A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

and Applied Analysis 3 To obtain the order of convergence of Legendre rational approximation, at first we define the space H r w,A (Λ) = {V : V is measurable and ‖V‖r,w,A < ∞} , (13) where the norm is induced by ‖V‖r,w,A = ( r ∑ k=0 󵄩󵄩󵄩󵄩󵄩󵄩󵄩 (t + 1) r/2+k d k dt V 󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 w ) , (14) and A is the Sturm-Liouville operator as follows: AV (t) = −w (t) d dt (t d dt V (t)) . (15) We have the following theorem for the convergence. Theorem 1. For any V ∈ H w,A (Λ) and r ≥ 0, 󵄩󵄩󵄩󵄩PNV − V 󵄩󵄩󵄩󵄩 ≤ cN −r ‖V‖r,w,A. (16) A complete proof of the theorem and discussion on convergence are given in [11]. 3. Linear Multipantograph System In this section, we propose the Legendre rational-Gauss collocation method to solve the following system of linear multipantograph equations: m ∑ s=1 β1,su 󸀠 s (t) = m ∑ s=1 a1,s (t) us (t) + l ∑ j=1 m ∑ s=1 b j 1,s (t) uj (λjt) + g1 (t) , m ∑ s=1 β2,su 󸀠 s (t) = m ∑ s=1 a2,s (t) us (t) + l ∑ j=1 m ∑ s=1 b j 2,s (t) uj (λjt) + g2 (t)


Introduction
Over the last three decades, the scientists have paid much attention to spectral methods due to their high accuracy (see, for instance, [1][2][3][4][5][6] and the references therein).On the other hand, spectral methods, in the context of numerical schemes for differential equations, generically belong to the family of weighted residual methods (WRMs) (cf.Finlayson [7]).WRMs represent a particular group of approximation techniques, in which the residuals (or errors) are minimized in a certain way and thereby leading to specific methods including Galerkin, Petrov-Galerkin, collocation, and tau formulations.WRMs are traditionally regarded as the foundation and cornerstone of the finite element, spectral, finite volume, boundary element, and some other methods.Many problems in science and engineering arise in unbounded domains (see, e.g., [8][9][10][11][12]).In general, the use of Jacobi rational functions has the advantage of obtaining the solutions in terms of the Jacobi rational parameters (see, e.g., [13][14][15][16]).Moreover, the authors of [17,18] proposed an efficient collocation schemes based on the operational matrices of rational Legendre and Chebyshev functions for solving problems in the half line.

Legendre Rational Interpolation
In this section, we present Legendre rational functions and Legendre rational approximation that will be used to construct the Legendre rational-Gauss collocation (LR-GC) method.
2.1.Legendre Rational Functions.The well-known Legendre polynomials   () are defined on the interval [−1, 1] with respect to the weight function () = 1.In order to use these polynomials on the interval  ∈ (0, ∞), we recall the Legendre rational functions by introducing the change of variable  = ( − 1)/( + 1).Let the Legendre rational functions   (( − 1)/( + 1)) be denoted by   ().Then   () can be obtained with the aid of the following recurrence formula: According to the properties of the standard Legendre polynomials, we have

Function Approximation.
Let () = 2/( + 1) 2 denote a nonnegative, integrable, real-valued function over the interval Λ = [0, ∞).We define where is the norm induced by the inner product of the space  2  (Λ): Thus, {  ()} ≥0 denotes a system which is mutually orthogonal under (6); that is, where   is the Kronecker delta function.This system is complete in  2  (Λ).For any function  ∈  2  (Λ) the following expansion holds: with The interpolating function of a smooth function  on a semi-infinite interval is denoted by   .It is an element of R  and is defined as is the orthogonal projection of  upon R  with respect to the inner product (6) and the norm (5).Thus by the orthogonality of Legendre rational functions we have [11] ⟨   − ,   ⟩  = 0, ∀  ∈ R  . ( Abstract and Applied Analysis 3 To obtain the order of convergence of Legendre rational approximation, at first we define the space   , (Λ) = {V : V is measurable and ‖V‖ ,, < ∞} , (13) where the norm is induced by and  is the Sturm-Liouville operator as follows: We have the following theorem for the convergence.
Theorem 1.For any V ∈   , (Λ) and  ≥ 0, A complete proof of the theorem and discussion on convergence are given in [11].

Linear Multipantograph System
In this section, we propose the Legendre rational-Gauss collocation method to solve the following system of linear multipantograph equations: subject to Let us first introduce some basic notation that will be used in the sequel.We define the discrete inner product and norm as follows: Obviously, The Legendre rational-Gauss collocation method for solving (17) and ( 18) is to seek We derive the algorithm for solving ( 17)- (18).To do this, let Now, substitution of ( 22) into ( 17) enables us to write Then, by virtue of (3), we deduce that Moreover, the initial condition (18)-with the aid of (2)yields If we collocate (24) at the () Legendre rational roots of  +1 (), then we get 2 Thus (26) with relation ( 25) generate ( + 1) of a set of algebraic equations which can be solved for the unknown coefficients  , , ( = 1, . . ., ;  = 0, 1, 2, . . ., ), by using any standard solver technique.

Nonlinear Multipantograph System
In this section, we consider the nonlinear multipantograph system of the form with initial conditions The Legendre rational-Gauss collocation method for solving (27) and ( 28) is to seek  , () ∈   (0, ∞), such that with (28) written in the form This constitute a system of ( + 1) nonlinear algebraic equations in the unknown expansion coefficients  , , ( = 1, . . ., ;  = 0, 1, 2, . . ., ), which can be solved by using any standard iteration technique, like Newton's iteration method.
Example 1.Consider the following linear multipantograph delay system: with the initial conditions  where The exact solution of the system is  1 () =  − and  2 () = 2 −  − .
Tables 1 and 2 list the results obtained by the Legendre rational-Gauss collocation method in terms of maximum absolute errors with different values of .The logarithmic graphs of absolute coefficients for Legendre rational functions are shown in Figures 1 and 2. This confirms that the proposed method has reasonable convergence rate.
In Tables 3 and 4, we list the absolute errors obtained by the Legedre rational-Gauss collocation method, with different values of .Figures 3 and 4 are plotted to compare the analytic solution with the approximate solution at  = 28.

Conclusion
In this paper, a collocation Legendre rational method has been proposed to obtain the approximate solutions of systems of multipantograph delay equations.The derivation of this method is essentially based on Legendre rational functions and Gauss quadrature formula.The main advantage of the developed method is that high accurate solutions were achieved using few numbers of the Legendre rational functions.Additionally, if  is increased, it can be seen that approximate solutions obtained by the method are close to the exact solutions.

Figure 1 :
Figure 1: Logarithmic graph of absolute coefficients |  | of Legendre rational functions at  = 60 of Example 1.

Figure 2 :
Figure 2: Logarithmic graph of absolute coefficients |  | of Legendre rational functions at  = 60 of Example 1.