scholarly journals Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
J. Calatayud ◽  
J.-C. Cortés ◽  
M. Jornet ◽  
L. Villafuerte
Keyword(s):  
Differential Equations ◽  
Analytic Solutions ◽  
Second Order ◽  
Statistical Properties ◽  
Linear Differential Equations ◽  
Mean Square ◽  
Order Linear ◽  
Linear Differential

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.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

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