Dealing with Dependent Uncertainty in Modelling: A Comparative Study Case through the Airy Equation

J.-C Cortés,J.-V Romero,R.-J Villanueva,M.-D Roselló

doi:10.1155/2013/279642

J.-C Cortés, J.-V Romero + Show 2 more

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https://doi.org/10.1155/2013/279642

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Abstract

The consideration of uncertainty in differential equations leads to the emergent area of random differential equations. Under this approach, inputs become random variables and/or stochastic processes. Often one assumes that inputs are independent, a hypothesis that simplifies the mathematical treatment although it could not be met in applications. In this paper, we analyse, through the Airy equation, the influence of statistical dependence of inputs on the output, computing its expectation and standard deviation by Fröbenius and Polynomial Chaos methods. The results are compared with Monte Carlo sampling. The analysis is conducted by the Airy equation since, as in the deterministic scenario its solutions are highly oscillatory, it is expected that differences will be better highlighted. To illustrate our study, and motivated by the ubiquity of Gaussian random variables in numerous practical problems, we assume that inputs follow a multivariate Gaussian distribution throughout the paper. The application of Fröbenius method to solve Airy equation is based on an extension of the method to the case where inputs are dependent. The numerical results show that the existence of statistical dependence among the inputs and its magnitude entails changes on the variability of the output.

Highlights

Introduction and MotivationDeterministic differential equations have demonstrated to be powerful tools for modelling numerous problems appearing in different areas including physics, chemistry, economy, engineering, and epidemiology
The consideration of uncertainty in models based on differential equations leads to random differential equations
For simplicity, most of these contributions rely on the assumption that random inputs are statistically independent, a hypothesis that could not be met in many applications

Summary

Introduction and Motivation

Deterministic differential equations (ddes) have demonstrated to be powerful tools for modelling numerous problems appearing in different areas including physics, chemistry, economy, engineering, and epidemiology Their practical application requires knowing their inputs (coefficients, forcing terms, initial/boundary conditions, etc.). We will compare, by means of several illustrative examples, the quality of the numerical approximations provided by the three approaches to compute the average and standard deviation of the solution sp to the initial value problem (ivp) (1) These examples will allow us to elucidate, through the random Airy differential equation, whether the statistical independence between the random inputs (initial conditions and coefficients), usually assumed in many applications, has a significant influence on the output.

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