Elsevier

Applied Numerical Mathematics

Volume 131, September 2018, Pages 174-189
Applied Numerical Mathematics

An effective numerical method for solving fractional pantograph differential equations using modification of hat functions

https://doi.org/10.1016/j.apnum.2018.05.005Get rights and content

Abstract

In this work, a spectral method based on a modification of hat functions (MHFs) is proposed to solve the fractional pantograph differential equations. Some basic properties of fractional calculus and the operational matrices of MHFs are utilized to reduce the considered problem to a system of linear algebraic equations. The greatest advantage of using MHFs is the large number of zeros in their operational matrix of fractional integration, product operational matrix and also pantograph operational matrix. This property makes these functions computationally attractive. Some illustrative examples are included to show the high performance and applicability of the proposed method and a comparison is made with the existing results. These examples confirm that the method leads to the results of convergence order O(h3).

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(y(t)β(t)y(qmt))(α)=a(t)y(t)+r=0lbry(αr)(qrt)+g(t),0tτ, with initial conditionsy(i)(0)=y0(i),i=0,1,,m1, using MHFs. Here, y(α)(t) is the Caputo derivative of order α, β(t), a(t) and g(t) are given known functions, y(t) is the unknown function to be determined, br, r=0,1,,l are real constants, m=α is the ceiling function of α, 0α0<α1<<αl<α and 0<qm,qr1, r=0,1,,l.

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 αR, m1<αm, mN and y(t) is a real valued continuous function defined on [0,), then the Caputo fractional derivative of order α>0 is defined byy(α)(t)={1Γ(mα)0t(ts)mα1dmdsmy(s)ds,m1<α<m,y(m)(t),α=m,

where Γ(x) is the gamma function which is defined asΓ(x)=0tx1etdt.

Definition 2.2

The Riemann–Liouville integral operator Itα of order α0 is defined byItαy(t)={1Γ(α)0t(ts)α1y(s)ds,α>0,y(t),α=0.

For α,β0

Properties of MHFs

MHFs consist of a set of linearly independent functions in L2[0,τ] which are obtained after modifying the hat functions [33] and replacing the domain [0,1] by [0,τ]. Considering an even integer number n2 and h=τn, these functions are defined as [20]ψ0(t)={12h2(th)(t2h),0t2h,0,otherwise, if i is odd and 1in1, we haveψi(t)={1h2(t(i1)h)(t(i+1)h),(i1)ht(i+1)h,0,otherwise, if i is even and 2in2, we haveψi(t)={12h2(t(i1)h)(t(i2)h),(i2)htih,12h2(t(i+1)h)(t(i+2)h),iht(i+2)h,

Numerical method

In this section, we suggest a numerical method to solve equation (1) with initial conditions (2). To this aim, suppose thatz(t)=y(t)β(t)y(qmt), then, equation (1) can be rewritten asz(α)(t)=a(t)y(t)+r=0lbry(αr)(qrt)+g(t). Applying the Riemann–Liouville integral operator Itα to both sides of equation (28) and employing (4) yieldz(t)i=0m1z(i)(0)tii!=Itα(a(t)y(t)+r=0lbry(αr)(qrt)+g(t)). On the other hand, using (27), we havez(i)(t)=y(i)(t)k=0i(ik)β(k)(t)qmiky(ik)(qmt). 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 l=0.

Theorem 5.1

[20] Assume that f(t)L2[0,τ] and fn(t) is the MHFs expansion of f(t) defined as fn(t)=i=0nf(ih)ψi(t). Then, the truncation error f(t)fn(t) can be estimated as follows|f(t)fn(t)|=O(h3).

Theorem 5.2

Let y(t)L2[0,τ] 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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