Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing

2022 ◽  
Vol 418 ◽  
pp. 126846
Author(s):  
C. Burgos ◽  
J.-C. Cortés ◽  
L. Villafuerte ◽  
R.J. Villanueva
Keyword(s):  
Laplace Transform ◽  
Linear Equations ◽  
Second Order ◽  
Statistical Computing ◽  
Mean Square ◽  
Order Linear ◽  
The Mean

Mean-Square Response of a Second-Order System to Nonstationary, Random Excitation

1970 ◽  
Vol 37 (3) ◽  
pp. 612-616 ◽  
Author(s):  
L. L. Bucciarelli ◽  
C. Kuo
Keyword(s):  
Narrow Band ◽  
Random Excitation ◽  
Second Order ◽  
Envelope Function ◽  
Order System ◽  
Mean Square ◽  
Forcing Function ◽  
Mean Square Response ◽  
Noise Function ◽  
The Mean

The mean-square response of a lightly damped, second-order system to a type of non-stationary random excitation is determined. The forcing function on the system is taken in the form of a product of a well-defined, slowly varying envelope function and a noise function. The latter is assumed to be white or correlated as a narrow band process. Taking advantage of the slow variation of the envelope function and the small damping of the system, relatively simple integrals are obtained which approximate the mean-square response. Upper bounds on the mean-square response are also obtained.

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

2018 ◽  
Vol 23 (4) ◽  
pp. 76
Author(s):  
Julia Gregori ◽  
Juan López ◽  
Marc Sanz
Keyword(s):  
Differential Equations ◽  
Homogeneous Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mean Square ◽  
Recent Contribution ◽  
Transform Method ◽  
Order Linear ◽  
Random Power Series ◽  
Linear Differential

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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