AM  Vol.4 No.10 B , October 2013
On the Numerical Solutions of One and Two-Stage Model of Carcinogenesis Mutations with Time Delay and Diffusion
Author(s) Ishtiaq Ali
ABSTRACT

In this paper, we focused on numerical solutions of carcinogenesis mutations models that are based on reaction-diffusion systems and Lotka-Volterra food chains. We consider the case with one and two-stages of mutations with appropriate initial conditions and the zero-flux boundary conditions. The main purpose is to construct a stable discretization scheme, which allows much accuracy than those of a standard approach. To this end, we use the spectral method to postprocess numerical solutions for the proposed model by using some classical methods for solving differential equations. The implementation of the algorithm is simple and it does not need to solve the linear or nonlinear system (in case the model is nonlinear). We simulate the one and two-stage carcinogenesis mutations model and compared the results with previously published ones.


Cite this paper
I. Ali, "On the Numerical Solutions of One and Two-Stage Model of Carcinogenesis Mutations with Time Delay and Diffusion," Applied Mathematics, Vol. 4 No. 10, 2013, pp. 118-124. doi: 10.4236/am.2013.410A2012.
References
[1]   J. C. Arciero, T. L. Jackson and D. E. Kirschner, “A Mathematical Model of Tumor-Immune Evasion and siRNA Treatment,” Discrete and Continuous Dynamical Systems, Series B, Vol. 4, No. 1, 2004, pp. 39-58.

[2]   S. Krus, “Pathological Anathomy,” PZWL, Warsaw, 2001.

[3]   R. Ahangar and X. B. Lin, “Multistage Evolutionary Model for Carcinogenesis Mutations,” Electronic Journal of Differential Equations, Conference 10, 2003, pp. 33-53.

[4]   M. Bodnar, U. Forys, M. J. Piotrowska and J. Poleszczuk, “A Simple Model of Carcinogenesis Mutations with Time Delay and Diffusion,” Mathematical Biosciences and Engineering, Vol. 10, No. 3, 2013, pp. 861-872.
http://dx.doi.org/10.3934/mbe.2013.10.861

[5]   U. Forys, “Biological Delay Systems and the Mikhailov Criterion of Stability,” Journal of Biological Systems, Vol. 12, No. 1, 2004, pp. 45-60.
http://dx.doi.org/10.1142/S0218339004001014

[6]   U. Forys, “Stability Analysis and Comparison of the Models for Carcinogenesis Mutations in the Case of TwoStages of Mutations,” Journal of Applied Analysis, Vol. 11, No. 2, 2005, pp. 200-281.
http://dx.doi.org/10.1515/JAA.2005.283

[7]   U. Forys, “Multi-Dimensional Lotka-Volterra System for Carcinogenesis Mutations,” Mathematical Methods in the Applied Sciences, Vol. 32, No. 17, 2009, pp. 2287-2308.
http://dx.doi.org/10.1002/mma.1137

[8]   U. Forys, “Comparison of the Models for Carcinogenesis Mutations—One-Stage Case,” The Proceeding of the 5th National Conference on Mathematics, Applied to Biology and Medicine, Swiety Krzyz, 2004.

[9]   J. D. Murry, “Mathematical Biology 1: An Introduction,” Springer, Berlin, 2002.

[10]   J. D. Murry, “Mathematical Biology 11: Special Models and Biochemical Applications, Interdisciplinary Applied Mathematics,” Springer, Berlin, 2003.

[11]   J. D. Murry, “Mathematical Biology 11: Special Models and Biochemical Applications, Interdisciplinary Applied Mathematics,” Springer, Berlin, 2003.

[12]   M. R. Garvie, “Finite-Difference Schemes for ReactionDiffusion Equations Modeling Redator-Prey Interactions in MATLAB,” Bulletin of Mathematical Biology, Vol. 69, No. 3, 2007, pp. 931-956.
http://dx.doi.org/10.1007/s11538-006-9062-3

[13]   T. Tang and X. Xu, “Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations,” Communications in Computational Physics, Vol. 5, 2009, pp. 779-792.

 
 
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