Analytical-Numerical Solution of a Parabolic Diffusion Equation Under Uncertainty Conditions Using DTM with Monte Carlo Simulations
DOI:
https://doi.org/10.17230/ingciencia.11.22.3Keywords:
random linear diffusion models, uncertainty conditions, fi- nite difference schemes, differential transformation method, analyticalnumerical solution.Abstract
A numerical method to solve a general random linear parabolic equation where the diffusion coefficient, source term, boundary and initial conditions include uncertainty, is developed. Diffusion equations arise in many fields of science and engineering, and, in many cases, there are uncertainties due to data that cannot be known, or due to errors in measurements and intrinsic variability. In order to model these uncertainties the corresponding parameters, diffusion coefficient, source term, boundary and initial conditions, are assumed to be random variables with certain probability distributions functions. The proposed method includes finite difference schemes on the space variable and the differential transformation method for the time. In addition, Monte Carlo method is used to deal with the random variables. The accuracy of the hybrid method is investigated numerically using the closed form solution of the deterministic associated.
MSC: 35K10, 65C05, 65M99, 35R60
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References
F. Boano, R. Revelli, and L. Ridolfi, “Stochastic modelling of DO and BOD components in a stream with random inputs,” Advances in Water Resources, vol. 29, no. 9, pp. 1341 – 1350, 2006.
B. Chen-Charpentier, B. Jensen, and P. Colberg, “Random Coefficient Differential Models of Growth of Anaerobic Photosynthetic Bacteria,” ETNA, vol. 34, pp. 44–58, 2009.
B. Oksendal, Stochastic Differential Equations. Springer, New York, 1995.
T. Soong, Probabilistic Modeling and Analysis in Science and Engineering. Wiley, New York, 1992.
B. M. Chen-Charpentier, J.-C. Cortés, J.-V. Romero, and M.-D. Roselló, “Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4208 – 4218, 2013.
G. González-Parra, B. Chen-Charpentier, and A. J. Arenas, “Polynomialchaos for random fractional order differential equations,” Applied Mathematics and Computation, vol. 226, pp. 123 – 130, 2014.
E. Vanden Eijnden, “Studying random differential equations as a tool for turbulent diffusion,” Phys. Rev. E, vol. 58, no. 5, pp. R5229–R5232, Nov 1998. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevE.58.R5229
B. Kegan and R. West, “Modeling the simple epidemic with deterministic differential equations and random initial conditions,” Math. Biosc, vol. 195, pp. 197–193, 2005.
H. Kim, Y. Kim, and D. Yoon, “Dependence of polynomial chaos on random types of forces of KdV equations,” Applied Mathematical Modelling, vol. 36, no. 7, pp. 3080 – 3093, 2012.
J. Wu, Y. Zhang, L. Chen, and Z. Luo, “A Chebyshev interval method for nonlinear dynamic systems under uncertainty,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4578 – 4591, 2013.
S. Bhatnagar and Karmeshu, “Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 3063 – 3079, 2011.
G. Gonzalez-Parra, L. Acedo, and A. Arenas, “Accuracy of analyticalnumerical solutions of the Michaelis-Menten equation,” Computational & Applied Mathematics, vol. 30, no. 2, pp. 445–461, 2011.
L. Villafuerte and B. Chen-Charpentier, “A random differential transform method: Theory and applications,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1490–1494, 2012.
F. Morrison, The Art of Modeling Dynamic Systems. John Wiley, 1991. 52 [15] Y. Chen, J. Liu, and G. Meng, “Incremental harmonic balance method for nonlinear flutter of an airfoil with uncertain-but-bounded parameters,” Applied Mathematical Modelling, vol. 36, no. 2, pp. 657 – 667, 2012.
V. Mallet and B. Sportisse, “Air quality modeling: From deterministic to stochastic approaches,” Comput. Math. Appl, vol. 55, no. 10, pp. 2329–2337, 5 2008.
S. D. Brown, R. Ratcliff, and P. L. Smith, “Evaluating methods for approximating stochastic differential equations,” Journal of Mathematical Psychology, vol. 50, no. 4, pp. 402–410, 8 2006.
S. Wu, “The Euler scheme for random impulsive differential equations,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 164–175, 2007.
S. R. Hanna, J. C. Chang, and M. E. Fernau, “Monte Carlo estimates of uncertainties in predictions by a photochemical grid model (uam-iv) due to uncertainties in input variables,” Atmospheric Environment, vol. 32, no. 21, pp. 3619–3628, 1998.
K. M. Hanson, “A framework for assessing uncertainties in simulation predictions,” Physica D: Nonlinear Phenomena, vol. 133, no. 1-4, pp. 179–188, 1999.
G. Pukhov, Differential Transformations of Functions and Equations. Naukova Dumka (in Russian), 1980.
J. Zhou, Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press, Wuhan (in Chinese), 1986.
J. Biazar and M. Eslami, “Differential transform method for quadratic Riccati differential equation,” International Journal of Nonlinear Science, vol. 9, no. 4, pp. 444–447, 2010.
A. J. Arenas, G. González-Parra, and B. M. Chen-Charpentier, “Dynamical analysis of the transmission of seasonal diseases using the differential transformation method,” Mathematical and Computer Modelling, vol. 50, no. 5–6, pp. 765 – 776, 2009. [Online]. Available: http://dx.doi.org/10.1016/j.mcm.2009.05.005
I. A.-H. Hassan, “Application to differential transformation method for solving systems of differential equations,” Applied Mathematical Modelling, vol. 32, pp. 2552–2559, 2008.
M. Jang and C. Chen, “Analysis of the response of a strongly nonlinear damped system using a differential transformation technique,” Applied Mathematics and Computation, vol. 88, pp. 137–151, 1997.
V. S. Ertürk, G. Zaman, and S. Momani, “A numeric–analytic method for approximating a giving up smoking model containing fractional derivatives,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3065–3074,2012.
C. Bervillier, “Status of the differential transformation method,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10 158 – 10 170, 2012.
I. Hwang, J. Li, and D. Du, “A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach,” International Journal of Control, vol. 81, no. 2, pp. 277–293, 2008. [Online]. Available: http://dx.doi.org/10.1080/00207170701556880
C. E. Mejía and D. A. Murio, “Numerical identification of diffusivity coef-ficient and initial condition by discrete mollification,” Comput. Math. Appl, vol. 12, pp. 35–50, 1995.
C.-K. Chen and S.-P. Ju, “Application of differential transformation to transient advective-dispersive transport equation,” Applied Mathematics and Computation, vol. 155, no. 1, pp. 25 – 38, 2004.
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