Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method

Khudair, Ayad R.; Haddad, S. A. M.; Khalaf, Sanaa L.

doi:10.4236/ojapps.2016.64028

Cited by 28 publications

(42 citation statements)

References 8 publications

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“…are continuous. By (16), the deterministic function k(t) is continuous on (t 0 − r, t 0 + r). This implies that k ∈ L 1 ([t 0 − r 1 , t 0 + r 1 ]) for each 0 < r 1 < r. By [21] (Th.…”

Section: Theoremmentioning

confidence: 99%

“…In this sense, some numerical experiments illustrating and demonstrating the potentiality of our main findings are also included. The study of random non-autonomous second order linear differential equations has been carried out for particular cases, such as Airy, Hermite, Legendre, Laguerre, and Bessel equations (see [8][9][10][11][12][13], respectively), and the general case [14][15][16][17]. Alternative approaches to study this class of random/stochastic differential equations include the so-called probabilistic transformation method [18] and stochastic numerical schemes [19,20], for example.…”

Section: Introductionmentioning

confidence: 99%

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Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

Gregori

López

Sanz

2018

MCA

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The objective of this paper is to complete certain issues from our recent contribution (Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villafuerte, L. Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Adv. Differ. Equ. 2018, 392, 1–29, doi:10.1186/s13662-018-1848-8). We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via a habitual Lipschitz condition that extends the classical Picard theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, the latter being a reformulation of our random Fröbenius method.

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“…are continuous. By (16), the deterministic function k(t) is continuous on (t 0 − r, t 0 + r). This implies that k ∈ L 1 ([t 0 − r 1 , t 0 + r 1 ]) for each 0 < r 1 < r. By [21] (Th.…”

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

Gregori

López

Sanz

2018

MCA

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“…In the concrete case of second-order random linear differential equations, the Fröbenius method has been successfully used to deal with particular equations: Airy [19], Hermite [20], Legendre [21], Bessel [22], etc. In [23,24,25], homotopy, Adomian decomposition and differential transformations techniques, respectively, have been extended to the random scenario to solve some particular second-order random linear differential equations.…”

Section: Introductionmentioning

confidence: 99%

Improving the Approximation of the First- and Second-Order Statistics of the Response Stochastic Process to the Random Legendre Differential Equation

2019

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In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient A and initial conditions X 0 and X 1 . In a previous study [Calbo G. et al, Comput. Math. Appl., 61(9), 2782-2792], a mean square convergent power series solution on (−1/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X 0 and X 1 , independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an L p solution (1 ≤ p ≤ ∞) to the random Legendre differential equation on the whole domain (−1, 1), as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of X 0 and X 1 . Moreover, the growth condition on the moments of A is characterized by the boundedness of A, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.

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“…Several numerical and semi-analytic techniques have been considered to approximate the solutions of random differential equations. Some of these methods include Adomian decomposition method, homotopy perturbation method, differential transformation method and variational iteration method [4][5][6][7]. In this paper, Galerkin finite element method is used to solve random nonlinear second-order ordinary differential equation (ODE) [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Finite Element Method for Solving Nonlinear Random Ordinary Differential Equations

Elkott¹,

El‐Kalla²,

Elsaid³

et al. 2019

Matrix sci. math.

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In this paper we utilize the finite element method for solving random nonlinear differential equations. In the proposed technique, the nodal coefficients are formulated as functions of the random variable. At certain values of random variable, curve fitting is used to construct the approximate nodal solution. Several numerical examples are presented, and the approximate mean solutions are compared with the exact mean solution to illustrate the ability and effectiveness of this method. KEYWORDSFinite element method, Random ordinary differential equations, Nonlinear equations.

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Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method

Cited by 28 publications

References 8 publications

Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

Improving the Approximation of the First- and Second-Order Statistics of the Response Stochastic Process to the Random Legendre Differential Equation

Finite Element Method for Solving Nonlinear Random Ordinary Differential Equations

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