Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique

El-Beltagy, Mohamed A.; Al-Johani, Amnah S.

doi:10.1155/2013/685137

Cited by 6 publications

(7 citation statements)

References 17 publications

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“…Also, WHE was combined with the perturbation theory to solve the perturbed NSDE by El-Tawil, M. and his co-workers [19] [20] [21]. Also, El-beltagy et al used this technique to study the higher order solution for many NSDEs [20] [21] [22] [23]. For, the second technique, it is called the WCE technique; it was developed by Cameron and Martin [24] in 1947.…”

Section: One Of Those Techniques Is the Whe Technique Which Was Suggementioning

confidence: 99%

Solution of Stochastic Van der Pol Equation Using Spectral Decomposition Techniques

hamed¹,

El‐Kalla²,

El-Beltagy³

et al. 2020

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In this paper, we introduce the study of the general form of stochastic Van der Pol equation (SVDP) under an external excitation described by Gaussian white noise. The study involves the use of Wiener-Chaos expansion technique (WCE) and Wiener-Hermite expansion (WHE) technique. The application of these techniques results in a system of deterministic differential equations (DDEs). The resulting DDEs are solved by the numerical techniques and compared with the results of Monte Carlo (MC) simulations. Also, we introduce a new formula that facilitates handling the cubic nonlinear term of van der Pol equations. The main results of this study are: 1) WCE technique is more accurate, programmable compared with WHE and for the same order, WCE consumes less time. 2) The number of Gaussian random variables (GRVs) is more effective than the order of expansion. 3) The agreement of the results with the MC simulations reflects the validity of the forms obtained through theorem 3.1.

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Section: One Of Those Techniques Is the Whe Technique Which Was Suggementioning

confidence: 99%

Solution of Stochastic Van der Pol Equation Using Spectral Decomposition Techniques

hamed¹,

El‐Kalla²,

El-Beltagy³

et al. 2020

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“…This model solution can be written directly into Mathematica for each kernel. The expectation and variance of the solution are obtained using the same formulae in (16). The advantage in using the analytical solution is that there are no restrictions on the solution convergence; that is, the solution can be always obtained and there are no limitations even on the values of the nonlinearity strength .…”

Section: The Analytical Solutionmentioning

confidence: 99%

“…The non-Gaussian behavior of the solution can be detected only with higher-order terms. Higher-order solutions are obtained using automated WHEP technique for the oscillatory equation in [16].…”

Section: Introductionmentioning

confidence: 99%

Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion

El-Beltagy

Al-Mulla

2014

Journal of Applied Mathematics

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In the current work, the Wiener-Hermite expansion (WHE) is used to solve the stochastic heat equation with nonlinear losses. WHE is used to deduce the equivalent deterministic system up to third order accuracy. The solution of the equivalent deterministic system is obtained using different techniques numerically and analytically. The finite-volume method (FVM) with Pickard iteration is used to solve the equivalent system iteratively. The WHE with perturbation technique (WHEP) is applied to deduce more simple and decoupled equivalent deterministic system that can be solved numerically without iterations. The system resulting from WHEP technique is solved also analytically using the eigenfunction expansion technique. The Monte-Carlo simulations (MCS) are performed to get the statistical properties of the stochastic solution and to verify other solution techniques. The results show that higher-order solutions are essential especially in case of nonlinearities where non-Gaussian effects cannot be neglected. The comparisons show the efficiency of the numerical WHE and WHEP techniques in solving stochastic nonlinear PDEs compared with the analytical solution and MCS.

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“…The aforementioned approximations correspond to the first statistical moments (mean and variance) because, as authors indicate in the introduction section, the computation of the probability density function (PDF) is usually very difficult to obtain. In [21], the authors extend the previous analysis to compute higher-order statistical moments of the oscillator response in the case the nonlinearity is only quadratic. The previous methodology is extended and algorithmically automated in [22].…”

Section: Introductionmentioning

confidence: 96%

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

et al. 2021

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We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

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Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique

Cited by 6 publications

References 17 publications

Solution of Stochastic Van der Pol Equation Using Spectral Decomposition Techniques

Solution of Stochastic Van der Pol Equation Using Spectral Decomposition Techniques

Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

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