Iterated solutions of linear operator equations with the Tau method

Mathematical and Computer Modelling

Soleymani

2013

a b s t r a c tIn this paper, we present a numerical solution of weakly singular Volterra integral equations including the Abel's equations by the Tau method with arbitrary polynomial bases. The Tau method produces approximate polynomial solutions of differential, integral and integro-differential equations. An extension of the Tau method has been done for the numerical solution of weakly singular Volterra integral equations, especially for Abel's equations. The integral part appearing in the equation is replaced by its operational Tau representation. An efficient error estimation of the Tau method is also introduced. Some numerical experiments are given to demonstrate the superior performance of the Tau method. The Chebyshev and Legendre bases are also introduced to improve the results.

Section: Introductionmentioning

confidence: 99%

Tau approximate solution of weakly singular Volterra integral equations

Mathematical and Computer Modelling

Soleymani

2013

“…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 [13] 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) [14]- [16], integral equations (IEs) [17], integro-differential equations (IDEs) [18] and etc. [19]- [22].…”

Section: Introductionmentioning

confidence: 99%

Operational Tau approximation for a general class of fractional integro-differential equations

Aminataei

2011

Comput. Appl. Math.

Abstract. In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, error estimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method.Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

“…The theoretical analysis and numerical applications of this method have been described in a series of papers see [19][20][21][22][23] for linear ordinary differential eigenvalue problems. The method was developed for the numerical solution of PDEs and their related eigenvalue problems, iterated solutions of linear operator equations [24][25][26][27][28] and for IDEs [29], too.…”

Section: Introductionmentioning

confidence: 99%

Tau approximate solution of fractional partial differential equations

Computers & Mathematics with Applications

Aminataei

2011

a r t i c l e i n f o Keywords: Spectral methods Operational approach of the Tau method Improved algebraic formulation Fractional partial differential equations a b s t r a c tIn this study, we improve the algebraic formulation of the fractional partial differential equations (FPDEs) by using the matrix-vector multiplication representation of the problem. This representation allows us to investigate an operational approach of the Tau method for the numerical solution of FPDEs. We introduce a converter matrix for the construction of converted Chebyshev and Legendre polynomials which is applied in the operational approach of the Tau method. We present the advantages of using the method and compare it with several other methods. Some experiments are applied to solve FPDEs including linear and nonlinear terms. By comparing the numerical results obtained from the other methods, we demonstrate the high accuracy and efficiency of the proposed method.