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A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation

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Abstract

In this paper a new version of the chain rule for calculating the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the corresponding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.

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Authors and Affiliations

  1. Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Building 8G, access C, 2nd floor, Camino de Vera s/n, 46022, Valencia, Spain

    J.-C. Cortés & C. Burgos

  2. Facultad de Ciencias en Física y Matemáticas, Universidad Autónoma de Chiapas, Ciudad Universitaria, 29050, Tuxtla Gutiérrez, Chiapas, Mexico

    L. Villafuerte

  3. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    L. Villafuerte

Authors
  1. J.-C. Cortés
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  2. L. Villafuerte
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  3. C. Burgos
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Correspondence to J.-C. Cortés.

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Cortés, JC., Villafuerte, L. & Burgos, C. A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation. Mediterr. J. Math. 14, 35 (2017). https://doi.org/10.1007/s00009-017-0853-6

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  • DOI: https://doi.org/10.1007/s00009-017-0853-6

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