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

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%.

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

“…3) and (4), respectively. Looking at the MSE (see [36]) of each model, Caputo derivative-based model has the best performance, with MSE = 0.03095, and α = 1.00360. The latest value is evidence that the error is minimized when the fractional order of the derivative doesn't coincide with the first-order classical derivative.…”

“…This section focuses on some FC concepts required for the rest of the paper. Definition 2.1 [35,36]. Given τ > 0, a positive real number, the one-parameter Mittag-Leffer function is defined as follows:…”

“…3) and (4), respectively. Looking at the MSE (see [36]) of each model, Caputo derivative-based model has the best performance, with MSE = 0.20992, and α = 1.22335. The Classical method and the Caputo-Fabrizio both occur as a second-best model with the same MSE = 0.63265.…”

Some Existence Results for a System of Nonlinear Sequential Fractional Differential Equations with Coupled Nonseparated Boundary Conditions

“…Other application examples are biology, finances, etc. (see [18][19][20] for illustration). Some interesting general results on boundary value problems are found in [3,[10][11][12][13][14][15][16][17] and [21][22][23].…”

On System of Mixed Fractional Hybrid Differential Equations

“…As a final example, the authors in [6] employed the fractional derivatives of the Caputo and Caputo-Fabrizio type by modeling the equation that gives the relationship between atmospheric pressure and altitude, and they were also able to show that the fractional equation gave less error in estimating atmospheric pressure at a certain altitude. There are many scientific papers in the literature that prove the superiority of fractional derivatives over classical ones.…”