Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials

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Abstract

In this article we propose a numerical scheme to solve the pantograph equation. The method consists of expanding the required approximate solution as the elements of the shifted Chebyshev polynomials. The Chebyshev pantograph operational matrix is introduced. The operational matrices of pantograph, derivative and product are utilized to reduce the problem to a set of algebraic equations. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results.

Highlights

► In the current paper a numerical technique is proposed. ► Numerical approximation is based on shifted Chebyshev polynomials. ► The operational matrices of derivative and product are utilized. ► The new technique is used to solve several test examples. ► Model is a delay differential equation which has many applications.

Introduction

Delay systems have been largely used to describe propagation and transport phenomena or population dynamics [31], [35]. In economical systems, delays appear in a natural way since decisions and effect (investment policy, commodity markets evolution, etc.) are separated by some time intervals [37]. In communication or neural network model, data or axon signal transmission is always accompanied by a nonzero time interval between the initial- and the delivery-time of a message or signals [37]. In a mathematical framework, such systems may often be described by differential equations on functional spaces [22].

The pantograph equation is one of the most important kinds of delay differential equations, and plays an important role in explaining many different phenomena. The delay pantograph differential equations arise in many applications such as electro-dynamic. The Doughty spring Pantograph has been designed to be lightweight and compact, with strength, safety and ease of use being the main features [47]. Loads from 5 to 36 kg can be suspended from overhead tracks or barrels and quickly positioned with height ranges of 2.2–5 m. A studio pantograph [11] is shown in Fig. 1, which is taken from [47].

The (quantitative and qualitative) theory of linear pantograph-type delay differential equations with multiple delays have been analyzed by many authors, but we briefly review a limited number of them. In [11], the pantograph equation was investigated using the Adomian decomposition method [12] and the convergence of the approach for this equation was established. The paper [32] studied the structure of the exact solution sets of multi-pantograph delay differential equations and proved the existence and uniqueness of an exact solution. Authors of [21] considered the stability of one-leg θ-methods for the solution of the pantograph equation. The properties of the analytic and numerical solutions of the multi-pantograph equation presented in [34]. The authors of [48] presented an analysis of the linear multi-step and one-leg methods applied to a scalar pantograph equation of neutral type. In [33] the asymptotical stability of the analytic and numerical solutions with constant step size for pantograph equations were investigated using the Razumikhin technique. Authors of [26] proposed a piecewise (2m,m)-rational approximation with quasi-uniform meshes which corresponds to the mth collocation method for the pantograph differential equation.

The paper [49] investigated the stability of the Runge–Kutta (RK) method for a family of pantograph equations of neutral type with different proportional delays. It is proved that the even-stage Gauss–Legendre methods are not asymptotically stable, but the Radau IA methods, Radau IIA methods and Lobatto IIIC methods are all asymptotically stable. Authors of [2] proposed a method for the integration of pantograph equation, by a super-convergent s-stage continuous RK method of discrete global order p and continuous uniform order q<p-1 for the approximation of the delayed term. Their proposed scheme is based on a particular choice of the mesh which was introduced by [4] in connection with the θ-method.

Authors of [25] proposed a piecewise (2m,m)-rational approximation with quasi-uniform meshes which corresponds to the mth collocation method for the pantograph differential equations. This method is more useful than the known collocation method in the case that a long time integration is needed; that is, if T is large, then the number of steps in this method will be less than that of the collocation method. Numerical experiments are also given in [25]. To compute long term integrations for the pantograph differential equation with proportional delay, Ishiwata and Muroya [25], presented two kinds of numerical methods by using special mesh distributions, that is, a rational approximant with quasi-uniform meshes and a Gauss collocation method with ’quasi-constrained meshes. In [38] the variational iteration method [13] is applied to solve the generalized pantograph equation. The technique of [38] provides a sequence of functions which converges to the exact solution of the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. In [30], the pantograph equation is investigated using the homotopy perturbation [39] and variational iteration methods.

The aim of [30] is based on the search for a solution in the form of a series with easily computed components. Authors of [27] proposed a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods [17]. A Legendre-collocation method employed to obtain highly accurate numerical approximations to the exact solution. Authors of [28] analyzed the convergence properties of the spectral method for pantograph-type differential and integral equations with multiple delays. Hsiao [23] first proposed Haar product matrix. By Haar wavelets Hsiao and Wu [24] solved time-varying systems. Motivated by these results, Wang [44] presented the general Legendre wavelets to solve the time-varying systems.

A numerical method based on the Taylor polynomials introduced in [41] for the approximate solution of the linear pantograph equation. Authors of [42] applied the Taylor method to approximate solution of the non homogenous multi-pantograph equation with variable coefficients. Also we refer the interested reader to [39], [40] for delay differential or integral equations. Our approach in the current paper is different.

In this paper we introduce a new direct computational method to solve pantograph equations. This method consists of reducing the pantograph problem to a set of algebraic equations by expanding the candidate function as a Chebyshev function with unknown coefficients.

The current paper is organized as follows: in Section 2, we describe the basic properties of the Chebyshev functions. Chebyshev operational matrix of pantograph is introduced. This matrix together with the operational matrix [16] of derivative and product are used to evaluate the coefficients of the Chebyshev function for the solution of pantograph equations. In Section 3, we apply the proposed numerical method to the pantograph equations. In Section 4, we provide error analysis for the proposed method. An error analysis is included in order to allow the number of polynomials to be selected based on some desired tolerance. In Section 5, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples. The conclusion is given in Section 6.

Section snippets

Properties of shifted Chebyshev polynomials

The well-known Chebyshev polynomials are defined on the interval z[-1,1] and can be determined with the aid of the following recurrence formula [18]:Tn(z)=2zTn-1(z)-Tn-2(z),n=2,3,,with T0(z)=1 and T1(z)=z. For practical use of Chebyshev polynomials on interval of interest x[a,b], it is necessary to shift the defining domain by means of the following substitution:z=2x-a-bb-a,axb.The shifted Chebyshev polynomials in x are then obtained as:T0(x)=1,T1(x)=2x-(b+a)b-a,Tn(x)=22x-a-bb-aTn-1(x)-Tn-2

Numerical solution

Consider the following pantograph equation of order m:u(x)=a(x)u(x)+r=1lbr(x)u(qrx),axb,with the initial conditionu(a)=ua,where 0qr1,a(x) and br(x) are known functions. It will be assumed that the given functions in (7) are smooth on their respective domains. This implies that the solution of (7) under the initial condition (8) is (globally) smooth on [a,b].

We now expand u(x),u(x) and u(qrx) by Chebyshev functions as follows:u(x)=TT(x)C.From (3) we haveu(x)=TT(x)MTCand by using (6), (9)

Error analysis

This section covers the error analysis of the proposed method. The error estimation of the method proposed in this work is based on those obtained in [8].

Numerical examples

In this section, some numerical examples will be demonstrated to compare the new method with the previous results.

Conclusion

The Chebyshev polynomials and the associated operational matrices of derivative M, product QC and the pantograph operational matrix D are used to solve the pantograph equations. The method is based upon reducing the system into a set of algebraic equations. The matrices M and D have many zeros, hence, the method is computationally attractive. They also reduce the CPU time and the computer memory, at the same time keeping the accuracy of the solution. The method has been successfully applied to

Acknowledgments

Authors are very grateful to one of the reviewers for carefully reading the paper and for his(her) comments and suggestions which have improved the paper.

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