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 = 1.005. Caputo–Fabrizio was the second best since it yielded an error rate value of 1.97% for a fractional order of derivative = 1.042, and finally the classical method produced an error rate of 4.36%.
1. Introduction
Natural phenomena are commonly used to indicate something that happens randomly or without human action. Besides, there exist many phenomena that occur as result of experiment. In either case, scientists are always interested in describing phenomena. Mathematical modeling is a powerful tool used over centuries for natural phenomenon study. Following time or space axes, occurrence of a phenomenon can be trendless, or it can show known mathematical trend. Whether or not a phenomenon shows a trend over elapsed time or space, there is always at least one mathematical tool that can be used for its description and modeling. In general, two mains approaches are used in mathematical modeling. These are deterministic and nondeterministic methods. Deterministic approach is mostly concerning with differential equations [1]. The method specificity is that given an initial value problem, other values can be predicted and computed using a function which is solution to a differential equation. Derivatives and Integrals are main tools used in finding solution to differential equations. Integer values were used as order of derivative until earlier 19th century [2] when the idea of fractional derivative was introduced. Earlier works on fractional differential equations (FDE) have focused on investigation of existence and uniqueness of the solution to designed models.
In recent decades, interest has increased among researchers who investigate efficiency of FDE in solving real-life problems. The approaches used are usually similar. Given a modeling problem that can be solved using differential equations, researchers will first check if there exists classical solution. Next, they will build FDE approach and finally the obtained results will be evaluated and discussed. Some interesting applied results are found in physics [3, 4], medical sciences [5], biology [6, 7], and economics [8], which represent a very tiny part of the existing literature.
In this work, a relationship between air pressure and altitude is studied using FDE. Caputo–Fabrizio fractional derivative is used to prove the existence and uniqueness of the proposed model. Moreover, the Caputo method is used in numerical simulation. Hence, it is first proven that an FDE model is appropriate to solve the problem, and secondly, experimental studies are carried out using two different types of fractional derivative and numerical solutions since there was no analytic form of the solution.
2. Elements of Fraction Calculus
This section is dedicated to the study of some elements of fractional calculus which are required in sequel of this work.
Definition 1. (see [9]). Consider a real number ; the one parameter Mittag–Leffler function is computed as where and represents the usual gamma function defined as .
The Mittag–Leffler function is useful in fractional calculus since it is often used for representing the solution of FDE.
Definition 2. (see [9]). Consider a real number representing a fractional order of derivative, the Caputo fractional derivative of order of a function is given by the formulawith being a strictly positive integer, that is .
Definition 3. (see [9]). Consider a function and ; it follows that a new version of the Caputo fractional derivative is defined asMoreover, if the function f is such that , then another new fractional derivative known as Caputo–Fabrizio fractional derivative is obtained as follows:
Definition 4. (see [9]). Consider a real number representing a fractional order of integral; the Caputo–Fabrizio fractional integral of a continuous function is defined as follows (see [10]):where is the normalization function having the property .
Definition 5. (see [9]). The Riemann–Liouville fractional integral of order for a function is defined asprovided that the right-hand side of the integral is pointwise defined on the open interval .
Definition 6. (see [9]). The Riemann–Liouville fractional derivative of order of a function is given bywhere .
3. Literature Review on the Classical Model
In this section, a survey of existing works carried out on the study of variation of air pressure as function of altitude is provided. Air resistance is highly considered in aerodynamic and mechanics. In August 1960, a US army captain known as Joseph Kittinger [11] has experimentally proven the effect of air resistance during a free fall he did from 102,800 feet above the earth surface. His main interest was to study the air resistance during his jump. From various studies on the air resistance, it was proven that given a free-falling object, the air resistance is not a perfect function of the object speed, but, in many cases, the moving speed is often enough as a variable to describe the resistance. In general, if the falling object has a speed , then air resistance is well approximated bywhere and are constants. It has been proven also that if the object speed is above 24 m/s but below the sound speed, the linear term of equation (7) can be neglected and the air resistance is approximated by
In the full report of the experiment by Joseph Kittinger [11], air pressure and resistance depend on altitude. In their work, Grigorie et al. [12] studied the measurement of an aircraft altitude using the air pressure. More important fact is that the authors described the atmospheric layers and how each layers’ temperature and pressure are characterized. Standard Atmosphere is a reference manual adopted by International Civil Aviation Organization [13] and the US National oceanic and atmospheric administration [14]. It provides standardized data for variation of air pressure as a function of altitude. West, in his work [15], stated two main reasons why the barometric air pressure declines with an increasing of altitude. The physical principles behind these reasons are given by the following hydrostatic equation:and the ideal gas law:where h represents altitude; is the density and it depends on altitude; is the gravity acceleration at height h; R is the universal gas constant; M is the molecular weight of the air; is the pressure; and T is the absolute temperature.
Note: it is common to denote the pressure and the density by and , respectively, as they are functions of altitude.
A solution to differential equation (9) is obtained prior to establishment of the relation between and . The gas state equation is given bywhere is the number of moles.
On the contrary, the relationships between the air mass m, volume V, mole mass M, and density are defined as
Equations (12) and (13) inserted into equation (11) leads to the following relation:
A substitution of equation (14) into equation (9) leads to
It is easy to obtain the solution of equation (15) through simple integration as follows:
Using the initial condition stated as at h = 0 , it follows that the analytic solution to equation (15) is
4. Modeling the Pressure with Fractional Derivative
This section describes a model of the relationship between air pressure and altitude using fractional derivative is proposed. Assume that the derivative term in the differential equation (15) is taken in the fractional sense. Without loss of generality, let it be the Caputo fractional derivative; then, the following model is obtained:where is the fractional order of derivative.
Existence and uniqueness to the problem equation (18) is proven as follows prior to numerical simulation.
Applying the Caputo–Fabrizio fractional integral to equation (18) leads to
For conformity and simplicity in notation, let the kernel function be chosen as
Denote by the Banach space of all functions f, such that f are continuous from the closed interval to ; moreover, let the space be endowed with the norm defined as . Let an operator be defined as follows:
Equations (20) and (21) are tools used in sequel process of proving existence and uniqueness of the solution. Beside these equations are also some lemmas which are also set as preliminaries tools in the process of proving the existence and uniqueness of the solution to the proposed fractional model equation (18).
Lemma 1. (Nonlinear alternative of Leray–Shauder type, see [6]).
Given the open subset of a Banach space , , and be a contraction such that is bounded then(i) has a fixed point in (ii) and such that holds.
Lemma 2. (Arzela–Ascoli Theorem, see [6]). is compact if and only if it is closed, bounded, and equicontinuous.
Lemma 3. (Krasnoselskii’s Theorem, see [6]). Given the Banach space , closed convex , is open, where , and , assume that can be written as . In addition, if it is a bounded set in satisfying(i) is continuous and completely continuous(ii) is a contraction, a continuous nondecreasing function , with , such that , for any Then, it follows that(i)(ii) and with
Lemma 4. (Banach’s contraction mapping principle, see [6]).
Let be a complete metric space; if is a contraction, then(i)G has a unique fixed point , that is, (ii), we have , with
4.1. Existence of Solutions
Theorem 1. Let be a continuous function defined in a way that the following two assumptions hold:(i) such that (ii)In addition, let it be assumed that the following relation holds on the given quantity . Having the above setting implies the existence of at least one solution to the initial value problem given by equation (18).
Proof. of Theorem 1. Consider the close set with the radius defined such that .
Moreover, let us introduce the following two operators, and , defined within the domain of and given as follows:Given two elements then it follows hence,
The next step at this point of the proof is to show that operator is a contraction. One can observe first of all that and the following relation holds:Then,Equation (24) clearly shows that operator is a contraction.
On the contrary, operator is continuous as a result of the continuity of . Operator is uniformly bounded as .
The compactness of operator is also proven as follows. For , it follows thatThe right-hand side of the inequality defined by equation (25) approaches zero as . Moreover, on can also observe that the quantity does not depend on , which implies the relative compactness of the operator . Recall Lemma 2 is enough to conclude that is a compact operator on the closed set . Moreover, recalling Lemma 3, the existence of a solution to equation (18) is thus proven. Q.E.D.
4.2. Uniqueness of the Solution
Theorem 2. Let be a continuous satisfying (i) and assume moreover that ; then, the solution to the initial value problem equation (18) is unique.
Proof. of Theorem 2. Consider the close set withThe first step of the proof is to show that . To do so, observe that , and it follows thatOn the contrary, the following inequality is derived from the quantity or norm of :Using equations (27) and (28) leads to the following relation:Equation (29) implies that which means that, in general, .
The second step of this proof is to show that the operator is a contraction. For that, observe that , and the following relation holds:The relation given by equation (30) leads toFrom equation (31), it is clearly observed that operator is a contraction. Hence, using Lemma 4, the solution to the initial value problem defined by equation (18) is unique over the closed interval Q.E.D.
Having proven the existence and the uniqueness of a solution to the problem defined by equation (18), it is important to investigate the shape of the said solution.
Lemma 5. (see page 135 in [9]). Given , denote by the integer part of and . Consider the initial value problem defined bywith , , and is a given function. Equation (32) has a solution expressible in the following form:withwhere , , and .
Note: if in Lemma 5 is such that , then solution to the problem equation (32) is expressed using equations (33) and (34) asUsing Lemma 5, the analytic form of the exact solution to equation (18), with the derivative taken in the Caputo sense is given by
Lemma 6. (see [16]). Let and be a natural number such that ; if , then the unique solution of the following initial value problemis given bywhere is the primitive of and .
Considering the Caputo–Fabrizio fractional derivative, the initial value problem equation (18) becomesRecalling Lemma 6, the analytic form of the exact solution to equation (39) is written as
5. Simulation Studies
The experimental dataset used in this section was retrieved from Sable Systems International website [16], where real data from an experiment that investigates the relationship between altitude and atmospheric pressure is given. The measurement unit of the pressure is KPa. Altitude is given in meters. The experimental dataset contains negative as well as positive values of altitude. However, we conducted the study using only data from the positive altitude. The constant values which are in equations (15), (36), and (39) are the following ; = 9.82; M = 29; R = 8.31; T = 273000; T0 = 293000; and . The Caputo model equation (36) and the Caputo–Fabrizio model equation (40) were used to fit the true dataset. Prior to fitting the data, optimization routines were used to determine the fractional order of derivative for which each model would best fit the dataset. The routine used here consists of using all the possible value of the fractional order of derivative in the interval [0.9, 1.2] with a moving step of 10-2, while recording alongside the error percentage for each value.
Figure 1 shows that the Caputo method has a singularity for = 0.999, in fact the error rate produced by = 0.999 suddenly jumped to 705.7%. Figure 2, on the contrary, shows the Caputo–Fabrizio error rate variation over the closed interval [0.9, 1.2]. Unlike the Caputo method, the curve depicting the error rate is smoothed over the interval. This means there is no singularity for this method. Interestingly, the minimum error, 1.97% appears at = 1.042. Recall that when = 1, this is the classical case.
Figures 3 and 4 depict the graphs of atmospheric pressure variation with altitude. Figure 3 shows the predicted data using the classical method alongside the true data, whereas Figure 4 shows the predicted data using the Caputo–Fabrizio method (= 1.042) alongside the true dataset. It is clearly observed from both figures that modeling with the Caputo–Fabrizio method is better than with the classical method.
Recall from Figure 1 that the Caputo method reaches the minimum error at = 1.005 and a singularity at = 0.99. Figure 5 shows the predicted data using the Caputo method (= 1.005) alongside the true data. One can clearly see that the error is very small. In order to observe prediction behavior at a singular point, Figure 6 was plotted using predicted data by the Caputo method (= 0.99) alongside the true dataset. The graph shows a weird behavior due to singularity.
6. Physical Meaning and Motivation of the Study
This section provides physical meaning of the relationship between altitude and atmospheric pressure. It also motivates the study. Recall that accuracy is the main metric of phenomena modeling being deterministic or stochastic. The relationship between altitude and atmospheric pressure and their consequences are well known. The said relationship finds its application in some of our daily activities. For instance, an airplane flies at high altitude and its atmospheric pressure decreases with altitude prompting pilot to use special pressurizing instruments to maintain the pressure of the plane at a convenient level. Failure to regulate a plane atmospheric pressure is a serious threat to the passenger’s lives. A sad history of aviation happened in 2005, precisely on Sunday 14th August 2005 when the crew of Helios flight 522 en route to Athens from Cyprus omitted to activate the pressurizing instrument on board. The plane was then deprived from oxygen and crashed [17, 18]. A morality of the Helios flight 522 case is that aviation needs to provide airplane with sophisticate barometers able produce the atmospheric pressure as a function of plane altitude with high accuracy.
Another field of application where controlling atmospheric pressure is required is foods processing and its supply chain. Indeed, we live in globalization era, thus human and goods are subject to long range traveling over the world. It is common, for instance, for one living in western part of the world to consume dates (fruit) that was produce in the Eastern part. That is just a basic example to highly the fact in foods cans supply chain, atmospheric pressure is a determine element to consider. In case there is high atmospheric pressure different between the manufacturing hub and the end market, special care must be taken for the goods to remain safe.
In this work, the statement “classical approach” is used to refer to the relationship between altitude and atmospheric pressure model as introduced in [11–15], whereas “fractional approach” refers to the proposed model built upon fractional derivative. The main result of this work is that “fractional approach” can best fit the model than “classical approach.” A real-life application of the results obtained in this work is that the proposed model can be used to produce pressure related devices such barometers.
7. Conclusion and Discussion
The aims of this work were all reached. In fact, fractional differential equation was used to build and to solve altitude-pressure relationship which is well known in the classical literature. Without loss of generality, Caputo–Fabrizio fractional derivative was used to prove the existence and uniqueness of a proposed fractional model of atmospheric pressure variation with respect to altitude.
After proving that the fractional differential equation model of the relationship altitude-pressure was well defined as well as that it has a unique solution, it was worth to find that solution. Caputo and Caputo–Fabrizio methods were both selected to express the solution of the defined problem.
In the numerical simulation, the performance of each of the approach used was evaluated. The Caputo method showed singularity points for some values of the fractional order of derivative which were close to 1. However, it has best fit the experimental data since it has yield an error rate which was as smaller as 1.74% for a fractional order of derivative = 1.005. This performance was followed by the Caputo–Fabrizio method which for = 1.042 has produced a minimum error rate of 1.97%. Finally, the classical approach has produced an error rate of 4.36%. As a matter of fact, the Caputo fractional derivative or the Caputo–Fabrizio fractional derivative performed better than their classical counterpart in solving the problem defined in this work. This result can be added to the nonexhaustive list of problems for which fraction calculus approach occurred to be more efficient than existing classical approach in finding their solutions.
Data Availability
The dataset used in this study was retrieved from the following website: https://www.sablesys.com/support/technical-library/barometric-pressure-vs-altitude-table/.
Disclosure
The authors planned to present preliminaries results of this work at Southeastern-Atlantic Regional Conference on Differential Equations on October 26-27, 2019, at Embry-Riddle Aeronautical University, Daytona Beach Campus [19]. However, due to reasons beyond control, the authors could not make it to the conference venue; hence, the project was aborted.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors, M. A, Y. Y. Y, and K. A., contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgments
The authors extend their appreciation to the Deputyship for Research & Innovation Ministry of Education in Saudi Arabia for funding this research work through the project no. IFT20135.