Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

Calatayud, Julia; López, Juan Carlos Cortés; Jornet, Marc; Villafuerte, L.

doi:10.1186/s13662-018-1848-8

Cited by 16 publications

(70 citation statements)

References 30 publications

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“…Regarding the rapidity of convergence of the power series X(t) = ∞ n=0 X n (t − t 0 ) n introduced in Theorem 2.1, some theoretical estimates were obtained in [19,Subsection 3.6], although no rate of convergence was derived. Fixed r > 0 finite, given ρ := |t − t 0 | < r and given an arbitrary s such that ρ < s < r, the following estimate holds:…”

Section: Stochastic Solutionmentioning

confidence: 99%

Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function

Jornet

Calatayud

Maître

et al. 2020

Journal of Computational and Applied Mathematics

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This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random non-autonomous second order linear differential equations with analytic data processes. The Fröbenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Fröbenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, we rely on the law of total probability to express it in closed-form as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a symbolic Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper.

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Section: Stochastic Solutionmentioning

confidence: 99%

Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function

Jornet

Calatayud

Maître

et al. 2020

Journal of Computational and Applied Mathematics

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“…This theorem is a generalization of the deterministic Fröbenius method to a random framework. As was demonstrated in [17], Theorem 1 has many applications in practice. It supposes a unified approach to study the most well-known second order linear random differential equations: Airy [8], Hermite [9], Legendre [10,11], Laguerre [12], and Bessel [13].…”

Section: Homogeneous Casementioning

confidence: 93%

“…In [17], some auxiliary theorems on random power series were stated and proven: differentiation of random power series in the L p (Ω) sense [17] (Th. 3.1) and Mertens' theorem for random series in the mean square sense [17] (Th. 3.2), which generalize their deterministic counterparts.…”

Section: Homogeneous Casementioning

confidence: 99%

“…The results established in these articles [8][9][10][11][12][13] are particular cases of Theorem 1. The main reason why this fact occurs is explained in [17] (Section 3.3): given a random variable Z, the fact that its centered absolute moments grow at most exponentially, E[|Z| n ] ≤ HR n for certain H > 0 and R > 0, is equivalent to Z being essentially bounded,…”

Section: Homogeneous Casementioning

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%

See 2 more Smart Citations

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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Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense

López

Jornet

2020

Math Methods in App Sciences

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In this paper, we deal with the randomized generalized diffusion equation with delay: u t (t, x) = a 2 u xx (t, x) + b 2 u xx (t − , x), t > , 0 ≤ x ≤ l; u(t, 0) = u(t, l) = 0, t ≥ 0; u(t, x) = (t, x), 0 ≤ t ≤ , 0 ≤ x ≤ l. Here, > 0 and l > 0 are constant. The coefficients a 2 and b 2 are nonnegative random variables, and the initial condition (t, x) and the solution u(t, x) are random fields. The separation of variables method develops a formal series solution. We prove that the series satisfies the delay diffusion problem in the random Lebesgue sense rigorously. By truncating the series, the expectation and the variance of the random-field solution can be approximated.

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Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

Cited by 16 publications

References 30 publications

Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function

Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function

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

Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense

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