Shifted Legendre series solution and parameter estimation of linear delayed systems

a b s t r a c tThe main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.

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“…(10) and (11). Thus for a fixed delay τ we have [29] Φ(x − τ ) = HΦ(x) x > τ , (17) where H is a constant lower triangular matrix defined as…”

Section: Properties Of Shifted Legendre Polynomialsmentioning

confidence: 99%

“…Recently there have been several published works in the literature on the applications of the tau method [25][26][27][28]. For more details of Legendre polynomials see [29,30] and also some technique for solving integrodifferential equations can be found in [31,32].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients

Saadatmandi

Dehghan

2010

Computers & Mathematics with Applications

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“…In general, the computed response of the delay systems via orthogonal functions is not in good agreement with the exact response of the system [2]. Special attention has been given to applications of Walsh functions [3], block pulse functions [4], Laguerre polynomials [5], Legendre polynomials [6], Chebyshev polynomials [7], Haar wavelets [8], Fourier series [9] and hybrid functions [10,11].…”

Section: Introductionmentioning

confidence: 99%

Analysis of time-varying delay systems via triangular functions

Hoseini

Marzban

2011

Applied Mathematics and Computation

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“…Using operational matrices, the technique is based on reduction of these problems to systems of algebraic equations. Special attention has been given to applications of Walsh functions [16], block-pulse functions [17], Laguerre polynomials [18], Legendre polynomials [19], Chebyshev polynomials [20], and Fourier series [21].…”

Section: Introductionmentioning

confidence: 99%

Optimal Control of a Delayed HIV Infection Model via Fourier Series

Ghanbari

Farahi

2014

Journal of Nonlinear Dynamics

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We present a delayed optimal control which describes the interaction of the immune system with the human immunodeficiency virus (HIV) and CD4+ T-cells. In order to improve the therapies, treatment and the intracellular delays are incorporated into the model. The optimal control in this model represents the efficiency of drug treatment in preventing viral production and new infections. The optimal pair of control and trajectories of this nonlinear delay system with quadratic cost functional is obtained by Fourier series approximation. The method is based on expanding time varying functions in the nonlinear delay system into their Fourier series with unknown coefficients. Using operational matrices for integration, product, and delay, the problem is reduced to a set of nonlinear algebraic equations.

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Shifted Legendre series solution and parameter estimation of linear delayed systems

Cited by 35 publications

References 9 publications

Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients

Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients

Analysis of time-varying delay systems via triangular functions

Optimal Control of a Delayed HIV Infection Model via Fourier Series

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