Numerical Simulation of Diffusion Equation by Means of He's Variational Iteration Method and Adomian's Decomposition Method

Abstract

In this study, we obtain approximate solutions for diffusion equation on a finite interval by the Adomian decomposition method (ADM) and variational iteration method (VIM) for three cases and then the numerical results are compared. These results show that the ADM leads to more accurate results, and they indicate that only a few terms are sufficient to obtain accurate solutions.

Keywords

• Diffusion operator,Adomian’s decomposition method,He’s variational iteration method

Numerical Simulation of Diffusion Equation by Means of He’s Variational Iteration Method and Adomian’s Decomposition Method

Abstract

Bu ¸calı¸smada, sonlu bir aralıkta ¨u¸c durumda dif¨uzyon denklemi i¸cin ADM ve VIM y¨ontemleri kullanılarak yakla¸sık ¸c¨oz¨umler elde edilmi¸s ve bulunan bu ¸c¨oz¨umler kar¸sıla¸stırılmı¸stır. Elde edilen sonu¸clar ADM’nin daha etkili sonu¸clar verdi˘gini g¨ostermi¸stir. Sayısal sonu¸clar, sadece birka¸c terimin tam ¸c¨oz¨umler elde etmek i¸cin yeterli oldu˘gunu g¨ostermi¸stir.

Keywords

• Difüzyon operatorü, Adomian ayrışım metodu, He’nin varyasyonel iterasyon metodu

References

1. [1] M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Commun. Math. Physics 28 (1972), 177–220.
2. [2] E. Bairamov, O. C¸ akar and A. O. C¸ elebi, Quadratic pencil of Shr¨odinger operators with spectral singularities, discrete spectrum and principal functions, J. Math. Anal. Appl. 216 (1997), 303–320.
3. [3] M. G. Gasymov and G. S. Guseinov, Determination diffusion operator on spectral data, SSSR Dokl. 37 (1981), 19–23.
4. [4] H. Koyunbakan and E. S. Panakhov, Half inverse problem for diffusion operators on the finite interval, J. Math. Anal. Appl. 326 (2007), 1024–1030.
5. [5] B. Chanane, Sturm-Liouville problems with parameter dependent potential and boundary conditions, Journal of Computational and Applied Mathematics 212 (2008), 282–290.
6. [6] A. M. Wazwaz, A first course in integral equations, World Scientific Publishing, Farrer Road, Singapore, 1997.
7. [7] H. Koyunbakan, Reconstruction of potential function for diffusion operator, Numerical Functional Analysis and Optimization 29 (2009), 826–835.
8. [8] G. Adomian, Nonlinear Stochastic Systems and Application to Physics, Kluwer Academic Publishers, Dortrecht, The Netherland, 1989.

9. [9] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, MA, 1994.
10. [10] T. Ozi¸s and A. Yıldırım, Comparison between Adomian’s Method and He’s homotopy perturbation method, Computer Mathematics with Applications 56 (2008), 1216–1224.
11. [11] M. Inc and D. J. Evans, An efficient approach to approximate solutions of eighth-order boundary-value problems, Int. J. Comput. Math. 81 (2004), 685–692.
12. [12] M. Inc and D. J. Evans, A different approach for soliton solution of improved Boussinesq equation, International Journal of Computer Mathematics 81 (2004), 313–323.
13. [13] Y. Cherruault, V. Seng, The resolution of non-linear integral equations of the first kind using the decompositional method of Adomian, Kybernetes 26 (1997), 198–206.
14. [14] M. N. Dour, Y. Cherruault, The decomposition method applied to a diffusion model, Kybernetes 26 (1997), 921–935.
15. [15] D. Lesnic, The Decomposition method for initial value problems, Applied Mathematics and Computation 181 (2006), 206–213.
16. [16] D. Lesnic, The Decomposition method for Cauchy reaction- diffusion problems, Applied Mathematics Letters 20 (2007), 412–418.
17. [17] D. Lesnic, Decomposition methods for non-linear, non-characteristic Cauchy heat problems, Communications in Nonlinear Science and Numerical Simulation 10 (2005), 581–596.
18. [18] E. Yılmaz, M. Inc and H. Koyunbakan, A numerical implementation of the decomposition method for Dirac system, Turkish Journal of Science and Technology 4 (2009), 111–119.
19. [19] J. H. He, Variational iteration method, a kind of nonlinear analytical technique, some examples, International Journal of Non-linear Mechanics 34 (1999), 699–708.
20. [20] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering 167 (1998), 57–68.
21. [21] J. H. He, A new approach to nonlinear partial equations, Commun. Nonlinear Sci. Numer. Simul. 2 (1997), 230–235.
22. [22] J. H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B 20 (2006), 1141–1199.
23. [23] J. H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation 114 (2000), 115–123.
24. [24] J. H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mechanics and Engineering 167 (1998), 69–73.
25. [25] A. Sadighi, D. D. Ganji, M. Gorji and N. Tolou, Numerical simulation of heat-like models with variable coefficients by the variational iteration method, Journal of Physics: Conference Series 96 (2008).
26. [26] H. Khaleghi, D. D. Ganji and A. Sadighi, Application of variational iteration and homotopy perturbation methods to nonlinear heat transfer equations with variable coefficients, Numerical Heat Transfer, Part A 52 (2007), 25–42.
27. [27] D. D. Ganji and A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics 207 (2007), 24–34.
28. [28] N. H. Sweilam and M. M. Khader, Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos Solitons & Fractals 32 (2007), 145–149.
29. [29] A. Yıldırım and T. Ozis, Solutions of Singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear Analysis Series A: Theory, Methods and Applications 70 (2009), 2480–2484.
30. [30] H. Ko¸cak and A. Yıldırım, Numerical solution of 3D Green’s function for the dynamic system of anisotropic elasticity, Physics Letters A 373 (2009), 3145–3150.
31. [31] T. Ozi¸s and A. Yıldırım, A study of nonlinear oscillators with ¨ u 1/3 force by He’s variational iteration method, Journal of Sound and Vibration 306 (2007), 372–376.
32. [32] E. Yusufoglu, Variational iteration method for construction of some compact and noncompact structures of Klein Gordon equations, International Journal of Nonlinear Science and Numerical Simulation 8 (2007), 153–158.
33. [33] N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method, and Adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Science and Numerical Simulation 7 (2006), 65–70.
34. [34] M. Inc, Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method, Chaos Solitons & Fractals 34 (2007), 1071–1084.