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Solution processes for second-order linear fractional differential equations with random inhomogeneous parts

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  • Villafuerte, L.

Abstract

The novelty of this paper is to derive a mean square solution of a second-order (Caputo) fractional linear differential equation in which the coefficients and initial conditions are random variables, and the forcing term is a second order stochastic process. Using the so-called mean square calculus and assuming mild conditions on the random variables of the equation together with an exponential growth condition on the forcing term, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and correlation obtained from the truncated power series solution are convergent as well. The theory is illustrated with several examples in which different kind of distributions on the input parameters are assumed.

Suggested Citation

  • Villafuerte, L., 2023. "Solution processes for second-order linear fractional differential equations with random inhomogeneous parts," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 17-48.
    • Handle: RePEc:eee:matcom:v:210:y:2023:i:c:p:17-48
    DOI: 10.1016/j.matcom.2023.03.001
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    References listed on IDEAS

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    1. Burgos, C. & Cortés, J.-C. & Villafuerte, L. & Villanueva, R.J., 2022. "Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    2. Corina D. Constantinescu & Jorge M. Ramirez & Wei R. Zhu, 2019. "An application of fractional differential equations to risk theory," Finance and Stochastics, Springer, vol. 23(4), pages 1001-1024, October.
    3. Burgos, C. & Cortés, J.-C. & Debbouche, A. & Villafuerte, L. & Villanueva, R.-J., 2019. "Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 15-29.
    4. Burgos, C. & Cortés, J.-C. & Villafuerte, L. & Villanueva, R.-J., 2017. "Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 305-318.
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