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Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 413–425 (2013) http://campus.mst.edu/adsa Solving Random Differential Equations by Means of Differential Transform Methods L. Villafuerte CEFyMAP, Universidad Autónoma de Chiapas Calle 4a. Ote. Nte. No. 1428, Tuxtla Gutiérrez Chiapas, C.P. 29000, México [email protected] and J.–C. Cortés Instituto Universitario de Matemática Multidisciplinar Universitat Politècnica de València Edificio 8G, 46022, Valencia, Spain [email protected] Abstract In this paper the random Differential Transform Method (DTM) is used to solve a time-dependent random linear differential equation. The mean square convergence of the random DTM is proven. Approximations to the mean and standard deviation functions of the solution stochastic process are shown in several illustrative examples. AMS Subject Classifications: 65L20, 60G15, 39A10. Keywords: p-stochastic calculus, random linear differential-difference equations, random differential transform method. 1 Introduction Differential transform method (DTM) is a semi-analytical method for obtaining mainly, the solution of differential equations in the form of a Taylor series. Although similar to the Taylor expansion which requires the computation of derivatives of the data functions, DTM constructs an analytical solution in the form of polynomials (truncated infinite Received October 29, 2012; Accepted January 16, 2013 Communicated by Eugenia N. Petropoulou 414 L. Villafuerte and J.–C. Cortés series) which involves less computational efforts. In the DTM, certain transformation rules are applied and the governing differential equation and initial/boundary conditions are transformed into a set of algebraic equations in terms of the differential transforms of the original functions. The solution of these algebraic equations gives the desired solution of the problem. The basic idea of DTM was introduced by Zhou [9]. The differential transform method (DTM) has been successfully applied to a wide class of deterministic differential equations arising in many areas of sciences and engineering. Since many physical and engineering problems are more faithfully modeled by random differential equations, a random DTM method based on mean fourth stochastic calculus has been developed in [8]. In this paper, we consider the random time-dependent linear differential equation: n Ẋ(t) = P (t)X(t) + B(t); P (t) := X A ti , 0 ≤ t ≤ T, n ≥ 0, n n i (1.1) i=0 X(0) = X , 0 where n is a nonnegative integer, B(t) is a stochastic process and Ai , 0 ≤ i ≤ n, and X0 are random variables satisfying certain conditions to be specified later. The aim of the paper is to give sufficient conditions on the data in such a way that a mean square solution of (1.1) can be expanded as X(t) = ∞ X X̂(k)tk , 0 ≤ t ≤ T, (1.2) k=0 where X̂(k) are random variables. In order to achieve our goal, a random DTM is applied to (1.1) taking advantage of results presented in paper [8]. Similar equations have been studied by the authors in [4,5], but using different approaches. As we will see later, a major advantage of having a solution of the form (1.2) is that it allows to obtain in a practical way, the main statistical functions, namely, expectation and variance (or equivalently standard deviation), of the solution stochastic process. The paper is organized as follows. Section 2 deals with some definitions, notations and results that clarify the presentation of the paper. Section 3 is addressed to establish sufficient conditions on the data of initial value problem (1.1) in order to obtain a mean square solution of the form (1.2) by applying the random DTM. Several illustrative examples are included in the last section. 2 Preliminaries This section begins by reviewing some important concepts, definitions and results related to the stochastic calculus that will play an important role throughout the paper. Further details can be found in [1, 6, 7]. Let (Ω, F, P) be a probability space, that is, a triplet consisting of a space Ω of points ω, a σ-field F of measurable sets A in Ω and, 415 Solving Random Differential Equations a probability measure P defined on F, [6]. Let p ≥ 1 be a real number. A real random variable U defined on (Ω, F, P) is called of order p (p-r. v.), if E[|U |p ] < ∞ , where E[ ] denotes the expectation operator. The space Lp of all the p-r. v.’s, endowed with the norm 1/p kU kp = (E[|U |p ]) , is a Banach space, [1, p.9]. Let {Un : n ≥ 0} be a sequence of p-r. v.’s. We say that it is convergent in the pth mean to the p-r. v. U ∈ Lp , if lim kUn − U kp = 0. n→∞ If q > p ≥ 1 and {Un : n ≥ 0} is a convergent sequence in Lq , that is, qth mean convergent to U ∈ Lq , then {Un : n ≥ 0} lies in Lp and it is pth mean convergent to U ∈ Lp , [1, p.13]. In this paper, we are mainly interested in the mean square (m. s.) and mean fourth (m. f.) calculus, corresponding to p = 2 and p = 4 respectively, but results related with p = 8, 16 will also be used. For p = 2, the space (L2 , k·k2 ) is not a Banach algebra, i. e., the inequality kU V k2 ≤ kU k2 kV k2 does not hold, in general. The space L2 is a Hilbert space and one satisfies the Schwarz inequality 1/2 2 1/2 hU, V i = E[|U V |] ≤ E U 2 E V = kU k2 kV k2 . (2.1) Clearly, if the p-r. v.’s U and V are independent, one gets: kU V kp = kU kp kV kp . However, independence cannot be assumed in many applications. The following inequalities are particularly useful to overcome such situation: kU V kp ≤ kU k2p kV k2p , ∀U, V ∈ L2p , which is a direct application of the Schwarz inequality, and m m Y m1 Y , with E [(Ui )mp ] < ∞, k(Ui )m kp Ui ≤ i=1 p (2.2) 1 ≤ i ≤ m, (2.3) i=1 which is proven in [2]. Let Ui be p-r. v.’s, the set of m-dimensional random vectors ~ = (U1 , U2 , . . . , Um )T together with the norm U ~ kp,v = max kUi kp kU 1≤i≤m (2.4) will be denoted by (Lm p , k · kp,v ). The set of m × m-dimensional random matrices A = (Ai,j ), which entries Ai,j ∈ Lp with the norm kAkp,m = m X m X i=1 j=1 kAi,j kp , (2.5) 416 L. Villafuerte and J.–C. Cortés will be denoted by (Lm×m , k · kp,m ). By using (2.2), it is easy to prove that p ~ kp,v ≤ kAk2p,m kU ~ k2p,v . kAU (2.6) Let T be an interval of the real line. If E[|U (t)|p ] < +∞ for all t ∈ T , then {U (t) : t ∈ T } is called a stochastic process of order p (p-s. p.). If there exists a dU (t) stochastic process of order p, denoted by U̇ (t), U (1) (t) or , such that dt U (t + h) − U (t) dU (t) → 0 as h → 0, − h dt p then {U (t) : t ∈ T } is said to be pth mean differentiable at t ∈ T . Furthermore, a p-s. p. {U (t) : |t| < c} is pth mean analytic on |t| < c if it can be expanded in the pth mean convergent Taylor series U (t) = ∞ X U (n) (0) n=0 n! tn , (2.7) where U (n) (0) denotes the derivative of order n of the s. p. U (t) evaluated at the point t = 0, in the pth mean sense. In connection with the pth mean analyticity, we state without proof the following result that can be found in [3]. Proposition 2.1. Let {U (t) : |t| < c} be a pth mean analytic s. p. given by U (t) = ∞ X Uk tk , Uk = k=0 U (k) (0) , k! |t| < c, where the derivatives are considered in the pth mean sense. Then, there exists M > 0 such that M kUk kp ≤ k , 0 < ρ < c, ∀k ≥ 0 integer. (2.8) ρ We end this section, with a short presentation of the random differential transform method and some of its important operational results that can be found in [8]. The random differential transform of an m. s. analytic s. p. U (t) is defined as 1 dk (U (t)) , k ≥ 0, integer, (2.9) Û (k) = k! dtk t=0 where Û is the transformed s. p. and dk denotes the m. s. derivative of order k. The dtk inverse transform of Û is defined as U (t) = ∞ X Û (k)tk . (2.10) k=0 The next result provides some operational rules for the random differential transform of stochastic processes that will be required later. 417 Solving Random Differential Equations Theorem 2.2. Let {F (t) : |t| < c}, {G(t) : |t| < c} be m. f. analytic s. p.’s. Then the following results hold: (i) If U (t) = F (t) ± G(t), then Û (k) = F̂ (k) ± Ĝ(k). (ii) Let A be a 4-r. v.. If U (t) = AF (t), then Û (k) = AF̂ (k). (iii) Let m be a nonnegative integer. dm (F (t)) If U (t) = , then Û (k) = (k + 1) . . . (k + m)F̂ (k + m). dtm (iv) If U (t) = F (t)G(t), then Û (k) = k X F̂ (n)Ĝ(k − n). n=0 3 Application of the Random DTM By applying the operational properties of random DTM given in Theorem 2.2 to the initial value problem (1.1), one obtains (k + 1)X̂(k + 1) = k X Pˆn (r)X̂(k − r) + B̂(k), (3.1) r=0 being Pˆn (r) = Ar , 0 ≤ r ≤ n, 0, r > n. (3.2) Notice that we have applied the deterministic DTM of ri , 1 ≤ i ≤ n, is just 1, [9]. Thus equation (3.1) writes (k + 1)X̂(k + 1) = n X Ar X̂(k − r) + B̂(k), k = n, n + 1, . . . , (3.3) r=0 for which the r. v’s X̂(0), . . . , X̂(n) are computed from (3.1). The initial value X̂(0) is computed directly from the initial condition X0 . Note that from equations (3.1)–(3.3) one can construct an approximation to the solution process of (1.1). In order to prove the m. s. convergence of the inverse transform of X̂(k) given in (1.2), it is defined: ~ Z(k) : = (Z1 (k), Z2 (k), . . . , Zn (k), Zn+1 (k))T = (X̂(k − n), X̂(k − n + 1), . . . , X̂(k − 1), X̂(k))T . (3.4) Then, recurrence (3.3) is written as ~ + 1) = A(k)Z(k) ~ ~ Z(k + H(k), (3.5) 418 L. Villafuerte and J.–C. Cortés where 0 0 .. . A(k) = 0 A n k+1 1 0 .. . 0 1 .. . 0 0 .. . ... ... ... 0 0 .. . 0 0 .. . ~ , H(k) = . 0 0 0 ... 1 B̂(k) An−2 A0 ... ... k+1 k + 1 (n+1)×(n+1) k + 1 n+1 (3.6) 0 An−1 k+1 Iterations of (3.5) yield ~ Z(k) = k−1 Y ! ~ A(i) Z(n) + i=n k−1 X k−1 Y r=n i=r+1 ! ~ A(i) H(r), (3.7) if m ≥ n0 , otherwise. (3.8) for k = n, n + 1, . . . where ( A(m)A(m − 1) . . . A(n0 ) A(i) = 1 i=n0 m Y 4 Mean Square Convergence This section is addressed to prove that the inverse transform of X̂(k) given in (1.2) is an m. s. solution of problem (1.1). This result is summarized as follows Theorem 4.1. Consider problem (1.1) and let us assume that: i) The polynomial coefficients Aj are r. v.’s satisfying the following condition: m ∃ M, K > 0 : E |Am < ∞, ∀m ≥ 0, j = 0, 1, . . . , n. j | ≤ KM ii) The initial condition X0 is a 16-r. v. iii) The forcing term {B(t) : |t| < c} is a 16th mean analytic stochastic process 1 . where c > n+1 Then there exists an m. s. solution of the form X(t) = ∞ X k=0 where X̂(k) are defined by (3.3) . X̂(k)tk , |t| < 1 , n+1 (4.1) 419 Solving Random Differential Equations Proof. Expression (3.7) will be used to show that ∞ X k ~ kZ(k)k 4,v |t| is convergent for k=0 |t| < d1 , where d1 is to be determined later. This fact will imply that ∞ X X̂(k)tk is m. f. k=0 ~ convergent for |t| < d2 because kZ(k)k 4,v ≥ kX̂(k)k4 . Note that the m. f. convergence of (4.1) is required because it was a ! condition for applying Theorem 2.2 to the ! equation k−1 k−1 k−1 Y X Y ~ 1 (k) = ~ ~ 2 (k) = ~ (1.1). Setting Z A(i) Z(n) and Z A(i) H(r) one r=n i=n ~ ~ 1 (k) + Z ~ 2 (k). Then, we will prove that gets: Z(k) =Z ∞ X i=r+1 ~ i (k)k4,v |t|k are convergent kZ k=0 for i = 1, 2 to achieve our goal. Following previous notation, let us define à := k−1 Y A(i) = Ãr,s . i=n From (3.8) it follows that Ãr,s = n+1 X Ar,s1 (k − 1)As1 ,s2 (k − 2)As2 ,s3 (k − 3) . . . Ask−n−1 ,s (n). (4.2) s1 ,s2 ,...,sk−n−1 =1 Note that à is an n + 1 × n + 1-dimension random matrix. Now, using the norm for (4.2) defined in (2.5) and inequality (2.3) yield n+1 n+1 k−1 Y X X Ar,s1 (k − 1)As1 ,s2 (k − 2) . . . As ≤ (n) A(i) , s k−n−1 2p r,s=1 s1 ,s2 ,...,sk−n−1 =1 i=n 2p,m 1 1 k−n n+1 k−n X k−n k−n . . . Ask−n−1 , s (n) . ≤ (Ar,s1 (k − 1)) r,s,s1 ,s2 ,...,sk−n−1 =1 2p 2p (4.3) Note that any component Ar,s (l) of the matrix A(l) can be either 0, 1 or j = 0, . . . , n and l = n, n + 1, . . . , k − 1. Then, one obtains 0 1 k−n 1 k−n 1 " = (Ar,s (l)) 2p(k−n) #! 2p(k−n) 2p A j . E l+1 Aj with l+1 or, or, (4.4) 420 L. Villafuerte and J.–C. Cortés By i) and (4.4) it follows that 1 " 1 2p(k−n) #! 2p(k−n) Aj K 2p(k−n) M E . ≤ l+1 l+1 (4.5) ~ On the other hand, by (3.3)–(3.4) and observing that the terms involved in Z(n) have γn−1 γn−1 γ0 γ0 the form A0 . . . An−1 X0 or A0 . . . An−1 B̂(k) for k = 0, . . . , n − 1, γi nonnegative integers such that 0 ≤ γ1 + . . . + γn−1 ≤ n, i) and ii) imply that there exists C(n) such ~ that Z(n) ≤ C(n) < ∞. 2p,v ~ 1 (k) and Z ~ 2 (k). To conduct the study, we distinguish Now we analyze the bounds of Z two cases. 1 k−1 Y K 2p(k−n) M Case I: If ≤ 1, for all l and k then A(i) ≤ (n + 1)k−n+1 . By l+1 i=n using (2.6) with p = 4 yields k−1 Y ~ Z1 (k) ≤ A(i) 4,v i=n ~ Z(n) 2p,m ≤ C(n)(n + 1)k−n+1 . 8,v 8,m ∞ X ~ Hence, in this case for each t, the series Z1 (k) k |t| can be majorized by 2p,v k=0 ∞ X αk1 , k=0 1 being αk1 = C(n)(n + 1)k−n+1 which is convergent for |t| < as it can be checked n+1 by D’Alembert’s k−1 test. Y Also A(i) ≤ (n + 1)k−r and by Proposition 2.1 it follows that i=r+1 2p,m B̂(r) M1 ~ = , (4.6) H(r) ≤ r + 1 (r + 1)ρr 2p,v 2p for 0 < ρ < c and M1 > 0. Thus, ∞ X ~ Z2 (k) k=n+1 p,v |t|k ≤ M1 ∞ X k−1 X k=n+1 r=n 1 r+1 1 (n + 1)ρ r ! (n + 1)k |t|k . (4.7) Using D’Alembert’s test, one proves that the series in the right-hand side of (4.7) is 1 1 convergent for |t| < for all n ≥ 1 and c > ρ ≥ . n + 1 n + 1 1 K 2p(k−n) M Case II: If ≥ 1, for all l and k, then l+1 k−1 Y k−n 1 M (n + 1)k−n+1 ≤ n!K 2p . A(i) k! i=n 2p,m 421 Solving Random Differential Equations ~ On the other hand, we know that Z(n) 2p,v ∞ X ~ ≤ C(n) and then Z1 (k) k=n+1 |t|k is 2p,v majorized by ∞ X 1 C(n)n!K 2p k=n+1 M k−n (n + 1)k−n+1 k |t| , k! which is convergent on the whole real line. Taking into account that k − r − 1 < k − n because of r = n, n + 1, . . ., it follows k−r−1 that < 1, hence for K > 1 we have that k−n k−r−1 1 K 2p(k−n) < K 2p . (4.8) By k−r k−1 Y A(i) i=r+1 (n + 1)k−r (r + 1)!K 2p(k−n) M k−r−1 ≤ k! 2p,m and (4.6), (4.8), it follows for K > 1 that ∞ X ∞ X k−1 X ! (n + 1)k k−1 k M |t| . k! r=n k=n+1 k=n+1 (4.9) It can be checked that the series in the right side of (4.9) is convergent for |t| < ρ. The same remains true for K < 1. By doing an analogous study of combining Case I and ∞ X ~ Case II, it is proved that the radios of convergence of the series Zi (k) |t|k , kZ2 (k)kp,v |t|k ≤ M1 K 1 2p r! ((n + 1)ρM )r k=0 2p,v 1 .† n+1 In order to obtain approximations of the mean and standard deviation functions by the DTM, we truncate the infinite sum given by (4.1): i = 1, 2 are greater or equal to XN (t) = N X X̂(k)tk . (4.10) k=0 Thus, N h i X E [XN (t)] = E X̂(k) tk (4.11) k=0 N X k−1 N h i 2 X X 2 2k E (XN (t)) = E X̂(k) t +2 E X̂(k)X̂(l) tk+l . k=0 This completes the proof. k=1 l=0 (4.12) 422 5 L. Villafuerte and J.–C. Cortés Examples and Conclusions In this section several illustrative examples in connection with the previous development are provided. The mean and standard deviation of the truncated solution are computed from (4.11)–(4.12). Example 5.1. Let us consider equation (1.1) with P1 (t) = At, where A is a standard normal r. v. (A ∼ N (0, 1)), B(t) = e−Bt , where B is an exponential r. v. with parameter λ = 1 (B ∼ Exp[λ = 1]) and X0 is a Beta r. v. with parameters α = 2 and β = 3, (X0 ∼ Be(α = 2; β = 3)). To simplify computations, in this example we will assume that A, B and X0 are independent r. v.’s. In order that i), iii) of Theorem 4.1 are satisfied, we truncated the r. v.’s A and B. Therefore, for those r. v.’s we select L1 , L2 > 0 sufficiently large so as to get P (B ≥ L1 ), P (|A| ≥ L2 ) arbitrarily small, see the truncated method in [6]. Let us say that Ā, B̄ are the corresponding truncated r. v.’s of A and B, respectively. It is easy to show that B̃(t) := e−B̄t is 16-th mean analytic on the whole real line and Ā satisfies i). Clearly, X0 is a 16-r. v. Hence, Theorem 4.1 holds and X(t) given by (4.1) is an m. s. solution of (1.1). Table 5.1 contains the numerical results of the mean E [XN (t)] and the standard deviation σ [XN (t)] using (4.11)–(4.12) C MC with N = 20 as well as the mean µM m (X(t)) and the standard deviation σm (X(t)) using Monte Carlo sampling with m = 100000 simulations. It was chosen L1 = 100 and L2 = 8 that made P (B ≥ L1 ) = 3.72008 × 10−44 and P (|A| ≥ L2 ) = 1.22125 × 10−15 . Notice that the approximations provided by the random DTM are in agreement with Monte Carlo simulations. t 0.00 0.10 0.20 0.30 0.40 0.50 C µM m (X(t)) m = 100000 0.249618 0.344950 0.432082 0.512552 0.587812 0.659367 E[XN (t)] 0.250000 0.345314 0.432392 0.512766 0.587884 0.659248 MC σm (X(t)) m = 100000 0.193389 0.193448 0.194243 0.197344 0.205102 0.220580 σ[XN (t)] 0.193649 0.193711 0.194511 0.197620 0.205386 0.220914 Table 5.1: Comparison of the expectation and standard function of Example 5.1 by using expressions (4.11)–(4.12) with N = 20 and Monte Carlo method with m = 100000 simulations at some selected points of the interval 0 ≤ t ≤ 1/2. Example 5.2. We now consider P1 (t) = At where A ∼ Be(α1 = 1, β1 = 2), B(t) = B where B ∼ Gamma(α2 = 2, β2 = 3) and X0 ∼ Unif([10, 14]). Condition i) of Theorem 4.1 is satisfied because A has codomain [0,1]. Clearly B is 16-th mean analytic and X0 is a 16th r. v. therefore X(t) given by (4.1) is an m. s. solution of equation (1.1). 423 Solving Random Differential Equations Using the same notation to the mean and standard deviation of Example 5.1, we show the numerical results in Table 5.2. Note that Monte Carlo simulations are in agreement with random DTM results. t 0.00 0.10 0.20 0.30 0.40 0.50 C µM m (X(t)) m = 150000 11.99464 12.61696 13.28357 13.99951 14.770583 15.603558 E[XN (t)] 12.00000 12.62069 13.28575 14.00020 14.76986 15.60148 MC σm (X(t)) m = 150000 1.15609 1.20764 1.35536 1.57876 1.86085 2.19295 σ[XN (t)] 1.15469 1.20759 1.35620 1.58004 1.86218 2.19409 Table 5.2: Comparison of the expectation and standard function of Example 5.2 by using expressions (4.11)–(4.12) with N = 10 and Monte Carlo method with m = 150000 simulations at some selected points of the interval 0 ≤ t ≤ 1/2. Example 5.3. The differential equation Ẋ(t) = (γ cos t)X(t) is a mathematical model for the population X(t) that undergoes yearly seasonal fluctuations [10]. In the deterministic scenario, γ is a positive parameter that represents the intensity of the fluctuations. However, in practice, it depends heavily on uncertain factors such as weather, genetics, prey and predator species and other characteristics of the surrounding medium. This motivates the consideration of parameter γ as a r. v. rather than a deterministic constant leading to a random differential equation. To apply Theorem 4.1, we use the γ 1 approximations cos t ≈ 1 − t2 which leads to equation Ẋ(t) = (γ − t2 )X(t). Thus 2 2 γ γ P2 (t) = γ − t2 and so A0 = γ, A1 = 0, A2 = − . Assuming that γ ∼ Be(α = 2, β = 2 2 5) and X0 is a r. v. such that X0 ∼ Unif([110, 140]), in Table 5.3 we collect the numerical approximations for the expectation and the standard deviation to the solution s. p. X(t) using both Monte Carlo with 106 simulations and random DTM with truncation order N = 10. Again, we observe that results provided by both approaches agree. We would like to point out that the main contribution of this paper is the rigorous construction of a mean square analytic solution of an important class of a timedependent random differential equation by the random differential transform method. The approach allowed to provide reliable approximations for the first and second moments of the solution stochastic process. 424 L. Villafuerte and J.–C. Cortés t 0.00 0.10 0.20 0.30 0.40 0.50 C µM m (X(t)) m = 100000 125.007 128.645 132.385 136.189 140.012 143.804 E[XN (t)] 125 128.6333 132.368 136.165 139.981 143.767 MC σm (X(t)) m = 100000 8.6580 9.1498 10.1174 11.4967 13.1918 15.1047 σ[XN (t)] 8.6602 9.1482 10.1079 11.4762 13.1584 15.0573 Table 5.3: Comparison of the expectation and standard function of Example 5.3 by using expressions (4.11)–(4.12) with N = 10 and Monte Carlo method with m = 100000 simulations at some selected points of the interval 0 ≤ t ≤ 1/2. Acknowledgements This work has been partially supported by the Ministerio de Economı́a y Competitividad grants: MTM2009–08587, Universitat Politècnica de València grant: PAID06–11– 2070, PIEIC–UNACH, Mexican Promep. References [1] L. Arnold. Stochastic differential equations: theory and applications. Wiley Interscience [John Wiley & Sons], New York–London–Sydney, 1974. Translated from the German. [2] G. Calbo, J.-C. Cortés, L. Jódar, and L. Villafuerte. Solving the random Legendre differential equation: mean square power series solution and its statistical functions. Comput. Math. Appl., 61(9):2782–2792, 2011. [3] J.-C. Cortés, L. Jódar, R. Company, and L. Villafuerte. Solving Riccati time-dependent models with random quadratic coefficients. Appl. Math. Lett., 24(12):2193–2196, 2011. [4] J.-C. Cortés, L. Jódar, M.-D. Rosello, and L. Villafuerte. Solving initial and two-point boundary value linear random differential equations: a mean square approach. Appl. Math. Comput., 219(4):2204–2211, 2012. [5] J.-C. Cortés, L. Jódar, and L. Villafuerte. Random linear-quadratic mathematical models: computing explicit solutions and applications. Math. Comput. Simulation, 79(7):2076–2090, 2009. [6] M. Loève. Probability theory I, volume 45 of Graduate Texts in Mathematics. Springer–Verlag, New York–Heidelberg, 4th edition, 1977. Solving Random Differential Equations 425 [7] T. T. Soong. Random differential equations in science and engineering, volume 103 of Mathematics in Science and Engineering. Academic Press [Harcourt Brace Jovanovich, Publishers], New York–London, 1973. [8] L. Villafuerte and B. M. Chen-Charpentier. A random differential transform method: theory and applications. Appl. Math. Lett., 25(10):1490–1494, 2012. [9] J. K. Zhou. Differential Transform Method and Its Applications for Electrical Circuits. Huqzhong University Press, Wuhan, China, 1986. (in chinese). [10] D. G. Zill. A First Course in Differential Equations with Modeling Applications. Brooks/Cole, Cengage Learning, Boston, 10th edition, 2012.