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 ...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.展开更多
This paper proposes a stochastic model to study the evolution of normal and excess weight population between 24 - 65 years old in the region of Valencia (Spain). An approximate solution process of the random model is ...This paper proposes a stochastic model to study the evolution of normal and excess weight population between 24 - 65 years old in the region of Valencia (Spain). An approximate solution process of the random model is obtained by taking advantage of Wiener-Hermite expansion together with a perturbation method (WHEP). The random model takes as starting point a classical deterministic SIS—type epidemiological model in order to improve it in several ways. Firstly, the stochastic model enhances the deterministic one because it considers uncertainty in its formulation, what it is considered more realistic in dealing with a complex problem as obesity is. Secondly, WHEP approach provides valuable information such as average and variance functions of the approximate solution stochastic process to random model. This fact is remarkable because other techniques only provide predictions in some a priori chosen points. As a consequence, we can compute and predict the expectation and the variance of normal and excess weight population in the region of Valencia for any time. This information is of paramount value to both doctors and health authorities to set optimal investment policies and strategies.展开更多
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 u...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.展开更多
In this paper, quadratic nonlinear oscillators under stochastic excitation are considered. The Wiener-Hermite expansion with perturbation (WHEP) method and the homotopy perturbation method (HPM) are used and compared....In this paper, quadratic nonlinear oscillators under stochastic excitation are considered. The Wiener-Hermite expansion with perturbation (WHEP) method and the homotopy perturbation method (HPM) are used and compared. Different approximation orders are considered and statistical moments are computed in the two methods. The two methods show efficiency in estimating the stochastic response of the nonlinear differential equations.展开更多
文摘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.
文摘This paper proposes a stochastic model to study the evolution of normal and excess weight population between 24 - 65 years old in the region of Valencia (Spain). An approximate solution process of the random model is obtained by taking advantage of Wiener-Hermite expansion together with a perturbation method (WHEP). The random model takes as starting point a classical deterministic SIS—type epidemiological model in order to improve it in several ways. Firstly, the stochastic model enhances the deterministic one because it considers uncertainty in its formulation, what it is considered more realistic in dealing with a complex problem as obesity is. Secondly, WHEP approach provides valuable information such as average and variance functions of the approximate solution stochastic process to random model. This fact is remarkable because other techniques only provide predictions in some a priori chosen points. As a consequence, we can compute and predict the expectation and the variance of normal and excess weight population in the region of Valencia for any time. This information is of paramount value to both doctors and health authorities to set optimal investment policies and strategies.
文摘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.
文摘In this paper, quadratic nonlinear oscillators under stochastic excitation are considered. The Wiener-Hermite expansion with perturbation (WHEP) method and the homotopy perturbation method (HPM) are used and compared. Different approximation orders are considered and statistical moments are computed in the two methods. The two methods show efficiency in estimating the stochastic response of the nonlinear differential equations.
