# zbMATH — the first resource for mathematics

Polynomial chaos for random fractional order differential equations. (English) Zbl 1354.34017
Summary: Fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models, where such effects are difficult to incorporate and often neglected. Also, many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations. Thus, to be able to deal with many real life problems, it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty. The aim of this paper is to study the application of the generalized Polynomial Chaos (gPC) to random fractional ordinary differential equations. The method of Polynomial Chaos has played an increasingly important role when dealing with uncertainties. The main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses. However, to apply Polynomial Chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations. In order to avoid more complex numerical computations and obtain accurate solutions we rely on Richardson extrapolation. It is shown that the application of generalized Polynomial Chaos method in conjunction with Richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations.

##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 34F05 Ordinary differential equations and systems with randomness 34C28 Complex behavior and chaotic systems of ordinary differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
Full Text:
##### References:
 [1] Almarashi, A. A.S., Approximation solution of fractional partial differential equations by neural networks, Advances in Numerical Analysis, 2012, 10, (2012), (ID 912810) · Zbl 1236.65110 [2] Arenas, A. J.; González-Parra, G.; Jódar, L., Randomness in a mathematical model for the transmission of respiratory syncytial virus (RSV), Mathematics and Computers in Simulation, 80, 5, 971-981, (2010) · Zbl 1184.92027 [3] Beckmann, P., Orthogonal polynomials for engineers and physicists, (1973), Goldem Press · Zbl 0253.42013 [4] Cameron, R. H.; Martin, W. T., The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals, Annals of Mathematics, 48, 2, 385-392, (1947) · Zbl 0029.14302 [5] Chen-Charpentier, B.; Jensen, B.; Colberg, P., Random coefficient differential models of growth of anaerobic photosynthetic bacteria, Electronic Transactions on Numerical Analysis, 34, 44-58, (2009) · Zbl 1173.60333 [6] Chen-Charpentier, B. M.; Cortés, J.-C.; Romero, J.-V.; Roselló, M.-D., Do the generalized polynomial chaos and frbenius methods retain the statistical moments of random differential equations?, Applied Mathematics Letters, 26, 5, 553-558, (2013) · Zbl 1266.60102 [7] Chen-Charpentier, B. M.; Cortés, J.-C.; Romero, J.-V.; Roselló, M.-D., Some recommendations for applying gPC (generalized polynomial chaos) to modeling: an analysis through the Airy random differential equation, Applied Mathematics and Computation, 219, 9, 4208-4218, (2013) · Zbl 1411.60084 [8] Danca, M.-F.; Garrappa, R.; Tang, W.; Chen, G., Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching, Computers & Mathematics with Applications, 66, 5, 702-916, (2013) · Zbl 1345.37095 [9] Garg, V.; Singh, K., An improved grunwald-Letnikov fractional differential mask for image texture enhancement, International Journal, 3, 3, 130-135, (2012) [10] Ghanem, R. G.; Spanos, P. D., Stochastic finite elements: A spectral approach, (1991), Courier Dover Publications Mineola, NJ · Zbl 0722.73080 [11] Hu, S.; Liao, Z.; Chen, W., Sinogram restoration for low-dosed x-ray computed tomography using fractional-order perona-malik diffusion, Mathematical Problems in Engineering, 2012, 13, (2012), (ID 391050) · Zbl 1264.94015 [12] Khudair, A., On solving non-homogeneous fractional differential equations of Euler type, Computational and Applied Mathematics Online Press, 1-8, (2013) [13] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, (2006), Elsevier The Netherlands · Zbl 1092.45003 [14] Kim, H.; Kim, Y.; Yoon, D., Dependence of polynomial chaos on random types of forces of KdV equations, Applied Mathematical Modelling, 36, 7, 3080-3093, (2012) · Zbl 1252.60063 [15] Konda, U.; Singh, T.; Singla, P.; Scott, P., Uncertainty propagation in puff-based dispersion models using polynomial chaos, Environmental Modelling & Software, 25, 12, 1608-1618, (2010) [16] Li, M., Approximating ideal filters by systems of fractional order, Computational and Mathematical Methods in Medicine, 2012, 6, (2012), (ID 365054) · Zbl 1233.92048 [17] Marchi, C.; Araki, L.; Alves, A.; Suero, R.; Gonalves, S.; Pinto, M., Repeated Richardson extrapolation applied to the two-dimensional Laplace equation using triangular and square grids, Applied Mathematical Modelling, 37, 7, 4661-4675, (2013) · Zbl 1426.65158 [18] Marchi, C. H.; Novak, L. A.; Santiago, C. D.; da Silveira Vargas, A. P., Highly accurate numerical solutions with repeated Richardson extrapolation for 2D Laplace equation, Applied Mathematical Modelling, 37, 12-13, 7386-7397, (2013) [19] Mazandarani, M.; Kamyad, A. V., Modified fractional Euler method for solving fuzzy fractional initial value problem, Communications in Nonlinear Science and Numerical Simulation, 18, 1, 12-21, (2013) · Zbl 1253.35208 [20] Pascual, B.; Adhikari, S., Hybrid perturbation-polynomial chaos approaches to the random algebraic eigenvalue problem, Computer Methods in Applied Mechanics and Engineering, 217, 153-167, (2012) · Zbl 1253.74028 [21] Pichugina, S., Application of the theory of truncated probability distributions to studying minimal river runoff: normal and gamma distributions, Water Resources, 35, 1, 23-29, (2008) [22] Pinto, C.; Tenreiro Machado, J., Fractional model for malaria transmission under control strategies, Computers & Mathematics with Applications, 66, 5, 908-916, (2013) [23] Podlubny, I., Fractional Differential Equations, vol. 204, (1999), Academic Press San Diego · Zbl 0918.34010 [24] Popovic, J. K.; Pilipovic, S.; Atanackovic, T. M., Two compartmental fractional derivative model with fractional derivatives of different order, Communications in Nonlinear Science and Numerical Simulation, 18, 9, 2507-2514, (2013) · Zbl 1304.34012 [25] Pulch, R., Polynomial chaos for boundary value problems of dynamical systems, Applied Numerical Mathematics, 62, 10, 1477-1490, (2012) · Zbl 1251.65114 [26] Rana, S.; Bhattacharya, S.; Pal, J.; NGuérékata, G. M.; Chattopadhyay, J., Paradox of enrichment: a fractional differential approach with memory, Physica A: Statistical Mechanics and its Applications, 392, 17, 3610-3621, (2013) · Zbl 1395.92185 [27] Salloum, M.; Alexanderian, A.; Matre, O. P.L.; Najm, H. N.; Knio, O. M., Simplified CSP analysis of a stiff stochastic ODE system, Computer Methods in Applied Mechanics and Engineering, 217220, 121-138, (2012) · Zbl 1253.65008 [28] Scheerlinck, N.; Peirs, A.; Desmet, M.; Schenk, A.; Nicola, B. M., Modelling fruit characteristics during apple maturation: a stochastic approach, Mathematical and Computer Modelling of Dynamical Systems, 10, 2, 149, (2004) · Zbl 1097.62147 [29] Scherer, R.; Kalla, S.; Tang, Y.; Huang, J., The grunwald-Letnikov method for fractional differential equations, Computers & Mathematics with Applications, 62, 3, 902-917, (2011) · Zbl 1228.65121 [30] Sepahvand, K.; Marburg, S.; Hardtke, H.-J., Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion, Journal of Sound and Vibration, 331, 1, 167-179, (2012) [31] Shi, W.; Zhang, C., Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations, Applied Numerical Mathematics, 62, 12, 1954-1964, (2012) · Zbl 1255.65025 [32] Song, L.; Wang, W., A new improved Adomian decomposition method and its application to fractional differential equations, Applied Mathematical Modelling, 37, 3, 1590-1598, (2013) · Zbl 1352.65219 [33] Soong, T., Random differential equations in science and engineering, (1973), Academic Press New York · Zbl 0348.60081 [34] Soong, T., Probabilistic modeling and analysis in science and engineering, (1992), Wiley New York [35] H. Tchelepi, H. Bazargan, M. Christie, Efficient Markov Chain Monte Carlo sampling using Polynomial Chaos expansion, in: 2013 SPE Reservoir Simulation, Symposium, 2013. [36] Wei, T.; Zhang, Z., Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Engineering Analysis with Boundary Elements, 37, 1, 23-31, (2013) · Zbl 1351.35267 [37] Wiener, N., The homogeneous chaos, American Journal of Mathematics, 60, 897-936, (1938) · JFM 64.0887.02 [38] Yzbasi, S., Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials, Applied Mathematics and Computation, 219, 11, 6328-6343, (2013) · Zbl 1280.65075 [39] Zhou, H.; Wu, Y.-J.; Tian, W., Extrapolation algorithm of compact ADI approximation for two-dimensional parabolic equation, Applied Mathematics and Computation, 219, 6, 2875-2884, (2012) · Zbl 1309.65102 [40] Zlatev, Z.; Farago, I.; Havasi, A., Richardson extrapolation combined with the sequential splitting procedure and the $$\theta$$-method, Central European Journal of Mathematics, 10, 1, 159-172, (2012) · Zbl 1250.65097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.