Response and Stability of a Random Differential Equation: Part I—Moment Equation Method

1989 ◽  
Vol 56 (1) ◽  
pp. 192-195 ◽  
Author(s):  
H. Benaroya ◽  
M. Rehak
Keyword(s):  
Differential Equation ◽  
Autocorrelation Function ◽  
Moment Equation ◽  
Random Force ◽  
Random Coefficients ◽  
Closed Form Solutions ◽  
Linear Stochastic Differential Equation ◽  
Random Differential Equation ◽  
Stability Bound ◽  
Present Solution

A linear stochastic differential equation of order N excited by an external random force and whose coefficients are white noise random processes is studied. The external force may be either white or colored noise random process. Given the statistical properties of the coefficients and of the force, equivalent statistics are obtained for the response. The present solution method is based on the derivation of the equation governing the response autocorrelation function. The simplifying assumption that the response is stationary when the coefficients and input force are stationary is introduced. Another simplification occurs with the assumption that the response is uncorrelated from the random coefficients. Closed-form solutions for the response autocorrelation function and spectral density are derived in conjunction with a stability bound.

Response and Stability of a Random Differential Equation: Part II—Expansion Method

1989 ◽  
Vol 56 (1) ◽  
pp. 196-201 ◽  
Author(s):  
H. Benaroya ◽  
M. Rehak
Keyword(s):  
Differential Equation ◽  
White Noise ◽  
Colored Noise ◽  
Expansion Method ◽  
Approximate Expression ◽  
Companion Paper ◽  
Random Coefficients ◽  
Random Input ◽  
Linear Stochastic Differential Equation ◽  
Random Differential Equation

A linear stochastic differential equation of order N with colored noise random coefficients and random input is studied. An approximate expression for the autocorrelation of the response is derived in terms of the statistical properties of the random coefficients and input. This is achieved by using an expansion method known as the Born expansion (Feynman, 1962). Feynman diagrams are used as a short hand notation. In the particular case where the coefficients are white noise processes, the expansion method yields identical results to those obtained using an alternate method in a companion paper (Benaroya and Rehak, 1989). The expansion method is also used to demonstrate that white noise coefficients are statistically uncorrelated from the response.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 369-388 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
A. OGROWSKY ◽  
B. SCHMALFUSS
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Random Dynamical System ◽  
Random Delay ◽  
Random Differential Equation ◽  
Autonomous Case ◽  
Random Differential Equations ◽  
Uniqueness Of A Solution

We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.

scholarly journals TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350007 ◽  
Author(s):  
HUIJIE QIAO ◽  
JINQIAO DUAN
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Topological Equivalence ◽  
Random Differential Equation ◽  
Jump Discontinuities ◽  
Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

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