9. [9] He, J.H.,”Approximate solution of nonlinear differential equations with convolution product nonlinearities”, Comput. Methods Appl. Mech. Eng.,. 167: 69–73 (1998).
10. [10] He, J.H., “Some asymptotic methods for strongly nonlinear equations”, Internat. J. Modern Phys. B., 20: 1141–1199 (2006).
11. [11] He, J.H., “A new approach to nonlinear partial differential equations”, Comm. Nonlinear Sci. Numer. Simul,. 2: 230–235 (1997).
12. [12] He, J.H., “Approximate analytical solution for seepage flow with fractional derivatives in porous media”, Comput. Methods Appl. Mech. Eng., 167: 57–68 (1998).
13. [13] He, J.H., “Variational iteration method for autonomous ordinary differential systems”, Appl. Math. Comput.,114: 115–123 (2000).
14. [14] He, J.H.,”Variational principle for some nonlinear partial differential equations with variable coefficients”, Chaos, Solitons and Fractals., 19: 847–851 (2004).
15. [15] Baker, A., “Essentials of Pad´e Approximants”, Academic Press, London, (1975).
16. [16] Baker, A. and Graves-Morris,P.,”Pade´ approdmants, Cambridge”, Cambridge University Press,(1996).
17. [17] Moustafa, E.S., “Application of differential transform method to non-linear oscillatory systems”, Comm. Nonlinear Sci. Numer. Simul.,13: 1714-1720 (2008).
18. [18] Momani, S. and Ertürk, V.S., “Solutions of non-linear oscillators by the modified differential transform method”, Computers and Mathematics with Applications.(2007) in press.
19. [19] Javidi, M. and Golbabai, A., “Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method”, Chaos Solitons Fractals,. 36: 309-313 (2008).
20. [20] Abdou, M.A. and Soliman, A.A., “Variational iteration method for solving Burger’s and coupled Burger’s equations”, J. Comput. Appl. Math., 181: 245–251 (2005).
21. [21] Abdou, M.A. and Soliman, A.A., “New applications of variational iteration method”, Physica D., 211: 1–8 (2005).
22. [22] Momani, S. and Abuasad, Momani, S., “Application of He’s variational iteration method to Helmholtz Equation”, Chaos, Solitons and Fractals., 27: 1119–1123 (2006).
23. [23] Odibat, Z.M. and Momani, S, “Application of variational iteration method to nonlinear differential equations of fractional order”, Internat. J. Nonlinear Sci. Numer. Simul,. 7: 27–34 (2006).
24. [24] Yusufoğlu,E.and Erbaş, B., “He’s variational iteration method applied to the solution of the prey and predator problem with variable coefficients”, Physics Letters A., 372: 3829–3835 (2008).
25. [25] Rafei, M., Daniali, H. And Ganji, D.D., “Variational iteration method for solving the epidemic model and the prey and predator problem”, Applied Mathematics and Computation., 186: 1701–1709 (2007).
26. [26] Ganji, D.D., Hosseini, M.J. and Shayegh, J., “Some nonlinear heat transfer equations solved by three approximate methods”, Internat Commun. Heat Mass Transfer., 34: 1003–1016 (2007).
27. [27] Rafei, M., Daniali, H. and Ganji, D.D., “Variational iteration method for solving the epidemic model and the prey and predator problem”, Appl. Math. Comput., 186: 1701–1709 (2007).
28. [28] Soliman, A.A. and Abdou, M.A., “Numerical solutions of nonlinear evolution equations using variational iteration method”, J. Comput. Appl. Math., 207: 111–120 (2007).
29. [29] Jiao, Y.C., Yamamoto, Y., Dang, C. and Hao, Y., “An aftertreatment technique for improving the accuracy of Adomian’s decomposition method”, Comput. Math. Appl,. 43: 783–798 (2002).
30. [30] Momani, S., “Analytical approximate solutions of non-linear oscillators by the modified decomposition method”, Int. J. Modern Phys C., 15: 967–979 (2004).
31. [31] Liao, S.J., “The proposed homotopy analysis technique for the solution of nonlinear Problems”, Ph.D. Thesis, Shanghai Jiao Tong University, (1992).
32. [32] Adomian, G., “Solving Frontier Problems of Physics: The Decomposition Method”, Kluwer Academic, Boston, (1994).
33. [33] Wazwaz, A.M., “Partial Differential Equations: Methods and Applications”, Balkema, Rottesdam, (2002).
34. [34] Ganji, D.D. and Rajabi, A., “Assessment of homotopy-perturbation and perturbation methods in heat radiation equations”, Internat Commun. Heat Mass Transfer., 33: 391–400 (2006).
35. [35] He, J.H., “Homotopy perturbation technique”, Comput. Methods Appl. Mech. Eng., 178: 257–262 (1999).
36. [36] He, J.H., “A coupling method of a homotopy technique and a perturbation technique for non-linear problems”, Internat J. Nonlinear Mech., 35: 37–43 (2000).
37. [37] He, J.H., “Application of homotopy perturbation method to nonlinear wave equations”, Chaos, Solitons Fractals., 26: 695–700 (2005).
38. [38] He, J.H., “Homotopy perturbation method for bifurcation of nonlinear problems”, Internat J. Nonlinear Sci. Numer. Simul., 6: 207–208 (2005).