Chebyshev polynomials

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1 - A Numerical Method For Solving Physiology Problems By Shifted Chebyshev Operational Matrix
E. Hashemizadeh F. Mahmoodi

In this study, a numerical solution of singular nonlinear differential equations, stemming from biology and physiology problems, is proposed. The methodology is based on the shifted Chebyshev polynomials operational matrix of derivative and collocation. To assess the ac چکیده کامل

In this study, a numerical solution of singular nonlinear differential equations, stemming from biology and physiology problems, is proposed. The methodology is based on the shifted Chebyshev polynomials operational matrix of derivative and collocation. To assess the accuracy of the method, five numerical problems, such as the human head, Oxygen diffusion and Bessel differential equation, were solved. The numerical results were compared with other existed methods in tables for verification. پرونده مقاله

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2 - An ‎E‎ffective Numerical Technique for Solving Second Order Linear Two-Point Boundary Value Problems with Deviating Argument
M. Khaleghi E. Babolian S. Abbasbandy

Based on reproducing kernel theory, an effective numerical technique is proposed for solving second order linear two-point boundary value problems with deviating argument. In this method, reproducing kernels with Chebyshev polynomial form are used (C-RKM). The convergen چکیده کامل

Based on reproducing kernel theory, an effective numerical technique is proposed for solving second order linear two-point boundary value problems with deviating argument. In this method, reproducing kernels with Chebyshev polynomial form are used (C-RKM). The convergence and an error estimation of the method are discussed. The efficiency and the accuracy of the method is demonstrated on some numerical examples. پرونده مقاله

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3 - Numerical solution of general nonlinear Fredholm-Volterra integral equations using Chebyshev ‎approximation
F. Fattahzadeh

A numerical method for solving nonlinear Fredholm-Volterra integral equations of general type is presented. This method is based on replacement of unknown function by truncated series of well known Chebyshev expansion of functions. The quadrature formulas which we use t چکیده کامل

A numerical method for solving nonlinear Fredholm-Volterra integral equations of general type is presented. This method is based on replacement of unknown function by truncated series of well known Chebyshev expansion of functions. The quadrature formulas which we use to calculate integral terms have been imated by Fast Fourier Transform (FFT). This is a grate advantage of this method which has lowest operation count in contrast to other early methods which use operational matrices (with huge number of operations) or involve intermediate numerical techniques for evaluating intermediate integrals which presented in integral equation or solve special case of nonlinear integral equations. Also rate of convergence are given. The numerical examples show the applicability and accuracy of the &lrm;method.&lrm; پرونده مقاله

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4 - Application of Chebyshev Polynomials for Solving Abel's Integral Equations of the First and Second Kind
Ahmad Shahsavaran M. M. Shamivand

In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interva چکیده کامل

In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval ofintegration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which leads to approximating the unknown function and it's derivatives in terms of Chebyshev polynomials of the first kind. The proposed method, transforms the Abel's integral equations of the first and second kind into a system of linear algebraic equations which can be solved by Gaussian elimination algorithm. Finally, some numerical examples are included to clarify the accuracy and applicability of the presented method which indicate that proposed method is computationally very attractive. In thispaper, all numerical computations were carried out on a PC executing some programs written in maple software. پرونده مقاله

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5 - Shifted Chebyshev Approach for the Solution of Delay Fredholm and Volterra Integro-Differential Equations via Perturbed Galerkin Method
Kazeem Issa Jafar Biazar Babatunde Yisa

The main idea proposed in this paper is the perturbed shifted Chebyshev Galerkin method for the solutions of delay Fredholm and Volterra integrodiﬀerential equations. The application of the proposed method is also extended to the solutions of integro-diﬀerential diﬀeren چکیده کامل

The main idea proposed in this paper is the perturbed shifted Chebyshev Galerkin method for the solutions of delay Fredholm and Volterra integrodiﬀerential equations. The application of the proposed method is also extended to the solutions of integro-diﬀerential diﬀerence equations. The method is validated using some selected problems from the literature. In all the problems that are considered, the new proposed approach performs better than many other methods. پرونده مقاله

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6 - A Chebyshev functions method for solving linear and nonlinear fractional differential equations based on Hilfer fractional derivative
M. H. Derakhshan A. Aminataei

The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we needan efficient and accurate computational method for the solution of fractional differential equa چکیده کامل

The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we needan efficient and accurate computational method for the solution of fractional differential equations.This paper presents a numerical method for solving a class of linear and nonlinear multi-order fractional differential equations with constant coefficients subject to initial conditions based on the fractional order Chebyshev functions that this function is defined as follows:\begin{equation*}\overline{T}_{i+1}^{\alpha}(x)=(4x^{\alpha}-2)\overline{T}_{i}^{\alpha}(x)\overline{T}_{i-1}^{\alpha}(x),\,i=0,1,2,\ldots,\end{equation*}where $\overline{T}_{i+1}^{\alpha}(x)$ can be defined by introducing the change of variable $x^{\alpha},\,\alpha>0$, on the shifted Chebyshevpolynomials of the first kind. This new method is an adaptation of collocationmethod in terms of truncated fractional order Chebyshev Series. To do this method, a new operational matrix of fractional order differential in the Hilfer sense for the fractional order Chebyshev functions is derived. By using this method we reduces such problems to those ofsolving a system of algebraic equations thus greatly simplifying the problem. At the end of this paper, several numerical experiments are given to demonstrate the efficiency and accuracy of the proposed method. پرونده مقاله

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7 - On a modication of the Chebyshev collocation method for solving fractional diffiusion equation
Hosein jalebbonab Hojatollah Adibi

In this article a modification of the Chebyshev collocation method is applied to the solution of space fractional differential equations.The fractional derivative is considered in the Caputo sense.The finite difference scheme and Chebyshev collocation method are used .T چکیده کامل

In this article a modification of the Chebyshev collocation method is applied to the solution of space fractional differential equations.The fractional derivative is considered in the Caputo sense.The finite difference scheme and Chebyshev collocation method are used .The numerical results obtained by this way have been compared with other methods.The results show the reliability and efficiency of the proposed method. پرونده مقاله

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