Approximate solution of generalized pantograph equations with variable coefficients by operational method

Article History: Received 15 February 2016 Accepted 09 October 2016 Available 12 December 2016 In this paper, we present an efficient direct solver for solving the generalized pantograph equations with variable coefficients. An approach is based on the second kind Chebyshev polynomials together with operational method. The main characteristic behind this approach is that it reduces such problem to ones of solving systems of algebraic equations. Only a small number of Chebyshev polynomials are needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving generalized pantograph equations with variable coefficients.


Introduction
Functional-differential equations with proportional delays are usually referred to as pantograph equations or generalized pantograph equations.Pantograph equations have gained more interest in many application fields such a biology, physics, engineering, economy, electrodynamics [1][2][3][4][5][6][7].In recent years, there has been a growing interest in the numerical treatment of pantograph equations of the retarded and advanced type.A special feature of this type of equation is the existence of compactly supported solutions [6].Pantograph equations are characterized by the presence of a linear functional argument and play an important role in explaining many different phenomena.In particular they turn out to be fundamental when ODEs-based model fail.In the literature, special attention has been given to applications of Taylor polynomials method, variation iteration method, Adomian decomposition method etc. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][25][26][27][28] Consider the generalized linear pantograph equations of the form H js (x)y (s) (α j x − β j ) = g(x) (1) for x ∈ [−1, 1], under the mixed condition, for 1 ≤ c j ≤ 1, i = 0, 1, 2, ..., m − 1 which is the y(x) an unknown function, the known function P k (x), H js (x), g(x) are defined on an interval and also c k ij are appropriate constant.
Our aim is to find an approximate solution expressed in terms of polynomial of degree N in the form *Corresponding Author where a r unknown coefficients and N is chosen any positive integer such that N ≥ m.

Chebyshev polynomial
Orthogonal functions, often used to represent an arbitrary time function, have received considerable attention in dealing with various problems of dynamical system.The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem.
Definition 1.The Chebyshev polynomial of the second kind U n (x) is a polynomial in x of degree n, defined by the relation If the range of the variable x is the interval [−1, 1], the range the corresponding variable θ can be taken [0, π].We suppose without lose of generality that the interval of Eq.( 1) is [−1, 1] which domain of the Chebyshev polynomial of the second kind, since any finite [a, b] can be transformed to interval [−1, 1] by linear maps [23,24].Using Moivre's Theorem we obtained the fundamental recurrence relation [22,23] U which together with the initial conditions These polynomials have the following properties: i) U n+1 (x) has exactly n + 1 real zeroes on the interval ii) These polynomials are orthogonal on [−1, 1] with respect to the weight function ω iii) It is well known that [23] the relation between the powers x n and the Chebyshev polynomials U n (x) is ] can be approximated as a sum of the second kind Chebyshev polynomials as: where, for n = 0, 1, ...

Fundamental matrix relations
Let us write Eq. ( 1) in the form where and We convert these parts and the mixed conditions in to the matrix form.Let us consider the Eq. ( 1) and find the matrix forms of each term of the equation.We first consider the solution y N (x) and its derivative y (k) N (x) defined by a truncated Chebyshev series.Then we can put series in the matrix form where By using (4), we obtained the corresponding matrix relation as follows: where for odd N , ) 0 ) . . .
Moreover it is clearly seen that the relation between the matrix X(x) and its derivative X (k) (x), where The derivative of the matrix U (x) defined in (8), by using the relation ( 9), can expressed as By substituting ( 10) into (8), we obtain, for ) Now, the matrix representation of differential part can be given by We know that; where Using relation (10), we can write In a similarly way as (12) , we obtain So that, the matrix representation of H(x) part can be given by

Method of solution
In this section, we presents the method for solving Eq.( 1) with conditions Eq.( 2).Firstly, we can write the Eq.( 1) follow as: Then, residual R N (x) can be written as Applying typical tau method [29][30][31][32][33], Eq.( 19) can be converted in (N − m) linear or nonlinear equations by applying for n = 0, 1, ..., N − m.The initial conditions are given by where Hence, we obtain the (N +1) sets of linear or nonlinear algebraic equation with (N + 1) unknowns by Eq.( 20) and Eq.( 21).Using the Maple program, we solve the (N + 1) sets of linear or nonlinear algebraic equations with (N + 1) unknowns and so approximate solution y N (x) can be calculated.

Checking of Solution
Likewise we can easily check the accuracy of the obtained solutions as follows: Since the obtained the Chebyshev polynomial of the second kind expansion is an approximate solution of Eq.( 1), when the function y N (x) and its derivatives are substituted in Eq.( 1), the resulting equation must be satisfied approximately; that is for [24] E N (α j x − β j ) − g(x) ∼ = 0

Illustrative example
In this section, several numerical examples are given to illustrate the accuracy and effectiveness of the properties of the method and all of them were performed on the computer using a program written in Maple 13.The absolute errors in tables are the values of N e = |y(x) − y N (x)| at selected points.
Table 2 shows numerical solutions Eq.( 23) with N = 5, 7 and 9 by present method.We see that the approximation solutions obtained by present method has good agreement with exact solution.
In Table 2 compare the absolute errors and E N (x) some selected points.with initial condition y(0) = γ arose in the mathematical modeling of the wave motion in the supply line to an overhead current collector (pantograph) of an electric locomotive [1][2].For values of a = −1,b = −1, q = 0.8 and γ = 1 [8], Table 3 shows solutions of Eq.( 24) with N = 8 by present method.Moreover, the previous results of Walsh series approach (WSA) [34], delayed unit step function series approach (DUSFA) [35], Laguerre series approach (LSA) [36], Taylor series method (TSM) [8] and present method (PM) are also given in Table 3 for comparison.The present method seems more rapidly convergent than Laguerre series and Taylor series and with errors more under control than Walsh or DUSFA series.The truncated errors for Eq.( 24) are O(9) and O( 15) for N = 8 and N = 15 respectively are also indicated.
Example 4. Consider the nonlinear pantograph equation of third order [11,20], which has the exact solution y(x) = sin(x).If we take N = 9, we get the difference between the exact and numerical solutions given in Table 4. Table 4 shows previous results of HPM [20], Adomian decomposition method (ADM) [11] and Present method (PM) for comparison.This shows that the errors are very small.Then, Figure 5  Example 5. consider the equation with y(0) = 1 Here µ 1 (x) = −exp −0.5x sin(0.5x),µ 2 (x) = −2exp −0.75x cos(0.5x)sin(0.25x).It can be seen that the exact solution of Eq.( 26) is y(x) = e −x cos(x).Using present method, we obtain the numerical solution for N = 10.In Figure 6 we give the exact solution and numerical solutions corresponding.

Conclusion
A new method based on the truncated Chebyshev series of the second kind is developed to numerical solve generalized pantograph equations with mixed conditions.Pantograph equations are usually difficult to solve analytically.In many cases, it is required to obtain the approximate solution.For this propose, the present method can be proposed.In this paper, the second kind Chebyshev polynomial approach has been used for the approximate solution of generalized pantograph equations.Thus the proposed method is suggested as an efficient method for generalized pantograph equations.Examples with the satisfactory results are used to demonstrate the application of this method.Suggested approximations make this method rather attractive and contributed to the good agreement between approximate and exact values in the numerical examples for only a few terms.Then examples shows truncated errors, absolute errors and E N (x) are coherent, and performed on the computer using a program written in Maple 13.Moreover, suggested method is applicable for the approximate solution of the pantograph-type integro-differential equations with variable delays.

Figure 1 .
Figure 1.Comparison of exact solution and approximate solutions of Example 1 for various N .

Figure 4 Figure 4 .Example 3 .
Figure 4. Comparison of error functions and E N (x)of Example 2 for various N.

Figure 5 .
Figure 5.Comparison of error function and E 15 (x)of Example 4.

Figure 6 .
Figure 6.Comparison of error function and E 10 (x)of Example 5.

Table 1 .
Mustafa Gülsu was born in Istanbul, Turkey.He received the B.Sc. degrees in Mathematics from Marmara University,Turkey, M.Sc.degrees in Mathematics in Northeastern University,USA, and the Ph.D. degree in mathematics from Ege University, Turkey in 2002.Currently, he is a full professor at Mathematics department of Mugla Sıtkı Koçman University, Turkey.His main research interests are ordinary and partial differential equations, integral and difference equations, and their numerical solutions.Numerical results of Example 1 for different N.

Table 2 .
Numerical results of Example 2 for different N.