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.

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 Efficient Statistical Extraction of the Per-Unit-Length Capacitance and Inductance Matrices of Cables with Random Parameters

2015 ◽  
Vol 4 (1) ◽  
pp. 22 ◽  
Author(s):  
P. Manfredi ◽  
F. Canavero
Keyword(s):  
Polynomial Chaos ◽  
Unit Length ◽  
Random Parameter ◽  
Random Parameters ◽  
Memory Consumption ◽  
Problem Size ◽  
Shielded Cable ◽  
Alternative Approach ◽  
Parameter Variations ◽  
The Impact

Cable bundles often exhibit random parameter variations due to uncertain or uncontrollable physical properties and wire positioning. Efficient tools, based on the so-called polynomial chaos, exist to rapidly assess the impact of such variations on the per-unit-length capacitance and inductance matrices, and on the pertinent cable response. Nevertheless, the state-of-the-art method for the statistical extraction of the per-unit-length capacitance and inductance matrices of cables suffers from several inefficiencies that hinder its applicability to large problems, in terms of number of random parameters and/or conductors. This paper presents an improved methodology that overcomes the aforementioned limitations by exploiting a recently-published, alternative approach to generate the pertinent polynomial chaos system of equations. A sparse and decoupled system is obtained that provides remarkable benefits in terms of speed, memory consumption and problem size that can be dealt with. The technique is thoroughly validated through the statistical analysis of two canonical structures, i.e. a ribbon cable and a shielded cable with random geometry and position.

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