Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density
[EN] A fractional forward Euler-like method is developed to solve initial value problems with uncertainties formulated via the Caputo fractional derivative. The analysis is conducted by using the so-called random mean square calculus. Under mild conditions on the data, the mean square convergence of the numerical method is proved. This type of stochastic convergence guarantees the approximations of the mean and the variance of the solution stochastic process, computed via the aforementioned numerical scheme, will converge to their corresponding exact values. Furthermore, from this probability information, we calculate reliable approximations to the first probability density function of the solution by taking advantage of the Maximum Entropy Principle. The theoretical analysis is illustrated by two examples. ; This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P and by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 20142020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010. ; Burgos, C.; Cortés, J.; Villafuerte, L.; Villanueva Micó, RJ. (2020). Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density. Journal of Computational and Applied Mathematics. 378:1-14. https://doi.org/10.1016/j.cam.2020.112925 ; S ; 1 ; 14 ; 378 ; Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6 ; Golmankhaneh, A. K., Arefi, R., & Baleanu, D. (2013). Synchronization in a nonidentical fractional order of a proposed modified system. Journal of Vibration and Control, 21(6), 1154-1161. doi:10.1177/1077546313494953 ; Lupulescu, V., O'Regan, D., & ur Rahman, G. (2014). Existence results for random fractional differential equations. Opuscula ...