Advanced Numerical and Semi-Analytical Methods for Differential Equations

19Differential Equations with Interval Uncertainty

19.1 Introduction

Generally, differential equations are the backbone of various physical systems occurring in a wide variety of disciplines viz. physics, chemistry, biology, economics, and engineering [1,2]. These physical systems are modeled either by ordinary or partial differential equations. Generally, in differential equations, the involved coefficients and variables are considered as deterministic or exact values. In that case, one may handle such differential equations (with deterministic coefficients or variables) by known analytic or numerical methods [3–5].

In actual practice, due to errors in experimental observation or due to truncation of the parametric values, etc. we may have only imprecise, insufficient, or incomplete information about the involved parameters of the differential equations. So, the parametric values involved in such differential equations are uncertain in nature. As such, there is a need of modeling different physical problems with uncertain parameters. In general, these uncertainties may be modeled through probabilistic, interval, or fuzzy approach [4, 5 ]. Probabilistic methods may not be able to deliver reliable results at the required condition without sufficient data. Therefore, in the recent years, interval analysis and fuzzy set theory have become powerful tools for uncertainty modeling.

In this chapter, we present different approaches to handle differential equations with interval uncertainty. ...