Constructing reliable approximations of the random fractional Hermite equation: solution, moments and density

CONSTRUCTING RELIABLE APPROXIMATIONS OF THE RANDOM FRACTIONAL HERMITE EQUATION: SOLUTION, MOMENTS AND DENSITY Clara Burgos1, Tom´as Caraballo2, Juan Carlos Cort´es3, Laura Villafuerte4, Rafael Jacinto Villanueva5 Abstract We extend the study of the random Hermite second-order ordinary differential equation to the fractional setting. We first construct a random generalized power series that solves the equation in the mean square sense under mild hypotheses on the random inputs (coefficients and initial conditions). From this representation of the solution, which is a parametric stochastic process, reliable approximations of the mean and the variance are explicitly given. Then, we take advantage of the Random Variable Transformation technique to go further and construct convergent approximations of the first probability density function of the solution. Finally, several numerically simulations are carried out to illustrate the broad applicability of our theoretical findings. MSC 2010: Primary 26A33; Secondary 37H10, 60H25, 30B20, 34F05, 49J55. Key Words and Phrases: random fractional Hermite differential equation; random mean square calculus; statistical moments; first probability density function 1. Introduction The extension of many classical results to the context of Fractional Calculus has allowed their successful application to a number of practical problems. In particular, fractional-order derivatives have demonstrated 2 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva to be powerful tools to better describe systems, media and fields characterized by non-local and memory of power-law type often met in models that appear in Physics, Control, Signal and Image Processing, Mechanics and Dynamic Systems, Biology, Environmental Science, Materials, Economic and Multidisciplinary in Engineering Fields [1]. The aforementioned extension is often done from relevant models formulated via classical differential equations that have been generalized using different fractional-order derivatives. Examples in this regard include the linear, logistic, Riccati, Gompertz, etc. [2, 3, 4, 5], just to mention a few models. On the other hand, the applications of fractional differential equations to modeling the dynamics of complex phenomena using real-world data involve the rigorous treatment of randomness coming from the combination of epistemic and aleatoric uncertainties [6]. Epistemic (or systematic) uncertainty appears because inaccurate measurements or because the model simplifies the true complexity of the phenomena under study neglecting certain effects, while aleatoric (or stochastic) uncertainty comes from the fact that different outcomes are obtained when we run or observe the same experiment. These facts lead to stochastic or random fractional differential equations. As it is accurately pointed out in [7, p.96], it is important to underline that there is a growing trend in the Uncertainty Quantification community to treat stochastic and random differential equations as synonymous terms, when in fact they require completely different approaches for analysis and approximation. In dealing with stochastic differential equations (SDEs), uncertainties are forced by an irregular process, such as the Brownian motion or, more generality, a Wiener process. SDEs are typically represented in terms of stochastic differentials, but they must be interpreted as Itˆo or Stratonovich stochastic integrals [7, p.97], [8]. The role of uncertainty is essentially different in random differential equations (RDEs). Indeed, in the setting of these equations, random effects are directly manifested through coefficients, initial/boundary conditions, and/or source term that are assumed to be well-behaved (e.g., continuous) with respect to time and/or space [7, p.97], [9]. As pointed out in [10, p.258], overall the theory of RDEs is much less advanced than that for SDEs. This fact is even more noticeable in the case of RDEs formulated by means of fractional-order derivatives. The aim of this paper is to continue contributing the realm of Fractional Calculus by extending the analysis of the Hermite differential equation in a two-fold sense, namely, introducing both fractional derivatives and uncertainties in its formulation. For the former goal, the mean square Caputo fractional derivative will be used, while for the later we will rely on the RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION 3 RDE approach. On the one hand, the fractional Hermite differential equation, based on Caputo operator, has been studied to introduce fractional Hermite polynomials and with applications to design special filters [11]. On the other hand, the random Hermite equation Y′′(t)−2tY ′(t) + λY (t)=0, Y (0) = Y0, Y ′(0) = Y1,(1.1) where Y0,Y1and λare random variables, has been studied in [12] using the so called mean square calculus [9]. In this latter contribution, one constructs a power series solution for the randomized classical Hermite differential equation and, then both the expectation and the variance of the solution are approximated. Apart from these above-mentioned contributions, and to the best of our knowledge, none contribution has dealt yet with the study of the random fractional Hermite differential equation. So, in some sense, the present paper is aimed at extending the results that are available so far. Even more, as it shall be seen, we will also give a method to calculate the first probability density function of the solution, that is a more ambitious goal. Hereinafter, we will work on the Lebesgue spaces Lp(D)≡Lp(D,dµ), 1≤p < ∞, whose elements are real-valued measurable functions h:D −→ Rwith the norm ∥h∥Lp(D)=RD|h|pdµ1/p <∞. In the case that p=∞, recall that the norm is defined as ∥h∥L∞(D)= inf{sup{|h(t)|:t∈ D\N} : µ(N) = 0}<∞. For p=∞, elements in the space L∞(D) are essentially bounded functions. Classically, D=T ⊂ Ris an interval and dµ= dtis the Lebesgue measure. Throughout the paper, as we shall also work with random variables and stochastic processes, we will implicitly take D= Ω (sample space) and µ=P(probability measure), and D=T × Ω and dµ= dt×dP, respectively. Notice that X∈Lp(Ω) if and only if ∥X∥Lp(Ω) = (E[|X|p])1/p <∞, where E[ ] denotes the expectation operator, and, X≡ X(t)∈Lp(T ×Ω) if and only if ∥X∥Lp(T ×Ω) =ERT|X(t)|pdt1/p <∞. Any stochastic process X(t) in Lp(T × Ω) can be interpreted as a set of random variables in Lp(Ω) indexed by t∈ T . An important result in the above probabilistic Lebesgue spaces is the so-called Liapunov’s inequality (E[|X|r])1/r ≤(E[|X|s])1/s,0< r ≤s, provided the expectation E[|X|s]<∞. This result indicates that Ls(Ω) ⊂ Lr(Ω), 0 < r ≤s, and as a consequence, in the probabilistic setting, it is preferred to establish results in the biggest space L2(Ω) whose elements 4 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva are real-valued random variables, X: Ω −→ R, with finite second-order moment E[X2]<∞(equivalently, finite variance). The elements of L2(Ω) are usually called second-order variables. It can be proven that L2(Ω) is a Hilbert space with the following inner product ⟨X, Y ⟩=E[XY ], from which one infers the so called 2-norm: ||X||2=p⟨X, X⟩=E[X2]1 2.Given a sequence of second-order random variables, {Xn:n≥0 integer}, is said to be mean square convergent to a random variable X∈L2(Ω) if and only if ||X−Xn||2−→ 0 as n→ ∞. In the case that the collection of second-order random variables is indexed with reference to an interval, say T ⊂ R, then {X(t) : t∈ T } is called a second-order stochastic process. The concepts of continuity, differentiability and integrability in the mean square sense are naturally inferred from the 2-norm. When trying to prove the mean square convergence of a sequence of second-order stochastic processes that defines the solution of a random fractional differential equation often is required to bound products of random variables. Unfortunately, the following inequality ∥XY ∥2≤ ∥X∥2∥Y∥2,X, Y ∈L2(Ω) does not hold, in general. However, H¨older inequality ||XY ||r≤ ||X||p||Y||q,0< p, q, r ≤ ∞,1 r=1 p+1 q,(1.2) applied to r=p= 2 and q=∞leads to ∥XY ∥2≤ ∥X∥2∥Y∥∞. This result, that relates the Lebesgue spaces L2(Ω) and L∞(Ω) will be very useful in our subsequent analysis to properly majorizing some quantities and then establishing the mean square convergence. After doing that, we will be interested in computing reliable approximations of the main moments of the solution, such as the expectation and the variance. To achieve this important goal, the following property of the mean square convergence will play a key role. Proposition 1.1.[9, Th 4.4.3] Let {Xn:n≥0}be a sequence of second-order random variables such that Xn−→ Xas n→ ∞ in the mean square sense. Then, E[Xn]−−−→ n→∞ E[X],V[Xn]−−−→ n→∞ V[X]. In this paper, we shall study the following random fractional initial value problem (RFIVP), that extends, to the fractional setting the random (classical) Hermite equation previously introduced in (1.1), (CD2α 0Y)(t)−2tα(CDα 0Y) + λY (t) = 0, Y (0) = Y0, Y ′(0) = Y1. (1.3) RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION 5 Here, (CDα 0Y)(t) stands for the Caputo mean square derivative of order α > 0 of the second-order stochastic process Y(t), and λ,Y0and Y1are secondorder random variables defined on a complete probability space (Ω,F,P). Let us recall that, given a second-order stochastic process, the random Caputo operator is defined by [13] (CDα 0Y)(t) := 1 Γ(n−α)Zt 0 (t−u)n−α−1Y(n)(u)du, (1.4) where n=−[−α], being [·] the ceiling function. As the classical setting the Hermite equation is a second-order differential equation, hereinafter we will assume that α∈]0,1] in (1.3). It is important to remark that throughout this paper, we take (CD2α 0Y)(t) := (CDα 0(CDα 0Y))(t). This paper is organized as follows. Section 2 is addressed to construct a mean square convergent solution of the RFIVP (1.3). In Section 3, we take advantage of Proposition 1.1 together with the results established in Section 2 to construct reliable approximations of mean and of the standard deviation (equivalently, the variance) functions for the solution of the RFIVP (1.3). To complete our probabilistic study, in Section 4 we will go further and, firstly, we will construct formal approximations of the probability density function of the solution in Subsection 4.1 and, secondly, in Subsection 4.2 we will rigorously prove they are convergent. In Section 5 we illustrate all our theoretical findings by means of two numerical examples, where a wide range of probability distributions for model parameters is considered to better illustrate the applicability of the results. 2. Obtaining a mean square convergent solution for the Hermite random fractional differential equation This section is devoted to construct a convergent solution of the random IVP (1.3) in the so called mean square sense [9]. The solution, which is a stochastic process, will be constructed, by means of a generalized random power series, by applying the extension of classical Fr¨obenius method to the stochastic setting. To guarantee the mean square convergence of the abovementioned series, we will impose some conditions, that will be specified later, on the random coefficient λ, and on the random initial conditions, Y0 and Y1. According to the random Fr¨obenius method, let us assume that the solution, Y(t), can be expanded via a generalized random power series, Y(t) = ∞ X m=0 Xmtαm,(2.5) 6 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva where {Xm}is a sequence of random variables in L2(Ω) to be determined. To calculate Xm, using the random Fr¨obenius method, we will impose that (2.5) is a solution of the random IVP (1.3). To this end, we need to determine the mean square Caputo fractional derivatives, (CDα 0Y)(t) and (CD2α 0Y)(t), of the stochastic process given in (2.5). We first deal with (CDα 0Y)(t) that, according to (1.4), is defined in terms of the first-order mean square derivative of Y(t), denoted by Y′(t). To rigorously do that, we will apply [32, Theorem 3.1]. Let us first denote Um(t) := Xmtαm, applying [9, Property 4.126] with the following identification: f(t) = tαm and X(t) = Xm(constant), one gets that Um(t) is mean square differentiable and U′ m(t) = αmXmtαm−1.Furthermore, by the assumption {Xm} ∈ L2(Ω), Um(t) and U′ m(t) are mean square continuous for each m≥0. Later, once the coefficients Xmhad been explicitly determined, we will justify that Y(t) = P∞ m=0 Um(t) is mean square convergent for all real t > 0 and P∞ m=0 U′ m(t) is mean square uniformly convergent on [−K, K] for any positive K. Then Y′(t) = ∞ X m=0 U′ m(t) = ∞ X m=1 αmXmtαm−1(2.6) will be justified, in the mean square sense, by [32, Theorem 3.1]. Now, we shall calculate the mean square Caputo derivative of the stochastic process Y(t), (CDα 0Y)(t), 0 < α ≤1. Recall that the Caputo derivative of the deterministic power function tνis given by (CDα 0)(tν) = (Γ(ν+1) Γ(ν+1−α)tν−αif ν > 0, 0 if ν= 0,(2.7) see [33, Example 3.1]. Then, taking into account (2.6) and (2.7), one gets (CDα 0Y)(t) = 1 Γ(1 −α)Zt 0 (t−u)−αY′(u)du =1 Γ(1 −α)Zt 0 (t−u)−α ∞ X m=0 Um(u)!′ du = ∞ X m=0 1 Γ(1 −α)Zt 0 (t−u)−αU′ m(u)du = ∞ X m=0 CDα 0(Um(t)) RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION 7 = ∞ X m=0 XmCDα 0(tαm) = ∞ X m=1 Xm Γ(αm + 1) Γ(α(m−1) + 1)tα(m−1) = ∞ X m=0 Xm+1 Γ(α(m+ 1) + 1) Γ(αm + 1) tαm.(2.8) Notice that we have used that P∞ m=0 U′ m(t) converges uniformly in the mean square sense to legitimate the commutation between this series and the integral. Now, we proceed to compute (CD2α 0Y)(t) by applying one more time Caputo’s fractional operator to (2.8), (CD2α 0Y)(t) = CDα 0(CDα 0Y)(t) = CDα 0 ∞ X m=0 Xm+1 Γ(α(m+ 1) + 1) Γ(αm + 1) tαm! = ∞ X m=0 Xm+1 Γ(α(m+ 1) + 1) Γ(αm + 1) CDα 0(tαm) = ∞ X m=1 Xm+1 Γ(α(m+ 1) + 1) Γ(αm + 1) Γ(αm + 1) Γ(αm + 1 −α)tαm−α = ∞ X m=1 Xm+1 Γ(α(m+ 1) + 1) Γ(α(m−1) + 1)tαm−α = ∞ X m=0 Xm+2 Γ(α(m+ 2) + 1) Γ(αm + 1) tαm =X2 Γ(2α+ 1) Γ(1) + ∞ X m=1 Xm+2 Γ(α(m+ 2) + 1) Γ(αm + 1) tαm =X2Γ(2α+ 1) + ∞ X m=0 Xm+3 Γ(α(m+ 3) + 1) Γ(α(m+ 1) + 1)tα(m+1). (2.9) Once (CDα 0Y)(t) and (CD2α 0Y)(t) have been computed, we formally plug expressions (2.8), (2.9) and (2.5) in the RFIVP (1.3), this gives 8 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva 0 =(CD2α 0Y)(t)−2tα(CDα 0Y) + λY (t) =X2Γ(2α+ 1) + ∞ X m=0 Xm+3 Γ(α(m+ 3) + 1) Γ(α(m+ 1) + 1)tα(m+1) −2 ∞ X m=0 Xm+1 Γ(α(m+ 1) + 1) Γ(αm + 1) tα(m+1) +λX0+λ ∞ X m=0 Xm+1tα(m+1) =Γ(2α+ 1)X2+λX0 + ∞ X m=0 Γ(α(m+ 3) + 1) Γ(α(m+ 1) + 1)Xm+3 −2Γ(α(m+ 1) + 1) Γ(αm + 1) Xm+1 +λXm+1tα(m+1). This relation fulfils choosing Xnsuch that X2=−λ Γ(2α+ 1)X0, and Xm+3 =Γ(α(m+ 1) + 1) Γ(α(m+ 3) + 1) 2Γ(α(m+ 1) + 1) Γ(αm + 1) −λXm+1, m ≥0. (2.10) Note that the terms X0and X1are obtained from the initial conditions given in (1.3), X0=Y(0) = Y0and X1=Y′(0) = Y1. As it can be observed from equation (2.10), odd and even terms, Xm, are independently defined. By recursion, it is easy to check that they can be explicitly expressed as follows Xm=Γ(α+ 1) Γ(mα + 1) m−3 2 Y k=0 2Γ((2k+ 1)α+ 1) Γ(2kα + 1) −λX1, m ≥3,m odd, and Xm=Γ(2α+ 1) Γ(mα + 1) m−2 2 Y k=1 2Γ(2kα + 1) Γ((2k−1)α+ 1) −λX2, m ≥4,m even, respectively. Then the solution (2.5) can be rewritten as Y(t) =X0+X1tα+X2t2α+ ∞ X m=1 X2m+1t(2m+1)α+ ∞ X m=2 X2mt2mα,(2.11) RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION 9 where X0=Y0, X1=Y1, X2=−λX0 Γ(2α+ 1) =−λY0 Γ(2α+ 1), X2m+1 =Γ(α+ 1) Γ((2m+ 1)α+ 1) m−1 Y k=0 2Γ((2k+ 1)α+ 1) Γ(2kα + 1) −λX1 =Γ(α+ 1) Γ((2m+ 1)α+ 1) m−1 Y k=0 2Γ((2k+ 1)α+ 1) Γ(2kα + 1) −λY1, X2m=Γ(2α+ 1) Γ(2mα + 1) m−1 Y k=1 2Γ(2kα + 1) Γ((2k−1)α+ 1) −λX2 =−1 Γ(2mα + 1) m−1 Y k=1 2Γ(2kα + 1) Γ((2k−1)α+ 1) −λλX0 =−1 Γ(2mα + 1) m−1 Y k=1 2Γ(2kα + 1) Γ((2k−1)α+ 1) −λλY0. (2.12) Substituting (2.12) into (2.11) and rearranging the terms yields Y(t) =Y0+Y1tα−λY0 Γ(2α+ 1)t2α +Y1 ∞ X m=1 "Γ(α+ 1) Γ((2m+ 1)α+ 1) m−1 Y k=0 2Γ((2k+ 1)α+ 1) Γ(2kα + 1) −λ#t(2m+1)α −λY0 ∞ X m=2 "1 Γ(2mα + 1) m−1 Y k=1 2Γ(2kα + 1) Γ((2k−1)α+ 1) −λ#t2mα =Y0ˆ Y1(t) + Y1ˆ Y2(t),(2.13) where ˆ Y1(t) := 1 −λ ∞ X m=1 "1 Γ(2αm + 1) m−1 Y k=1 2Γ(2kα + 1) Γ((2k−1)α+ 1) −λ#t2mα (2.14) 16 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva +E[Y1] tα+ M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 E[λi](−1)iˆ Gm,i!#!. (3.24) Recall that the standard deviation of YM(t), σ[YM(t)], is defined by σ[YM(t)] = qE[Y2 M(t)] −(E[YM(t)])2.(3.25) Note that Y2 M(t) =Y2 0 1 + M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!#!2 | {z } :=A +Y2 1 tα+ M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#!2 | {z } :=B + 2Y0Y1 1 + M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!#! | {z } C · tα+ M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#! | {z } :=C . (3.26) Now, for the sake of clarity, we separately compute the above three terms, denoted by A, B and C. RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION17 A := 1 + M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!#!2 =1 + 2 M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!# + M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!#!2 =1 + 2 M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!# +  M X m=1 M X n=1  t2α(m+n) Γ(2αm + 1)Γ(2αn + 1)   m−1 X i=0 n−1 X j=0 λi+j+2(−1)i+j+2Gm−1,iGn−1,j   , B := tα+ M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#!2 =t2α+ 2tα M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!# + M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#!2 =tα+ 2tα M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!# + M X m=1 M X n=1  Γ(α+ 1)2t(2n+2m+2)α Γ((2m+ 1)α+ 1)Γ((2n+ 1)α+ 1)   m X i=0 n X j=0 λi+j(−1)i+jˆ Gm,i ˆ Gn,j   18 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva and C := 1 + M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!#! · tα+ M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#! =tα+tα M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!# + M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!# + M X m=1 M X n=1  Γ(α+ 1)t2αmt(2n+1)α Γ((2n+ 1)α+ 1)Γ(2αm + 1)   m−1 X i=0 n X j=0 λi+j+1(−1)i+j+1Gm−1,i ˆ Gn,j  . Substituting A, B and C in (3.26), YM(t)2can be expressed as Y2 M(t) =Y2 0 1+2 M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!# +  M X m=1 M X n=1  t2α(m+n) Γ(2αm + 1)Γ(2αn + 1)   m−1 X i=0 n−1 X j=0 λi+j+2(−1)i+j+2Gm−1,iGn−1,j     +Y2 1 tα+ 2tα M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!# + M X m=1 M X n=1  Γ(α+ 1)2t(2n+2m+2)α Γ((2m+ 1)α+ 1)Γ((2n+ 1)α+ 1)   m X i=0 n X j=0 λi+j(−1)i+jˆ Gm,i ˆ Gn,j    + 2Y0Y1 tα+tα M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!# + M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!# + M X m=1 M X n=1  Γ(α+ 1)t2αmt(2n+1)α Γ((2n+ 1)α+ 1)Γ(2αm + 1)   m−1 X i=0 n X j=0 λi+j+1(−1)i+j+1Gm−1,i ˆ Gn,j   . (3.27) RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION19 Applying the expectation operator on (3.27), one gets E[Y2 M(t)] =E[Y2 0] 1+2 M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 E[λi+1](−1)i+1Gm−1,i!# +  M X m=1 M X n=1  t2α(m+n) Γ(2αm + 1)Γ(2αn + 1)   m−1 X i=0 n−1 X j=0 E[λi+j+2](−1)i+j+2Gm−1,iGn−1,j     +E[Y2 1] tα+ 2tα M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 E[λi](−1)iˆ Gm,i!# + M X m=1 M X n=1  Γ(α+ 1)2t(2n+2m+2)α Γ((2m+ 1)α+ 1)Γ((2n+ 1)α+ 1)   m X i=0 n X j=0 E[λi+j](−1)i+jˆ Gm,i ˆ Gn,i    + 2E[Y0]E[Y1] tα+tα M X m=1 "t2αm Γ(2αm + 1) m−1 X i=0 E[λi+1](−1)i+1Gm−1,i!# + M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 E[λi](−1)iˆ Gm,i!# + M X m=1 M X n=1  Γ(α+ 1)t2αmt(2n+1)α Γ((2n+ 1)α+ 1)Γ(2αm + 1)   m−1 X i=0 n X j=0 E[λi+j+1](−1)i+j+1Gm−1,i ˆ Gn,j   . (3.28) From the previous expressions, it is interesting to observe that the approximation of order Mof the mean, E[YM(t)], depends on E[Y0], E[Y1] and E[λm], m= 1, . . . , M, while the approximation of the second order moment, E[Y2 M(t)] (and hence, by (3.25), of σ[YM(t)])), depends on the above quantities together with E[Y2 0], E[Y2 1] and E[λm], m= 1,...,2M, as expected. Finally, notice that Theorem 2.1 ensures the mean square convergence of YM(t), and according to Proposition 1.1, E[YM(t)] and E[Y2 M(t)] converge to their corresponding exact values, E[Y(t)] and E[Y2(t)], respectively. 4. Convergent approximations for the 1-PDF of the solution So far, convergent approximations for the mean, E[YM(t)], and for the standard deviation, σ[YM(t)], of the solution, Y(t), given in (3.22) have been computed from its truncation of order M,YM(t), given in (3.23). Nevertheless, sometimes it is required further statistical information of Y(t). On the one hand, computing higher-order one-dimensional statistical moments, E[(YM(t))k], allow us to approximate additional statistical properties, such as the asymmetry, the kurtosis, etc., of Y(t) that are useful 20 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva functions to better describing the solution from a probabilistic standpoint. On the other hand, the probability that the solution lies within an interval of interest is, obviously, a relevant information in practice. Approximations for both quantities can be calculated by integrating the so called first probability density function (1-PDF) of YM(t), say fYM(t)(y), E[(YM(t))k] = Z∞ −∞ ykfYM(t)(y) dy, k = 1,2,..., and P[l1≤YM(t)≤l2] = Zl2 l1 fYM(t)(y)dy. Of course the above approximations will be legitimated provided fYM(t)(y)−→ fY(t)(y) as M→ ∞, where fY(t)(y) stands for the 1-PDF of the exact solution Y(t), given in (3.22). In this section, we first formally construct the approximations fYM(t)(y) and, then, we establish sufficient conditions so that the foregoing convergence fulfils. 4.1. Constructing formal approximations for the 1-PDF. In the extant literature there exist different approaches to obtain, exact or approximately, the 1-PDF of a stochastic process. Most of these methods are natural extensions of their corresponding counterpart for calculating the PDF of a random variable. As we have previously obtained approximations for the two first moments of the solution, a natural approach would be to apply the principle of maximum entropy (PME). This method constructs the PDF taking into account the available information of the random variable (in our case, the two first moments) by maximizing the concept of Shannon’s entropy, which defines the lack of knowledge of a random variable [34]. In the setting of ordinary and fractional differential equations with randomness, this approach has been recently applied in [29] and [35], respectively. Although, the method provides well-founded approximations to calculate the 1-PDF, the results heavily depend on the accuracy of the approximations of the first statistical moments. Moreover, according to the PME method, the approximations of the 1-PDF are limited to certain specific classes of densities depending on the number of statistical moments that have been pre-calculated. For example, if it is only known the mean and that the solution is positive, the PDF will be an exponential distribution; if both the mean and the variance are known, the approximation of the PDF will be Gaussian; etc. [34]. Non-standard distributions can be achieved at expenses of pre-calculating higher statistical moments that could be cumbersome, as can be guess from the expressions of the two first moments (see expressions (3.24) and (3.28)). RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION21 To avoid these drawbacks, we here propose to obtain the 1-PDF by an alternative method termed the Probabilistic Transformation Method (PTM), which is based on the following result. Theorem 4.1 (PTM).[9, p. 25]. Let us consider Z= (Z1, . . . , Zk) and X= (X1, . . . , Xk)two k-dimensional absolutely continuous random vectors defined on a common complete probability space (Ω,FΩ,P). Let r:Rk→Rkbe a one-to-one deterministic transformation of Zinto X, i.e., X=r(Z). Assume that ris continuous in Zand has continuous partial derivatives with respect to each Zi,1≤i≤k. Then, if fZ(z)denotes the joint probability density function of random vector Z, and s=r−1= (s1(x1, . . . , xk), . . . , sk(x1, . . . , xk)) represents the inverse mapping of r= (r1(z1, . . . , zk), . . . , rk(z1, . . . , zk)), the joint probability density function of random vector Xis given by fX(y) = fZ(s(x)) |J|, where |J|, which is assumed to be different from zero, is the absolute value of the Jacobian defined by the following determinant J= det ∂s ∂x= det     ∂s1(x1,...,xk) ∂x1··· ∂sk(x1,...,xk) ∂x1 . . ..... . . ∂s1(x1,...,xk) ∂xk··· ∂sk(x1,...,xk) ∂xk    . In our setting the key idea to take advantage of the above results is to note that, for t > 0 fixed, the approximate solution, YM(t), given in (3.23) is described by means of a transformation, r, of the input parameters Y0,Y1and λ, whose PDFs, fY0,fY1and fλare known. Observe that, according to hypothesis H1, the joint PDF of (Y0, Y1, λ) is given by fY0,Y1,λ =fY0fY1fλ. Applying Theorem 4.1 to YM(t), we first shall obtain the approximations, fYM(t)(y) and, later, we will establish sufficient condition so that fYM(t)(y)−→ fY(t)(y) as M→ ∞. The PTM (also referred to as RVT-Random Variable Transformation) method has been successfully applied to obtain the 1-PDF of the solution of some classes of differential equations with uncertainties. In [36] the authors have obtained the 1-PDF of the solution of a logistic random differential equation. In [37], the PTM method is applied to approximate the 1-PDF of the solution of a delay random differential equation. The PTM method has also been applied to numerically solve PDEs [38]. In [39], some of the authors of this contribution, approximate the 1-PDF of a linear autonomous 22 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva random fractional differential equation, whose order of fractional differentiation is 0 < α ≤1, by taking advantage of the PTM technique. Let us apply Theorem (4.1) with the following identification, k= 3 and Z= (Z1, Z2, Z3)=(Y0, Y1, λ). The vector X= (X1, X2, X3) is defined by the following deterministic transformation r= (r1, r2, r3), of Z, i.e. X=r(Z), where x1=r1(y0, y1, λ) = y0 1 + M X m=1 "t2mα Γ(2αm + 1) m−1 X i=0 λi+1(−1)i+1Gm−1,i!#!, +y1 tα+ M X m=1 "Γ(α+ 1)t(2m+1)α Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#!, x2=r2(y0, y1, λ) = y1 x3=r3(y0, y1, λ) = λ. It can be seen that the inverse mapping of r,s=r−1, is given by y0=s1(x1, x2, x3) = x1−x2tα+ M P m=1 Γ(α+1)t(2m+1)α Γ((2m+1)α+1) m P i=0 xi 3(−1)iˆ Gm,i 1 + M P m=1 t2αm Γ(2αm+1) m P i=0 xi+1 3(−1)i+1Gm−1,i , y1=s2(x1, x2, x3) = x2, λ=s3(x1, x2, x3) = x3. The absolute value of the Jacobian of the transformation sis given by |J|= ∂s1(x1, x2, x3) ∂x1=1 1 + M P m=1 t2αm Γ(2αm+1) m−1 P i=0 xi+1 3(−1)i+1Gm−1,i . Applying Theorem (4.1), the PDF of the random vector X= (X1, X2, X3) is given by fX1,X2,X3(x1, x2, x3) =fY0,Y1,λ      x1−x2tα+ M P m=1 Γ(α+1)t(2m+1)α Γ((2m+1)α+1) m P i=0 xi 3(−1)iˆ Gm,i 1 + M P m=1 t2αm Γ(2αm+1) m−1 P i=0 xi+1 3(−1)i+1Gm−1,i , x2, x3     ·1 1 + M P m=1 t2αm Γ(2αm+1) m−1 P i=0 xi+1 3(−1)i+1Gm−1,i . RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION23 Marginalizing with respect X2=Y1and X3=λ, we can obtain the 1-PDF of the approximate solution, YM(t), fYM(t)(y) = fX1(y) = Z∞ −∞ Z∞ −∞ fX1,Y1,λ(y, y1, λ) dy1dλ =Z∞ −∞ Z∞ −∞ fY0     y−y1tα+ M P m=1 Γ(α+1)t(2m+1)α Γ((2m+1)α+1) m P i=0 λi(−1)iˆ Gm,i 1 + M P m=1 t2αm Γ(2αm+1) m−1 P i=0 λi+1(−1)i+1Gm−1,i      ·fY1(y1)fλ(λ)1 1 + M P m=1 t2αm Γ(2αm+1) m−1 P i=0 λi+1(−1)i+1Gm−1,i dy1dλ. (4.29) 4.2. Convergence of approximations of the 1-PDF. This subsection is addressed to show that fYM(t)(y)−→ fY(t)(y) as M→ ∞ under mild conditions. Note that fYM(t)(y) is given by (4.29), while the limit is given by fY(t)(y) = Z∞ −∞ Z∞ −∞ fY0     y−y1tα+ ∞ P m=1 Γ(α+1)t(2m+1)α Γ((2m+1)α+1) m P i=0 λi(−1)iˆ Gm,i 1 + ∞ P m=1 t2αm Γ(2αm+1) m−1 P i=0 λi+1(−1)i+1Gm−1,i      ·fY1(y1)fλ(λ)1 1 + ∞ P m=1 t2αm Γ(2αm+1) m−1 P i=0 λi+1(−1)i+1Gm−1,i dy1dλ. (4.30) For the sake of clarity in the subsequent development, we first introduce the following notation. 24 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva SM 0(t) = 1 + M X m=1 "t2αm Γ(2αm + 1) m X i=0 λi+1(−1)i+1Gm−1,i!#, S0(t) = 1 + X m≥1"t2αm Γ(2αm + 1) m X i=0 λi+1(−1)i+1Gm−1,i!#, SM 1(t) = tα+ M X m=1 "Γ(α+ 1) Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#, S1(t) = tα+X m≥1"Γ(α+ 1) Γ((2m+ 1)α+ 1) m X i=0 λi(−1)iˆ Gm,i!#. (4.31) Then, expressions (4.29) and (4.30) read fYM(t)(y) = ZR2 fY0y−y1SM 1(t) SM 0(t)fY1(y1)fλ(λ) 1 SM 0(t)dy1dλ, fY(t)(y) = ZR2 fX0y−y1S1(t) S0(t)fY1(y1)fλ(λ) 1 S0(t)dy1dλ. (4.32) Before proceeding with the proof, it is important to remark the following observations. Note that with the notation of (4.31), the solution (3.22) is given by Y(t) = Y0S0(t) + X1S1(t). If Y0= 0 then Y0=Y(0) = Y0S0(0) + Y1S1(0) = Y0S0(0), and S0(0) = 1 with probability 1, because S1(0) = 0. Taking into account that S0(t) is a power series evaluated at t2αand consequently continuous, we can guarantee that ∃δ0>0 : 0 < ms,0≤min{|SM 0(t)|,|S0(t)|},∀t:|t| ≤ δ0,∀integer M≥0. (4.33) Moreover, by the definition of equations (4.31), it is known that SM 0(t) and SM 1(t) are convergent series in the whole real line. Thus, these series are almost surely uniform convergent in every compact subset of R. This guarantees that, for j= 0,1, ∃Ms,j >0 : max{|SM j(t)|,|Sj(t)|} ≤ Ms,j,∀t:|t| ≤ δ0,∀integer M≥0. (4.34) Finally, it is note that SM 0(t) and SM 1(t) converge uniformly to S0(t) and S1(t) on [−δ0, δ0], respectively. So, taken εj>0, j = 0,1,arbitrarily RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION25 but fixed, there exists Mj 0>0 integer, so that |SM j(t)−Sj(t)|< εj,∀M≥Mj 0integer and ∀t:|t| ≤ δ0.(4.35) To complete the proof, we fix tassuming that it lies within a neighbourhood about t0= 0, where the RFIVP is formulated and the bounds (4.33) and (4.34) fulfil. To proof that fYM(t)(y)−→ fY(t)(y) as M→ ∞, besides assuming hypotheses H1 and H2, we will assume that •H3: The PDF, fY0, of the initial condition Y0is Lipschitz on the whole real line, R, i.e. there exists L0>such that |fY0(x)−fY0(z)| ≤ L0|x−z|,∀x, z ∈R. To prove the convergence, we fix tand calculate the difference fY(t)(y)−fYM(t)(y) using (4.32). fY(t)(y)−fYM(t)(y) =ZR2fY0y−y1S1(t) S0(t)1 |S0(t)|−fY0y−y1SM 1(t) SM 0(t)1 |SM 0(t)|fY1(y1)fλ(λ) dλdy1 ≤ZR2fY0y−y1S1(t) S0(t)1 |S0(t)|−fY0y−y1SM 1(t) SM 0(t)1 |SM 0(t)|fY1(y1)fλ(λ) dλdy1 =ZR2fY0y−y1S1(t) S0(t)1 |S0(t)|−fY0y−y1SM 1(t) SM 0(t)1 |S0(t)| +fY0y−y1SM 1(t) SM 0(t)1 |S0(t)|−fY0y−y1SM 1(t) SM 0(t)1 |SM 0(t)|fY1(y1)fλ(λ)dλdy1 ≤ZR2         fY0y−y1S1(t) S0(t)−fY0y−y1SM 1(t) SM 0(t) | {z } (I) 1 |S0(t)| | {z } (II) +fY0y−y1SM 1(t) SM 0(t) | {z } (III)  1 |S0(t)|−1 |SM 0(t)| | {z } (IV)          fY1(y1)fλ(λ) dλdy1. (4.36) Now we proceed to bound the terms (I)–(IV) in (4.36). Let us start with term (III). Firstly, let us denote F0:= fY0(0), then using hypothesis H3 32 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 0.92 0.94 0.96 0.98 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 0.92 0.94 0.96 0.98 4 5 6 7 8 M=7 M=10 M=12 M=15 M=17 Figure 3. Mean and standard deviation, (3.24) and (3.28) respectively, in the context of Example 5.2 considering different order of truncation M∈ {7,10,12,15,17}on the interval t∈[0,1]. t∈ {0.25,0.5,0.75}considering different order of truncation have been included. A zoom has been added at the maximum of each plot to better show graphically the convergence proved in Section 4.2. 6. Conclusions In this paper we have presented a comprehensive analysis of the fractional Hermite differential equation with uncertainties in all its data (coefficient and initial conditions). Our study has been based on the so called random differential equation approach. To perform the study, we firstly have constructed a random generalized power series and we have proved that this solution is mean square convergent by assuming mild hypotheses on the data. Secondly, we have taken advantage of a key property of the mean square convergence to approximate the mean and the variance of the solution. Afterwards, we have constructed approximations of the first probability density function of the solution using the so called Probability RANDOM FRACTIONAL HERMITE DIFFERENTIAL EQUATION33 -5 0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 0.25 0.3 1.3 1.4 1.5 1.6 1.7 0.254 0.256 0.258 0.26 0.262 0.264 0.266 M=3 M=4 M=5 M=7 M=10 -5 0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 0.25 1.6 1.8 2 2.2 2.4 0.205 0.21 0.215 0.22 0.225 0.23 M=3 M=4 M=5 M=7 M=10 -5 0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 1 2 3 4 0.08 0.1 0.12 0.14 0.16 0.18 M=4 M=5 M=7 M=10 M=12 Figure 4. 1-PDF of the solution, fYM(t)(y), given in (4.29), for different t∈ {0.25,0.5,0.75}in the context of Example 5.2, considering different order of truncation M∈ {4,5,7,10,12}. Transformation Method. We have also shown that these approximations are also convergent under some assumptions that fulfill in many practical applications. The main spirit of the paper is to continue developing new results in the setting of Fractional Calculus with uncertainty, where results for random fractional differential equations are still scarce. In this sense, the results presented in this paper for the random fractional Hermite equation can inspire to extend our analysis to other significant random fractional secondorder differential equations in forthcoming contributions. Furthermore, the ideas developed in this contribution may help to extend the deterministic theory for other types of polynomials, such as Chebyshev [40], Humbert [41], Laguerre [42], and Bernoulli [43], to the fractional random framework. Acknowledgements This work has been supported by the grant PID2020-115270GB–I00 funded by MCIN/AEI/10.13039/501100011033 (Spanish “Agencia Estatal 34 C. Burgos, T. Caraballo, J.C. Cort´es, L. Villafuerte, R.J. Villanueva de Investigaci´on”), the grant AICO/2021/302 (Generalitat Valenciana) and the posdoctoral grant from Universitat Polit`ecnica de Valencia Margarita Salas funded by the Spanish Ministry of Universities and Next-Generation, EU. Conflict of Interest Statement The authors declare that there is no conflict of interests regarding the publication of this article. References [1] S. Honguang, Z. Yong, D. Baleanu, C. Wen, C. Yangquan. A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation. Elsevier 2018, 64, 213-231. https://doi.org/10.1016/j. cnsns.2018.04.019 [2] M. Rivero, L. Rodr´ıguez-Germ´a, J.J. Trujillo. Linear fractional differential equations with variable coefficients. Applied Mathematics Letters. Elsevier 2008. 21 (9), 892-897. https://doi.org/10.1016/j. aml.2007.09.010 [3] J. J. Nieto. Solution of a fractional logistic ordinary differential equation. 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On an operational matrix method based on generalized Bernoulli polynomials of level m. Calcolo. 2018, 55, 30. https://doi.org/10.1007/s10092-018-0272-5. 1Instituto Universitario de Matem´atica Multidisciplinar Universitat Polit`ecnica de Val`encia Camino de vera s/n, Valencia, 46022, Spain e-mail: [email protected] 2Departamento de ecuaciones diferenciales y an´alislsl num´erico, Universidad de Sevilla Avenida Reina Mercedes s/n, Sevilla, 41012, Spain e-mail: carabal[email protected] 3Instituto Universitario de Matem´atica Multidisciplinar Universitat Polit`ecnica de Val`encia Camino de vera s/n, Valencia, 46022, Spain e-mail: jcc[email protected] (Corr. author) 4Department of Mathematics University of Texas at Austin USA e-mail: [email protected]du 5Instituto Universitario de Matem´atica Multidisciplinar Universitat Polit`ecnica de Val`encia Camino de vera s/n, Valencia, 46022, Spain e-mail: rjvil[email protected]