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

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.

Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

“…A key result, that will be used in this paper to construct a random generalized power series solution to the random fractional linear differential equation with a random initial condition, is the following chain rule [29]. This rule allows us to compute the mean square derivative of a…”

Solving Second-Order Linear Differential Equations with Random Analytic Coefficients about Regular-Singular Points

“…The RVT method has been successfully applied to obtain exact or approximate representations of the 1-PDF of the solution SP to random ordinary/partial differential and difference equations and systems [1][2][3][4]. In particular, the RVT technique has also been applied to conduct the study for the random IVP (10) and (11) in the case that t 0 is an ordinary point [6]. Thus, the present contribution can be considered as a natural continuation of our previous paper [6].…”

“…Therefore, S 1 (t; A) and S 2 (t; A) do. On the other hand, taking into account the uniqueness of the solution of IVP (10)- (11), both series expansions (as powers of t − t 0 and as powers of a(ω) − a 0 (ω)) match. Henceforth, the series X 1 (t; A) and X 2 (t; A), given by (14), are convergent in t 0 -deleted neighborhoods N X 1 (t 0 ; a 0 (ω)) and N X 2 (t 0 ; a 0 (ω)), respectively, for all a 0 (ω) ∈ D A , ω ∈ Ω.…”

“…It is important to emphasize that the analysis of second-order linear random differential equations has been performed in previous contributions using the random mean square calculus [5] (Ch. 4) (see [7][8][9][10][11][12] for the Airy, Hermite, Legendre, Laguerre, Chebyshev, and Bessel equations, respectively) and using other approaches like the random homotopy method [13,14], the Adomian method [15], the differential transformation method [16], the variational iteration method [17], etc. However, in all these contributions, only approximations for the two first statistical moments (mean and variance) of the solution were calculated.…”

“…where t ∈T 0 , and the coefficients K 0 (A, Y 0 , Y 1 ) and K 1 (A, Y 0 , Y 1 ) can be obtained in terms of the random ICs given in (11), which are established at the time instant t 1 ∈T 0 . In the subsequent development, we will take advantage of the above representation along with the application of the random variable transformation (RVT) technique [5] (p. 25), [6] (Th.…”