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Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations

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  • Oraby, T.
  • Suazo, E.
  • Arrubla, H.

Abstract

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the Riesz–Feller sense. Lastly, using Riccati equation, we study families of fractional PDEs with variable coefficients which allow explicit solutions. Those solutions connect Lie symmetries to fractional PDEs.

Suggested Citation

  • Oraby, T. & Suazo, E. & Arrubla, H., 2023. "Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:chsofr:v:166:y:2023:i:c:s0960077922010803
    DOI: 10.1016/j.chaos.2022.112901
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    References listed on IDEAS

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    1. Mark Craddock & Eckhard Platen, 2003. "Symmetry Group Methods for Fundamental Solutions and Characteristic Functions," Research Paper Series 90, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Md Rafiul Islam & Angela Peace & Daniel Medina & Tamer Oraby, 2020. "Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles," IJERPH, MDPI, vol. 17(6), pages 1-19, March.
    3. Francesco Mainardi & Armando Consiglio, 2020. "The Wright Functions of the Second Kind in Mathematical Physics," Mathematics, MDPI, vol. 8(6), pages 1-26, June.
    4. Benoit Mandelbrot & Howard M. Taylor, 1967. "On the Distribution of Stock Price Differences," Operations Research, INFORMS, vol. 15(6), pages 1057-1062, December.
    5. Feng, Zhaosheng, 2008. "Traveling wave behavior for a generalized fisher equation," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 481-488.
    6. Marjorie Hahn & Kei Kobayashi & Sabir Umarov, 2012. "SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations," Journal of Theoretical Probability, Springer, vol. 25(1), pages 262-279, March.
    7. Yanovsky, V.V. & Chechkin, A.V. & Schertzer, D. & Tur, A.V., 2000. "Lévy anomalous diffusion and fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 282(1), pages 13-34.
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