Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation

In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Numerical solution of time-varying delay systems by Chebyshev wavelets

The solution of time-varying delay systems is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets consisting of wavelets and Chebyshev polynomials are presented. The method is based upon expanding various time functions in the system as their truncated Chebyshev wavelets. The operational matrix of delay is introduced. The operational matrices of integration and delay are utilized to reduce the solution of time-varying delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations

This paper presents a computational method for solving a class of system of nonlinear singular fractional Volterra integro-differential equations. First, existences of a unique solution for under studying problem is proved. Then, shifted Chebyshev polynomials and their properties are employed to derive a general procedure for forming the operational matrix of fractional derivative for Chebyshev wavelets. The application of this operational matrix for solving mentioned problem is explained. In the next step, the error analysis of the proposed method is investigated. Finally, some examples are included for demonstrating the efficiency of the proposed method.     

Numerical Simulation of the Variable Order Fractional Integro-Differential Equation via Chebyshev Polynomials

Mathematical Notes - In this paper, the Chebyshev polynomials method is applied to solve a space-time variable fractional order integro-differential equation. Using operational matrices of...     

The operational matrix of fractional integration for shifted Chebyshev polynomials

A new shifted Chebyshev operational matrix (SCOM) of fractional integration of arbitrary order is introduced and applied together with spectral tau method for solving linear fractional differential equations (FDEs). The fractional integration is described in the Riemann–Liouville sense. The numerical approach is based on the shifted Chebyshev tau method. The main characteristic behind the approach using this technique is that only a small number of shifted Chebyshev polynomials is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient for linear multi-term FDEs.     

Numerical technique based on generalized Laguerre and shifted Chebyshev polynomials

In this study, we present a numerical scheme for solving a class of fractional partial differential equations. First, we introduce psi -Laguerre polynomials like psi-shifted Chebyshev polynomials and employ these newly introduced polynomials for the solution of space-time fractional differential equations. In our approach, we project these polynomials to develop operational matrices of fractional integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provide us the desired results. The convergence of the proposed method is analyzed. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.     

Multi-variable Hermite matrix polynomials: Properties and applications

In this paper, the multi-variable Hermite matrix polynomials are introduced by algebraic decomposition of exponential operators. Their properties are established using operational methods. The matrix forms of the Chebyshev and truncated polynomials of two variable are also introduced, which are further used to derive certain operational representations and expansion formulae.     

Numerical treatment of a well-posed Chebyshev Tau method for Bagley-Torvik equation with high-order of accuracy

The main purpose of this study is to develop and analyze a new high-order operational Tau method based on the Chebyshev polynomials as basis functions for obtaining the numerical solution of Bagley-Torvik equation which has a important role in the fractional calculus. It is shown that some derivatives of the solutions of these equations have a singularity at origin. To overcome this drawback we first change the original equation into a new equation with a better regularity properties by applying a regularization process and thereby the operational Chebyshev Tau method can be applied conveniently. Our proposed method has two main advantages. First, the algebraic form of the Tau discretization of the problem has an upper triangular structure which can be solved by forward substitution method. Second, Tau approximation of the problem converges to the exact ones with a highly rate of convergence under a more general regularity assumptions on the input data in spite of the singularity behavior of the exact solution. Numerical results are presented which confirm the theoretical results obtained and efficiency of the proposed method.     

Adomian decomposition method with orthogonal polynomials: Legendre polynomials

This paper illustrates the using of orthogonal polynomials to modify the Adomian decomposition method. The method of employing Legendre polynomials to improve the Adomian decomposition method is presented here and compared to the method of using Chebyshev polynomials. The presented modified Adomian decomposition method is validated through an example and advantage as well as efficiency of this method is verified through investigating and comparing the results. In this paper, it is concluded that both orthogonal polynomials: Chebyshev and Legendre polynomials can be successfully used for the Adomian decomposition method and comparatively the Chebyshev expansion provides the better estimation.     

Operational Matrix of Fractional Integration Based on the Shifted Second Kind Chebyshev Polynomials for Solving Fractional Differential Equations

This paper is devoted to studying a computational method for solving multi-term differential equations based on new operational matrix of shifted second kind Chebyshev polynomials. The properties of the operational matrix of fractional integration are exploited to reduce the main problem to an algebraic equation. We present an upper bound for the error in our estimation that leads to achieve the convergence rate of O(M ^−κ). Numerical experiments are reported to demonstrate the applicability and efficiency of the proposed method.

     

Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method

In this article we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation. The method consists of expanding the required approximate solution as the elements of shifted Chebyshev polynomials. Using the operational matrices of integral and derivative, we reduce the problem to a set of linear algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.     

A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.     

Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.     

A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation

This paper presents a direct solution technique for solving the generalized pantograph equation with variable coefficients subject to initial conditions, using a collocation method based on Bernoulli operational matrix of derivatives. Only small dimension of Bernoulli operational matrix is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for generalized pantograph equations.     

A note on Chebyshev polynomials

New families of generating functions and identities concerning the Chebyshev polynomials are derived. It is shown that the proposed method allows the derivation of sum rules involving products of Chebyshev polynomials and addition theorems. The possiblity of extending the results to include gnerating functions involving products of Chebyshev and other polynomials is finally analyzed.

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Sunto Si derivano nuove famiglie di funzioni generatrici e di identità relative ai polinomi di Chebyshev. Si dimostra che il metodo proposto permette la derivazione di regole di somma relative a prodotti di polinomi di Chebyshev e teoremi di addizione. La possibilità di estendere i risultati includendos funzioni generatrici di prodotti di polinomi di Chebyshev ed altri polinomi è infine analizzata.

     

A Bessel collocation method for numerical solution of generalized pantograph equations

This article is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, we introduce a collocation method based on the Bessel polynomials for the approximate solution of the pantograph equations. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

关于勒让德多项式与契贝谢夫多项式间的关系

主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用．     

Natural Boundaries for Solutions to a Certain Class of Functional Differential Equations

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation. The pantograph equation contains a linear functional argument. In this paper we generalize this functional argument to include nonlinear polynomials. In contrast to the entire solutions generated by the pantograph equation for the retarded case, we show that in the nonlinear case natural boundaries occur. These boundaries are linked to the Julia set of the polynomial functional argument.     

Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models

A new efficient type of Chebyshev wavelet is used to find the optimal solutions of general linear, continuous-time, multi-delay systems with quadratic performance indices and also to obtain the responses of linear time-delay systems. According to the new definition of Chebyshev wavelets, the operational matrices of integration, product, delay and inverse time and the integration matrix are derived. Furthermore, new operational matrices as the piecewise delay operational matrix and the stretch operational matrix of the desired Chebyshev wavelets are introduced to analyze systems with, in turn, piecewise constant delays and stretched arguments or proportional delays. Two novel algorithms based on newly Chebyshev wavelet method are proposed for the optimal control and the analysis of delay models. Some examples are solved to establish that the accuracy and applicability of Chebyshev wavelet method in delay systems are increased.