scholarly journals Computing Invariant Sets of Random Differential Equations Using Polynomial Chaos

2020 ◽  
Vol 19 (1) ◽  
pp. 577-618
Author(s):  
Maxime Breden ◽  
Christian Kuehn
Keyword(s):  
Differential Equations ◽  
Polynomial Chaos ◽  
Invariant Sets ◽  
Random Differential Equations

Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

2017 ◽  
Vol 7 (2) ◽  
pp. 306-324
Author(s):  
Chengjian Zhang ◽  
Wenjie Shi
Keyword(s):  
Differential Equations ◽  
Polynomial Chaos ◽  
Piecewise Constant ◽  
Aliasing Error ◽  
Piecewise Constant Argument ◽  
Finite Dimensional ◽  
Discretisation Error ◽  
Random Differential Equations ◽  
Projection Error ◽  
Constant Argument

AbstractWe propose a class of numerical methods for solving nonlinear random differential equations with piecewise constant argument, called gPCRK methods as they combine generalised polynomial chaos with Runge-Kutta methods. An error analysis is presented involving the error arising from a finite-dimensional noise assumption, the projection error, the aliasing error and the discretisation error. A numerical example is given to illustrate the effectiveness of this approach.

scholarly journals Numerical Procedures for Random Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-23 ◽  
Author(s):  
Mohamed Ben Said ◽  
Lahcen Azrar ◽  
Driss Sarsri
Keyword(s):  
Differential Equations ◽  
Polynomial Chaos ◽  
Methodological Approach ◽  
Random Parameter ◽  
Random Coefficients ◽  
Superposition Method ◽  
Random Parameters ◽  
Matrix Formulation ◽  
Random Differential Equations ◽  
Linear Differential

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

On generalized random differential equations

1983 ◽  
Vol 16 (2) ◽  
pp. 469-476
Author(s):  
Andrzej Nowak
Keyword(s):  
Differential Equations ◽  
Random Differential Equations

On the stability of invariant sets of functional differential equations

2003 ◽  
Vol 55 (6) ◽  
pp. 641-656 ◽  
Author(s):  
Stephen R. Bernfeld ◽  
Constantin Corduneanu ◽  
Alexander O. Ignatyev
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Invariant Sets ◽  
Functional Differential ◽  
The Stability
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