In this paper, the Adomian decomposition method (ADM) is applied to obtain the approximate solution of a mathematical model of carcinogenesis which is a Riccati differential equation derived by Moolgavkar and Venzon (see [9]). The numerical solution obtained by this way have been compared with the exact solution which obtained by Moolgavkar and Venzon (see [11]). This comparison show that the (ADM) is a powerful method for solving this differential equations. The method does not need weak nonlinearity assumptions or perturbation theory, the decomposition procedure of Adomian will be obtained easily without linearization the problem by implementing the decomposition method rather than the standard methods for the exact solutions.
| Published in |
American Journal of Theoretical and Applied Statistics (Volume 6, Issue 5-1)
This article belongs to the Special Issue Statistical Distributions and Modeling in Applied Mathematics |
| DOI | 10.11648/j.ajtas.s.2017060501.16 |
| Page(s) | 40-45 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2017. Published by Science Publishing Group |
Adomian Decomposition Method, Riccati Differential Equation, Carcinogenesis, Error Absolute
| [1] | Mahmoud M. El-Borai and Khairia El-Said El-Nadi, A Stochastic model of the growth and diffusion of brain tumor cancer, Jokull journal, 111-122, (2014). |
| [2] | Mahmoud M. El-Borai, A. Z. H. El-Banna and W. H. Ahmed, On some fractional- integro partial differential equations, international journal of Basic &Applied science 13, (2013). |
| [3] | W. Y. Tan, Stochastic models of carcinogenesis, Marcel Dekker, New York, (1991). |
| [4] | Medhi Dehghan and Mehdi Tatari, The use of Adomian decomposition method for solving problems in calculus of variation. Mathematical Problems in Engineering, (2006). |
| [5] | T. R. Ramesh Rao, The use of Adomian decomposition method for solving generalized Riccati differential equations, Kuala Lumpur, Malaysia, (ICMSA), (2010). |
| [6] | Cherrault. Y, Convergence of Adomian's decomposition method, Math. Comp. model.14, 83-86 (1990). |
| [7] | Tan, W. Y. and Brown, A non homogeneous two-stage model of carcinogenesis, Math modeling 9, 631-642, (1987). |
| [8] | I. Hashim, R. Ahmad, A. M. Zakaria, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos, Solitons Fractals, Vol. 28, no. 5,pp. 1149-1185, (2006). |
| [9] | Moolgavkar, S. H, Knudson, A model for human carcinogenesis, Jour. Nat. Cancer Inst. 66, (1998). |
| [10] | G. Adomian, R. Rach, Analytical solution of nonlinear boundary value problems in several dimensions by decompositions, J. Math. Anal, 174, (1993). |
| [11] | Moolgavkar, S. H. and Venzon, D. J., Two event model for carcinogenesis: Incidence Curves for childhood and adult cancer, Math. Biosciences 47, 183-214, (1986). |
| [12] | Park, Sang-Hyeon; Kim, Jeong-Hoon, "Homotopy analysis method for option pricing under stochastic volatility", Applied Mathematics Letters, 24, 1740–1744, (2011). |
| [13] | Liao, S. J., Homotopy Analysis Method in Nonlinear Differential Equations, Berlin & Beijing: Springer & Higher Education, ISBN 978-7-04-032298-9, (2012). |
| [14] | E. Babolian and J. Biazar. Solving the problem of biological species living together by Adomian decomposition method. Applied Mathematics and Computation, 129:339–343, (2002). |
| [15] | E. Babolian and J. Biazar. On the order of convergence of Adomian method. Applied Mathematics and Computation, 130:383–387, (2002). |
| [16] | Y. cherruault, G. Adomian, K. Abbaoui, and R. Rach. Further remarks on convergence of de composition method. International Journal of Bio-Medical Computing, 38:89–93, (1995). |
| [17] | D. Lesnic. Convergence of Adomian’s decomposition method: periodic temperatures. computers and Mathematics with Applications, 44:13–24, (2002). |
| [18] | Reid, William T., Riccati Differential Equations, London: Academic Press, (1972). |
| [19] | H. Jafari and V. Daftardar-Gejji. Revised Adomian decomposition mehtod for solving a system of nonlinear equations. Applied Mathematics and Computation, 175:1–7, (2006). |
| [20] | A. Wazwaz. Adomian decomposition method for a reliable treatment of the Emden- Fowler equation. Applied Mathematics and Computation, 161:543–560, (2005). |
| [21] | 21Mahmoud M. El-Borai, Wagdy G. El-Sayed and Faez N. Ghaffoori, Existence solution for a fractional nonlinear integral equation of Volterra type, Aryabhatta Journal of Mathematics and informatics, July 2016, Vol.8, Issue2, 1-15. |
| [22] | M. M. El-Borai, H. M. El-Owaidy, Hamdy M. Ahmed and A. H. Arnous, Soliton solutions of Hirota equation and Hirota-Maccari system, New Trends in Mathematical Sciences, (NTMSCI), July 2016, Vol.4, No. 3, 231-238. |
| [23] | M. M. El-Borai, H. M. El-Owaidy, Hamdy M. Ahmed and A. H. Arnous. Soliton solutions of the nonlinear Schrodinger equation by three integration schemes, Nonlinear Sci. Lett. A, 2017, 32-40. |
| [24] | M. M. El-Borai, H. M. El-Owaidy, Hamdy M. Ahmed,A. H. Arnous, Seithuti Moshokao, Anjan Biswas, Milivoj Belic, Topological and singular soliton solution to Kundu-Eckhaus equation with extended Kundryashv method, International Journal for Light and Electron Optics, OPTIK, 2017, 128, 57-62. |
| [25] | M. M. El-Borai, H. M. El-Owaidy, Hamdy M. Ahmed,A. H. Arnous, Seithuti Moshokao, Anjan Biswas, Milivoj Belic, Dark and singular optical solitons solutions with saptio-temporal dispersion, International Journal for Light and Electron Optics, Volume 130, February 2017, 324-331. |
| [26] | M. M. El-Borai, H. M. El-Owaidy, Hamdy M. Ahmed,A. H. Arnous, Exact and soliton solutions to nonlinear transmission line model, Nonlinear Dyn, 2017, 87, 767-773. |
| [27] | M. M. El-Borai, H. M. El-Owaidy, Hamdy M. Ahmed,A. H. Arnous and M. Mirzazadah, Soliton and other solutions to coupled nonlinear Schrödinger type, Nonlinear Engineering 2016, aop, 1-8. |
| [28] | Mahmoud M. El-Borai, Wagdy G. El-Sayed, Noura N. Khalefa, Solvability of some nonlinear integral functional equations, American J. of Theoritical and Applied Statistics, 2017, 6(5-1): 13-22. |
| [29] | Mahmoud M. El-Borai, Wagdy G. El-Sayed, Afaf E. Abduelhafid, A generalization of some lag synchronization of system with parabolic differential equation, American J. of Theoritical and Applied Statistics, 2017, 6(5-1): 8-12. |
APA Style
Mahmoud M. El-borai, M. A. Abdou, E. M. Youssef. (2017). Approximate Solutions for Mathematical Model of Carcinogenesis Using Adomian Decomposition Method. American Journal of Theoretical and Applied Statistics, 6(5-1), 40-45. https://doi.org/10.11648/j.ajtas.s.2017060501.16
ACS Style
Mahmoud M. El-borai; M. A. Abdou; E. M. Youssef. Approximate Solutions for Mathematical Model of Carcinogenesis Using Adomian Decomposition Method. Am. J. Theor. Appl. Stat. 2017, 6(5-1), 40-45. doi: 10.11648/j.ajtas.s.2017060501.16
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajtas.s.2017060501.16,
author = {Mahmoud M. El-borai and M. A. Abdou and E. M. Youssef},
title = {Approximate Solutions for Mathematical Model of Carcinogenesis Using Adomian Decomposition Method},
journal = {American Journal of Theoretical and Applied Statistics},
volume = {6},
number = {5-1},
pages = {40-45},
doi = {10.11648/j.ajtas.s.2017060501.16},
url = {https://doi.org/10.11648/j.ajtas.s.2017060501.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2017060501.16},
abstract = {In this paper, the Adomian decomposition method (ADM) is applied to obtain the approximate solution of a mathematical model of carcinogenesis which is a Riccati differential equation derived by Moolgavkar and Venzon (see [9]). The numerical solution obtained by this way have been compared with the exact solution which obtained by Moolgavkar and Venzon (see [11]). This comparison show that the (ADM) is a powerful method for solving this differential equations. The method does not need weak nonlinearity assumptions or perturbation theory, the decomposition procedure of Adomian will be obtained easily without linearization the problem by implementing the decomposition method rather than the standard methods for the exact solutions.},
year = {2017}
}
TY - JOUR T1 - Approximate Solutions for Mathematical Model of Carcinogenesis Using Adomian Decomposition Method AU - Mahmoud M. El-borai AU - M. A. Abdou AU - E. M. Youssef Y1 - 2017/04/05 PY - 2017 N1 - https://doi.org/10.11648/j.ajtas.s.2017060501.16 DO - 10.11648/j.ajtas.s.2017060501.16 T2 - American Journal of Theoretical and Applied Statistics JF - American Journal of Theoretical and Applied Statistics JO - American Journal of Theoretical and Applied Statistics SP - 40 EP - 45 PB - Science Publishing Group SN - 2326-9006 UR - https://doi.org/10.11648/j.ajtas.s.2017060501.16 AB - In this paper, the Adomian decomposition method (ADM) is applied to obtain the approximate solution of a mathematical model of carcinogenesis which is a Riccati differential equation derived by Moolgavkar and Venzon (see [9]). The numerical solution obtained by this way have been compared with the exact solution which obtained by Moolgavkar and Venzon (see [11]). This comparison show that the (ADM) is a powerful method for solving this differential equations. The method does not need weak nonlinearity assumptions or perturbation theory, the decomposition procedure of Adomian will be obtained easily without linearization the problem by implementing the decomposition method rather than the standard methods for the exact solutions. VL - 6 IS - 5-1 ER -
Email: