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Operational Tau Approximation for Neutral Delay Differential Systems



J. Sedighi Hafshejani, S. Karimi Vanani and J. Esmaily
 
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ABSTRACT

Neutral Delay Differential Systems (NDDSs) arise in many areas of various mathematical modeling. Infectious diseases, population dynamics, physiological and pharmaceutical kinetics and chemical kinetics, the navigational control of ships and mechanical systems, chemical process simulation and optimal control are the main field concerning with NDDSs. The purpose of this study was to present an extension of the algebraic formulation of the Operational Tau Method (OTM) for the numerical solution of NDDSs. The proposed method converts the delay parts of the desired NDDS to some operational matrices. Then the NDDS reduces to a set of algebraic equations. Some orthogonal bases including shift Chebyshev and shifted Legendre polynomials are used to decrease the volume of computations. Two illustrative linear and nonlinear experiments are included to show the high accuracy and efficiency of the proposed method.

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  How to cite this article:

J. Sedighi Hafshejani, S. Karimi Vanani and J. Esmaily, 2011. Operational Tau Approximation for Neutral Delay Differential Systems. Journal of Applied Sciences, 11: 2585-2591.

DOI: 10.3923/jas.2011.2585.2591

URL: https://scialert.net/abstract/?doi=jas.2011.2585.2591
 
Received: March 18, 2011; Accepted: May 04, 2011; Published: June 07, 2011



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:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

where, Image for - Operational Tau Approximation for Neutral Delay Differential Systems and ω (x) is a weight function. Let L2ω [a, b] be the space of all functions f: [a, b] →R with ||f||2ω < ∞.

The main idea of the method is to seek a polynomial to approximate u (x) ε L2ω [a,b]. Let Image for - Operational Tau Approximation for Neutral Delay Differential SystemsΦX be a set of arbitrary orthogonal polynomial bases defined by a lower triangular matrix Φ and Xx = [1, x, x2, ...]T.

Lemma 1: Suppose that u (x) is a polynomial as Image for - Operational Tau Approximation for Neutral Delay Differential Systems, then we have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(1)

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(2)

and:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(3)

where, u = [u0, u1, ....,un, ....], xa = [1, a, a2, ...]T, a ε Image for - Operational Tau Approximation for Neutral Delay Differential Systems and M, N and P are infinite matrices with only nonzero elements:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

Proof: (Liu and Pan, 1999).

Let us consider:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(4)

To be an orthogonal series expansion of the exact solution where, Image for - Operational Tau Approximation for Neutral Delay Differential Systems is a vector of unknown coefficients, Φ Xx is an orthogonal basis for polynomials in Image for - Operational Tau Approximation for Neutral Delay Differential Systems .

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 Xx = [1, x, x2, ...]T, u = [u0, u1, u2, ....] be infinite vectors and Φ = [φ012|...], φi are infinite columns of matrix Φ. Then, we have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(5)

where, U is an upper triangular matrix as:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(6)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(7)

is valid.

Proof: We have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

Therefore:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

So, Eq. 7 is proved.

Application on NDDSs: Let us consider the following NDDS:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(8)

Where:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(9)

is the state vector and:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(10)

Such that Image for - Operational Tau Approximation for Neutral Delay Differential Systems and Image for - Operational Tau Approximation for Neutral Delay Differential Systems are delay functions; A (x), B (x) and C (x) are m-dimensional matrices which their entries are complex functions of x. Also:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(11)

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(12)

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

Now the aim is to write uii (x)) and Image for - Operational Tau Approximation for Neutral Delay Differential Systemsi (x)) i = 0, 1..., m; in operational forms. Using Eq. 4, we have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(13)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

By approximating each power of αi (x) as Image for - Operational Tau Approximation for Neutral Delay Differential Systems we obtain:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

If Image for - Operational Tau Approximation for Neutral Delay Differential Systems be the last coefficient matrix then we have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(14)

Substituting Eq. 14 in Eq. 13 we get:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(15)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(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:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(17)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(18)

Therefore:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(19)

In the same way, we have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(20)

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(21)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(22)

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(23)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(24)

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(25)

Where:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

Now, we set the residual matrix Image for - Operational Tau Approximation for Neutral Delay Differential Systems or we use the following inner products:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(26)

For supplementary conditions of Eq. 8 we have:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(27)

Therefore, imposing supplementary conditions and setting Image for - Operational Tau Approximation for Neutral Delay Differential Systems, 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:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(28)

or:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

and shifted Chebyshev polynomials are defined as:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(29)

Now, we consider the following lemma.

Lemma 3:. Suppose that T and T* are coefficient matrices of Chebyshev polynomials {Ti (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:

T* = TQ

Where:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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

Proof: Definition of T states that:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

We know that Image for - Operational Tau Approximation for Neutral Delay Differential Systems, thus:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

so

T* = TQ

Therefore, the lemma is valid.

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(30)

and we define shifted Legendre polynomials as:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems
(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):

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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):

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

The exact solution is:

Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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
Image for - Operational Tau Approximation for Neutral Delay Differential Systems

Table 2: Exact and approximate solution of u3 (x) of experiment 1
Image for - Operational Tau Approximation for Neutral Delay Differential Systems

Table 3: Exact and approximate solution of u1 (x) and u2 (x) of experiment 2
Image for - Operational Tau Approximation for Neutral Delay Differential Systems

Table 4: Exact and approximate solution of u1 (x) and u2 (x) of experiment 2
Image for - Operational Tau Approximation for Neutral Delay Differential Systems

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.

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