Approximate solution of a mixed nonlinear stochastic oscillator

El-Tawil, Magdy A.; Al-Johani, Amna S.

doi:10.1016/j.camwa.2009.03.057

Cited by 10 publications

(13 citation statements)

References 31 publications

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“…The technique was successfully applied to several nonlinear stochastic equations; see for example [4,5,10]. The WHEP technique for general nonlinear exponent ( ), general order ( ), and general number of corrections (NC) follows the following steps [16].…”

Section: Whep Techniquementioning

confidence: 99%

“…The application of the WHE [4][5][6][7][8][9][10] aims at finding a truncated series solution to the solution process of a stochastic differential equation. The truncated series is composed of two major parts: the first is the Gaussian part which consists of the first two terms, while the rest of the series constitutes the non-Gaussian part.…”

Section: Introductionmentioning

confidence: 99%

“…According to [5,6], quadrate oscillation arises through many applied models in applied sciences and engineering when studying oscillatory systems [22]. These systems can be exposed to a lot of uncertainties through the external forces, the damping coefficient, the frequency, and/or the initial or boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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

El-Beltagy

Al-Johani

2013

Journal of Applied Mathematics

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This paper introduces higher-order solutions of the stochastic nonlinear differential equations with the Wiener-Hermite expansion and perturbation (WHEP) technique. The technique is used to study the quadratic nonlinear stochastic oscillatory equation with different orders, different number of corrections, and different strengths of the nonlinear term. The equivalent deterministic equations are derived up to third order and fourth correction. A model numerical integral solver is developed to solve the resulting set of equations. The numerical solver is tested and validated and then used in simulating the stochastic quadratic nonlinear oscillatory motion with different parameters. The solution ensemble average and variance are computed and compared in all cases. The current work extends the use of WHEP technique in solving stochastic nonlinear differential equations.

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Section: Whep Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

El-Beltagy

Al-Johani

2013

Journal of Applied Mathematics

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“…The application of the WHE [4][5][6][7][8][9][10] aims at finding a truncated series solution to the solution process of a stochastic differential equation. The truncated series composes of two major parts; the first is the Gaussian part which consists of the first two terms, while the rest of the series constitute the non-Gaussian part.…”

Section: Introductionmentioning

confidence: 99%

“…Solutions up to second order are obtained by solving the equivalent deterministic system by an iterative scheme. M. El-Tawil and his coworkers [4][5][6][7][8][9][10] used the WHE together with the perturbation theory (WHEP technique) to solve a perturbed nonlinear stochastic diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation

El-Beltagy

Al-Johani²

2013

ENG

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This paper introduces higher-order solutions of the quadratic nonlinear stochastic oscillatory equation. Solutions with different orders and different number of corrections are obtained with the WHEP technique which uses the WienerHermite expansion and perturbation technique. The equivalent deterministic equations are derived for each order and correction. The solution ensemble average and variance are estimated and compared for different orders, different number of corrections and different strengths of the nonlinearity. The solutions are simulated using symbolic computation software such as Mathematica. The comparisons between different orders and different number of corrections show the importance of higher-order and higher corrected WHEP solutions for the nonlinear stochastic differential equations.

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Solving the random Legendre differential equation: Mean square power series solution and its statistical functions

Calbo¹,

López²,

Jódar³

et al. 2011

Computers & Mathematics with Applications

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In this paper we construct, by means of random power series, the solution of second order linear differential equations of Legendre-type containing uncertainty through its coefficients and initial conditions. By assuming appropriate hypotheses on the data, we prove that the constructed random power series solution is mean square convergent. In addition, the main statistical functions of the approximate solution stochastic process generated by truncation of the exact power series solution are given. Finally, we apply the proposed method to some illustrative examples to compare the numerical results for the average and the variance with respect to those obtained by Monte Carlo approach.

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Approximate solution of a mixed nonlinear stochastic oscillator

Cited by 10 publications

References 31 publications

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

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

Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation

Solving the random Legendre differential equation: Mean square power series solution and its statistical functions

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