INTRODUCTION

Spectral methods and spectral modelings provide a computational approach which achieved substantial popularity in the last three decades (Ghafarian et al., 2011; Taiwo and Abubakar, 2011; Tasci, 2003). Tau method is one of the most important spectral methods which is extensively applied for numerical solution of many problems. This method was invented by Lanczos (1938) for solving Ordinary Differential Equations (ODEs) and then the expansion of the method were done for many different problems such as Partial Differential Equations (PDEs) (Matos et al., 2004; Kong and Wu, 2009; Doha and Abd-Elhameed, 2005), Integral Equations (IEs) (El-Daou and Khajah, 1997), Integro-Differential Equations (IDEs) (Pour-Mahmoud et al., 2005) and etc. (Rahimi-Ardabili and Shahmorad, 2007; Garcia-Olivares, 2002; Parand and Razzaghi, 2004).

A time delay phenomenon is encountered in a wide variety of scientific and engineering applications, including circuit analysis, computer-aided design, real-time simulation of mechanical systems, chemical process simulation, optimal control, population dynamics and vibrating masses attached to an elastic bar (Hale and Verduyn Lunel, 1993; Taiwo and Odetunde, 2010; Rao et al., 2011; Vanani and Aminataei, 2009, 2010; Kolmanovskii and Myshkis, 1992; Salamon, 1984).

In this study, we are interested in solving NDDSs with an operational approach of the Tau method. Because in the Tau method, we are dealing with a system of equations wherein the matrix of unknown coefficients is sparse and can be easily invertible. Also, the delay parts appearing in the equation are replaced by their operational Tau representation. Then, we obtain a system of algebraic equations wherein its solution is easy.

Operational tau method: In this section, we state some preliminaries and notations using in this study.

For any integrable functions Ψ (x) and φ (x) on (a, b), we define the scalar product <, > by:

where, and ω (x) is a weight function. Let L²_ω [a, b] be the space of all functions f: [a, b] →R with ||f||²_ω < ∞.

The main idea of the method is to seek a polynomial to approximate u (x) ε L²_ω [a,b]. Let ΦX be a set of arbitrary orthogonal polynomial bases defined by a lower triangular matrix Φ and X_x = [1, x, x², ...]^T.

Lemma 1: Suppose that u (x) is a polynomial as , then we have:

(1)

(2)

and:

(3)

where, u = [u₀, u₁, ....,u_n, ....], x_a = [1, a, a², ...]^T, a ε and M, N and P are infinite matrices with only nonzero elements:

Proof: (Liu and Pan, 1999).

Let us consider:

(4)

To be an orthogonal series expansion of the exact solution where, is a vector of unknown coefficients, Φ X_x is an orthogonal basis for polynomials in .

In the Tau method, the aim is to convert the linear and nonlinear terms to an algebraic system using some operational matrices. Therefore, we state the following lemma.

Lemma 2: Let X_x = [1, x, x², ...]^T, u = [u₀, u₁, u₂, ....] be infinite vectors and Φ = [φ₀|φ₁|φ₂|...], φ_i are infinite columns of matrix Φ. Then, we have:

(5)

where, U is an upper triangular matrix as:

(6)

In addition, if we suppose that u (x) = uΦ X_x represents a polynomial, then for any positive integer p, the relation:

(7)

is valid.

Proof: We have:

Therefore:

If we call the last upper triangular coefficient matrix as U, then we have:

Now, in order to prove Eq. 7, we apply induction. For p = 1, it is obvious that u (x) = u Φ X_x. For p = 2 we rewrite u² (x) = u Φ X_x u Φ X_x, = u Φ (X_x u Φ X_x) and using Eq. 5, we have:

Therefore, Eq. 7 is hold for p = 2. Now, suppose that Eq. 7 is hold for p = k, then we must prove that the relation is valid for s = k + 1 . Thus:

So, Eq. 7 is proved.

Application on NDDSs: Let us consider the following NDDS:

(8)

Where:

(9)

is the state vector and:

(10)

Such that and are delay functions; A (x), B (x) and C (x) are m-dimensional matrices which their entries are complex functions of x. Also:

(11)

(12)

represent the initial vector function and known vector function, respectively.

Now the aim is to write u_i (α_i (x)) and (β_i (x)) i = 0, 1..., m; in operational forms. Using Eq. 4, we have:

(13)

We know that X_x = [1, x, x²,...]^T, therefore:

By approximating each power of α_i (x) as we obtain:

If be the last coefficient matrix then we have:

(14)

Substituting Eq. 14 in Eq. 13 we get:

(15)

Also, from Eq. 1 and 4, it is obvious that:

In the same manner from Eq. 13 to 15, there exist the coefficient matrices Δ_i such that:

(16)

In next step we desire to approximate each elements of matrices A (x), B (x) and C (x) in operational forms. Since, each elements of A (x), B (x) and C (x) are smooth functions therefore we can approximate them as follows:

(17)

Substituting above equations in Eq. 8 and using Eq. 2, we obtain:

(18)

Therefore:

(19)

In the same way, we have:

(20)

(21)

The vectors U (x) and F (x) also can be considered as:

(22)

(23)

Thus Eq. 8 is replaced by the following algebraic system:

(24)

So, the residual matrix R (x) of Eq. 8, can be written as:

(25)

Where:

Now, we set the residual matrix or we use the following inner products:

(26)

For supplementary conditions of Eq. 8 we have:

(27)

Therefore, imposing supplementary conditions and setting , a system of algebraic equations is obtained. Since, somewhere we require finite terms of approximation, then we must truncate the series solution to finite number of terms. This is the so-called operational Tau method which is applicable for finite, infinite, regular and irregular domains.

Some shifted orthogonal polynomials: We have considered OTM based on arbitrary orthogonal polynomials. Orthogonal functions can be used to obtain a good approximation for transcendental functions. Since shifted Chebyshev and Legendre polynomials are more applicable orthogonal functions for a wide range of problems therefore, we consider them, briefly.

Shifted chebyshev polynomials: The Chebyshev polynomials are defined on [-1, 1] as:

(28)

or:

and shifted Chebyshev polynomials are defined as:

(29)

Now, we consider the following lemma.

Lemma 3:. Suppose that T and T* are coefficient matrices of Chebyshev polynomials {T_i (x)| x ε [-1, 1], i = 0, 1, 2,...} and shifted Chebyshev polynomials {T*_i (x)| x ε [a, b], i = 0, 1, 2,...} , respectively. Hence, we have:

Where:

with v = 2/b-a and w = a + b/a-b.

Proof: Definition of T states that:

We know that , thus:

If we let Q to be the last coefficient matrix, then:

so

Therefore, the lemma is valid.

Shifted legendre polynomials: The Legendre polynomials on [-1, 1] are defined as:

(30)

and we define shifted Legendre polynomials as:

(31)

In a similar manner with lemma 3 we can prove P* = PC, where, P and P* are coefficient matrices of Legendre and shifted Legendre polynomials, respectively.

Illustrative numerical experiments: In this section, two experiments of NDDSs are given to illustrate the efficiency of the method. In all experiments, we consider the shifted Chebyshev and Legendre polynomials as basis functions and have compared the obtained results with the exact solutions. The computations associated with the experiments discussed above were performed in Maple 14 on a PC with a CPU of 2.4 GHz.

Experiment 1: Consider the following NDDS (Vanani and Aminataei, 2010):

The exact solution in the interval [0, 1] is:

We have solved this experiment by OTM for n = 20 with shifted Chebyshev and Legendre bases and compared with the exact solution. Results are given in Table 1 and 2 for u₁ (x), u₂ (x) and u₃ (x), respectively.

From the numerical results in Table 1 and 2, it is easy to conclude that obtained results by OTM are in good agreement with the exact solution. Also, during the running of programs we find out the run time of OTM is 0.952 sec. Therefore, the algorithm of OTM is fast.

Experiment 2: Consider the following nonlinear NDDS (Vanani and Aminataei, 2009):

The exact solution is:

We have solved this experiment by OTM for n = 20 with shifted Chebyshev and Legendre bases. Results are given in Table 3 and 4 for u₁ (x), u₂ (x), u₃ (x) and u₄ (x), respectively. Numerical results in Tables 3 and 4 illustrate a good agreement between OTM solutions and exact solutions. In this experiment, the run time of OTM is 1.607 sec. Again, we can conclude that OTM is a fast method.

Table 1:	Exact and approximate solutions of u1 (x) and u2 (x) of experiment 1

Table 2:	Exact and approximate solution of u3 (x) of experiment 1

Table 3:	Exact and approximate solution of u1 (x) and u2 (x) of experiment 2

Table 4:	Exact and approximate solution of u1 (x) and u2 (x) of experiment 2

CONCLUSION

In the present study, OTM is proposed for solving NDDSs. Reducing the NDDSs to algebraic equations is the first characteristic of the proposed method. The main idea of the proposed method is to convert the NDDS including linear and nonlinear terms to an algebraic system to simplify the computations. Arbitrary orthogonal polynomial bases were applied as basis functions to reduce the volume of computations. Furthermore, this method yields the desired accuracy only in a few terms in a series form of the exact solution. All of these advantages of the OTM to solve nonlinear problems assert the method as a convenient, reliable and powerful tool.