In the last few decades, it has been shown that fractional calculus appears
in modeling of many real-world phenomena, in different branches of science
where nonlocality plays an essential role, such as physics, chemistry, biology,
economics, engineering, signal and image processing, and control theory
Agarwal et al. (2015); Baleanu et al. (2012),
Malinowska et al. (2015); Morales-Delgado et al. (2019),
Rekhviashvili et al. (2019); Tariboon et al. (2015). This is mainly due to the fact that fractional operators consider the evolution of a system, by taking the global correlation, and not only local characteristics Almeida et al. (2019). As a result, fractional systems have attracted the attention of many researchers in different areas. Nevertheless, obtaining analytic solutions for such problems is very difficult. So, in most cases, the exact solution is not known, and it is necessary to find a numerical approximation. Therefore, many researchers have worked on numerical methods to obtain some numerical solutions of fractional dynamic systems (see, e.g., Ali et al. (2019); Jafarian et al. (2018, 2017); El-Sayed and Agarwal (2019); Nigmatullin and Agarwal (2019)).
Hat basis functions consist of a set of piecewise continuous functions with shape of hats,
when plotted in two dimensional planes. These functions are usually defined on the
interval and a generalization of them is obtained by extending the interval
into with any arbitrary positive number . Hat basis functions have
shown to be a powerful mathematical tool in solving many different kinds of equations.
For example, in Babolian and Mordad (2011), hat functions have been used to solve linear and
nonlinear integral equations of second kind. These functions have been also
efficiently employed to solve fractional differential equations (FDEs) Tripathi et al. (2013),
while in Heydari et al. (2014) generalized hat functions are used for solving stochastic
Itô–Volterra integral equations. Recently, a modification
of hat functions has been introduced and used in order to solve a variety of problems.
To mention some of these problems, we refer to two-dimensional linear Fredholm integral
equations Mirzaee and Hadadiyan (2015), integral equations of Stratonovich–Volterra Mirzaee and Hadadiyan (2016a)
and Volterra–Fredholm type Mirzaee and Hadadiyan (2016b), and fractional integro-differential
Nemati and Lima (2018), fractional pantograph nonlinear differential equations
Nemati et al. (2018) and fractional optimal control problems Nemati et al. (2019). The aim of our work is to have a comparison between these two classes of hat basis functions in solving systems of FDEs.
The paper is organized as follows. Section 2 is devoted to the required preliminaries for presenting our numerical technique. In Section 3, we present the new numerical method, which is based on two classes of hat functions for solving systems of FDEs. Section 4 is concerned with an application of the method for solving a problem in epidemiology related to human respiratory syncytial virus. Finally, concluding remarks are given in Section 5.
In this section, some necessary definitions and properties of fractional calculus are presented. Moreover, two classes of hat functions and some of their properties are recalled.
2.1 Preliminaries of fractional calculus
In this work, we employ fractional differentiation in the sense of Caputo, which is defined via the Riemann–Liouville fractional integral.
Podlubny (1999) The (left) Riemann–Liouville fractional integral operator with order of a given function is defined as
where is the Euler gamma function.
Podlubny (1999) The (left) Caputo fractional derivative of order of a function is defined as
For , , we recall two important properties of the Caputo derivative and Riemann–Liouville integral:
2.2 Hat functions
We consider both generalized and modified hat functions.
Generalized hat functions. The interval is divided into subintervals , , of equal length , where . Then, the generalized hat functions (GHFs) are defined as follows Tripathi et al. (2013):
These functions form a set of linearly independent continuous functions in , satisfying the property
An arbitrary function can be approximated using the GHFs as follows:
be the GHFs basis vector given by (4) and . Then,
where is a matrix of dimension called the operational matrix of fractional integration of order of the GHFs. This matrix is given as Tripathi et al. (2013)
Modified hat functions. By considering an even integer number , the interval is divided into subintervals , , with equal length . The modified hat functions (MHFs) form a set of linearly independent functions in . These functions are defined as follows Nemati and Lima (2018); Nemati et al. (2018):
is odd and, then
if is even and , then
The following property is satisfied for the set of MHFs:
Any function may be approximated using the MHFs as
3 Main results
We consider the following general initial value problem, described by a system of FDEs of order :
The aim is to seek functions , …, solution of (12) on the interval .
3.1 Numerical method
In order to find a numerical solution of (12), we consider approximations of the fractional derivative of the unknown functions as follows:
where are the coefficients vectors with the unknown elements , ,
and is given by (6) for the GHFs, or (11) for the MHFs. Therefore, by setting in (15) and employing (2) or (7), we obtain . Now, we can obtain approximations of the functions , by using the considered basis functions, as follows:
where or , and
which gives the following system:
This system includes nonlinear algebraic equations with unknown parameters, which are the elements of , . By solving this system, approximations of the functions are given by (15).
3.2 Complexity of the resulting systems
The speed of the numerical method given above depends on the speed of solving system (18). Therefore, the form of this system is an important aspect of our method. Here, we display the form of the system given in (18) in detail, for each of the basis functions.
Case 1: GHFs.
According to the form of the operational matrix of fractional integration of the GHFs, given by (6), we can rewrite this matrix as follows:
. As it can be seen in (20a), the values of the unknown parameters , , are obtained easily by using the initial conditions. By substituting the given into (20b), we have a system of nonlinear algebraic equations in unknown parameters , . After solving this system, and substituting the obtained results for into the next equations, a system of equations in , , is obtained. This process continues until , , are found by solving (20c), in which the results for , , …, have been substituted. Therefore, our method, based on GHFs, reduces the main problem to the solution of systems of nonlinear algebraic equations.
Case 2: MHFs.
In a similar way as for GHFs, we rewrite the operational matrix of fractional integration of MHFs as follows:
. It is seen that the values of the unknown parameters , , are given easily by substituting the initial conditions into (23a). By substituting the given results of into (23b) and (23c), a system of nonlinear algebraic equations in the unknown parameters and , , is obtained. After solving this system, and substituting the obtained results for and into the next equations, a system of algebraic equations in and , , is given. By continuation of this process, we find and , , by solving (23d) and (23e), in which the results for , , …, have been substituted. Our method based on MHFs reduces the system of nonlinear FDEs to solving systems of nonlinear algebraic equations.
3.3 Test problems
In order to illustrate the efficiency and accuracy of the proposed method, we apply it to two test problems whose exact solutions are known. Consider the following system of FDEs:
on the interval , which has the exact solution
We have solved this problem by the suggested method, based on GHFs and MHFs, with different values of , and display the numerical results in Figure 1, Table 1, and Figure 2. In Figure 1, the approximate solutions of and , obtained by employing our method with , are shown. In this figure, the numerical results of and , obtained by GHFs, are, respectively, displayed by and . Also, the numerical results of the unknown functions given by MHFs are represented by and . In Table 1, we see the numerical results for the functions and , with different values of , together with the CPU time (in seconds), which have been obtained on a 2.5 GHz Core i7 personal computer with 16 GB of RAM using Mathematica 11.3. For solving the resulting systems of algebraic equations, the Mathematica function FindRoot was used. In this table, the following notations are used for introducing the error and the convergence order of the method:
where and are the exact solutions, and are the approximate solutions obtained with GHFs, and and are the approximate solutions obtained with MHFs. These results confirm the accuracy order of the numerical method with GHFs and the accuracy order of the numerical method with MHFs. Finally, in Figure 2 (left), the results for the errors obtained by employing our method, for some selected values of , are plotted in a logarithmic scale. Moreover, the CPU times of the method are plotted in Figure 2 (right). It can be seen that the computational complexity of the resulting systems of GHFs and MHFs are similar.
|CPU Time||CPU Time|
Consider the following system of linear FDEs:
on the interval . The exact solution of this problem, when , is Atkinson et al. (2009)
We set and solve the problem. By considering the same notations as introduced in Example 3.3, we report the numerical results in Table 2. Furthermore, in Figures 3 and 4, the numerical solutions based on GHFs and MHFs, obtained by different values of and , together with the exact solution with , are plotted. As it could be expected, the numerical solution is close to the exact solution of the corresponding first order problem when is close to .
4 Application to the SEIRS- epidemic model
4.1 Description of the model
Human respiratory syncytial virus (HRSV) is a virus that causes respiratory tract infections. We refer a reader interested in this virus to Rosa and Torres (2018) and references therein. Here we just mention that HRSV is a principal cause of lower respiratory tract infections and hospital visits during infancy and childhood. There is an annual epidemic in temperate climates during the winter season, while in tropical climates this infection is most common throughout the rainy season. Since protective immunity is induced by natural infection with HRSV more than many other respiratory viral infections, people can be infected multiple times even within a single HRSV season.
A mathematical model can show how the infectious disease with HRSV progresses and what are the outcomes of an epidemic of this virus. Recently, a compartmental model was proposed in Rosa and Torres (2018), based on stratifying the population into four health states: susceptible to the infection, denoted by ; a group of individuals who have been infected but are not infectious yet, which become infectious at a rate ; infected and infectious, denoted by ; and recovered individuals . A particular property of HRSV is that immunity after infection is temporary, i.e., the recovered individuals become susceptible again Weber et al. (2001), hence, the model is called a SEIRS model. The authors of Rosa and Torres (2018) considered that the annual recruitment rate is seasonal due to schools opening/closing periods and proposed the following system of FDEs:
with given initial conditions
where denotes the birth rate, which was assumed equal to the mortality rate, is the rate of loss of immunity, is the rate of loss of infectiousness, denotes the transmission parameter, which is modeled by the cosine function as , in which is the mean of and is the amplitude of the seasonal fluctuation, is the recruitment rate, which includes newborns and immigrants, with as the amplitude of the seasonal fluctuation, and where denotes the left Caputo derivative of order . In the parameters and , the parameter is an angle that is chosen in agreement with real data. Note that by introducing the group , a latency period is included in the model, which is assumed equal to the time between infection and the first symptoms.
|CPU Time||CPU Time|