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Burgos, C.; Cortés, J.-C.; Villafuerte, L.; Villanueva, R.-J.

Mean square convergent numerical solutions of random fractional differential equations: approximations of moments and density. (English) Zbl 1503.65009

J. Comput. Appl. Math. 378, Article ID 112925, 13 p. (2020).
Summary: 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.

Cited in 10 Documents

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
34A08 Fractional ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H25 Random operators and equations (aspects of stochastic analysis)
65L05 Numerical methods for initial value problems involving ordinary differential equations

Keywords:

fractional differential equations with randomness; random mean square calculus; random mean square Caputo fractional derivative; random numerics; maximum entropy principle
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\textit{C. Burgos} et al., J. Comput. Appl. Math. 378, Article ID 112925, 13 p. (2020; Zbl 1503.65009)
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