An effective numerical method for solving fractional pantograph differential equations using modification of hat functions
Introduction
In recent years, fractional calculus has been regarded as an effective tool for investigating the behavior of many phenomena in science and engineering. Some examples of applications of fractional calculus have been given in [1], [19], [22], [23]. Among all of the fractional models, fractional differential equations have attracted an increasing attention because of their widely applications. Some remarkable works on the theoretical analysis for this class of equations have been considered in [2], [9], [34]. Nevertheless, it is very difficult to obtain analytic solutions for fractional differential equations. Therefore, many researchers have developed numerical methods to obtain numerical solutions for them (see for example [5], [6], [10], [11], [12], [13], [14], [16], [17], [26], [27], [36], [37], [39]).
A delay differential equation (DDE) is a differential equation where the state variable appears with delayed argument [15]. Fractional DDEs appear in many fields of science such as economy and biology, automatic control, biology and hydraulic networks and long transmission lines [18]. One of the most important classes of DDEs is the class of pantograph differential equations. Many numerical methods have been proposed to solve integer-order and fractional-order DDEs. Dehghan and Salehi in [7] have applied semi-analytical approaches to solution of a nonlinear time-delay model in biology. In [8], the Adomian decomposition method has been considered for solving a DDE arising in electrodynamics. Also, this method has been employed for solving the pantograph equation of order m in [30]. An analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation has been given in [31] for nonlinear systems of Volterra integro-differential equations with delay arguments. Sedaghat et al. [29] have used Chebyshev polynomials to solve integer-order DDEs of pantograph type. A collocation method based on Bernoulli polynomials basis has been presented for solving integer-order pantograph DDEs in [32]. Yu [38] considered a variational iterative method for solving the multi-pantograph DDEs. Hermite wavelet methods have been suggested to solve fractional DDEs in [28]. Yang and Huang [35] have used spectral collocation methods to obtain the numerical solution for fractional pantograph delay integro-differential equation. Legendre multiwavelet collocation method has been proposed for solving linear fractional time delay systems. Rahimkhani et al. have considered Bernoulli wavelets for solving fractional DDEs [25] and generalized fractional-order Bernoulli wavelet for solving pantograph DDEs [24].
The main aim of this work is to propose a numerical method to solve the following fractional pantograph differential equation with initial conditions using MHFs. Here, is the Caputo derivative of order α, , and are given known functions, is the unknown function to be determined, , are real constants, is the ceiling function of α, and , .
The current paper is organized as follows: in Section 2, we give some definitions and preliminaries of fractional calculus. Section 3 is devoted to the properties of MHFs and introducing their operational matrices. In Section 4, we suggest a new method to solve problem (1)–(2) using the properties of MHFs. In Section 5, a criterion to test the reliability of the results obtained by the proposed method is given. In Section 6, the numerical results given by implementing the new method are reported for some examples. Finally, conclusion is given in Section 7.
Section snippets
Preliminaries of fractional calculus
In this section, we give some preliminaries and definitions of fractional calculus.
Definition 2.1 Consider that , , and is a real valued continuous function defined on , then the Caputo fractional derivative of order is defined by Definition 2.2 The Riemann–Liouville integral operator of order is defined by
For
Properties of MHFs
MHFs consist of a set of linearly independent functions in which are obtained after modifying the hat functions [33] and replacing the domain by . Considering an even integer number and , these functions are defined as [20] if i is odd and , we have if i is even and , we have
Numerical method
In this section, we suggest a numerical method to solve equation (1) with initial conditions (2). To this aim, suppose that then, equation (1) can be rewritten as Applying the Riemann–Liouville integral operator to both sides of equation (28) and employing (4) yield On the other hand, using (27), we have Utilizing (27),
Error estimate
The purpose of this section is to obtain an estimate of the error of the numerical solution obtained by the presented method in the previous section. For ease of exposition in this paper, we will describe error analysis only for the case of .
Theorem 5.1 [20] Assume that and is the MHFs expansion of defined as . Then, the truncation error can be estimated as follows Theorem 5.2 Let be the exact solution of the fractional
Illustrative examples
This section is devoted to solving some examples in order to show the applicability, accuracy and high performance of the proposed method. The computations were performed on a personal computer using a 2.50 GHz processor and the codes were written in Mathematica 11. The final systems have been solved by employing the Mathematica Solve function and the computing time consumed for solving these systems (in seconds) is reported in the examples. For the sake of introducing the convergence rate of
Conclusion
The MHFs and their operational matrices have been used to introduce a new numerical method to solve the fractional pantograph differential equations. The operational matrix of fractional integration and the pantograph operational matrix are Hessenberg matrices and the product operational matrix is a diagonal matrix, therefore, the method is computationally attractive (even with a large number of basis functions). Moreover, the system matrix has a simple structure, described by (36). As a
Acknowledgements
P. Lima acknowledges support from FCT, through grant SFRH/BSAB/135130/2017.
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