Modeling the Dependence of Barometric Pressure with Altitude Using Caputo and Caputo–Fabrizio Fractional Derivatives

Muath Awadalla,Kinda Abu Asbeh,Sumit Chandok,Yves Yameni Noupoue Yannick

doi:10.1155/2020/2417681

Abstract

This work is dedicated to the study of the relationship between altitude and barometric atmospheric pressure. There is a consistent literature on this relationship, out of which an ordinary differential equation with initial value problems is often used for modeling. Here, we proposed a new modeling technique of the relationship using Caputo and Caputo–Fabrizio fractional differential equations. First, the proposed model is proven well-defined through existence and uniqueness of its solution. Caputo–Fabrizio fractional derivative is the main tool used throughout the proof. Then, follow experimental study using real world dataset. The experiment has revealed that the Caputo fractional derivative is the most appropriate tool for fitting the model, since it has produced the smallest error rate of 1.74% corresponding to the fractional order of derivative α = 1.005. Caputo–Fabrizio was the second best since it yielded an error rate value of 1.97% for a fractional order of derivative α = 1.042, and finally the classical method produced an error rate of 4.36%.

Highlights

Natural phenomena are commonly used to indicate something that happens randomly or without human action
Mathematical modeling is a powerful tool used over centuries for natural phenomenon study
Two mains approaches are used in mathematical modeling

Summary

Introduction

Natural phenomena are commonly used to indicate something that happens randomly or without human action. Earlier works on fractional differential equations (FDE) have focused on investigation of existence and uniqueness of the solution to designed models. Given a modeling problem that can be solved using differential equations, researchers will first check if there exists classical solution. They will build FDE approach and the obtained results will be evaluated and discussed. Consider a real number c > 0 representing a fractional order of derivative, the Caputo fractional derivative of order c of a function h: [0, +∞] ⟶ R is given by the formula. Consider a real number c > 0 representing a fractional order of integral; the Caputo–Fabrizio fractional integral of a continuous function f: [0, +∞) ⟶ R is defined as follows (see [10]): CF Ic0+.

Literature Review on the Classical Model

Classical method True data

Findings

Caputo method True data