Singularity Treatment in the Bidimensional Tau Method with an Application to Problems Defined on L-Shaped Domains

1988 ◽  
pp. 179-187
Author(s):  
Eduardo L. Ortiz
Keyword(s):  
Tau Method

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

An efficient Chebyshev-tau method for solving the space fractional diffusion equations

2013 ◽  
Vol 224 ◽  
pp. 259-267 ◽  
Author(s):  
Rong-fen Ren ◽  
Hou-biao Li ◽  
Wei Jiang ◽  
Ming-yan Song
Keyword(s):  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Tau Method ◽  
Fractional Diffusion Equations

scholarly journals Note on the Solution of Transport Equation by Tau Method and Walsh Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Abdelouahab Kadem ◽  
Adem Kilicman
Keyword(s):  
Transport Equation ◽  
Legendre Polynomials ◽  
Neutron Transport ◽  
Three Dimensional ◽  
Walsh Function ◽  
Dimensional Case ◽  
Angular Variable ◽  
Tau Method ◽  
Algebraic Equations ◽  
Neutron Transport Equation

We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem.

A tau method based on Jacobi operational matrix for solving fractional telegraph equation with Riesz-space derivative

2020 ◽  
Vol 39 (4) ◽  
Author(s):  
Samira Bonyadi ◽  
Yaghoub Mahmoudi ◽  
Mehrdad Lakestani ◽  
Mohammad Jahangiri Rad
Keyword(s):  
Operational Matrix ◽  
Telegraph Equation ◽  
Riesz Space ◽  
Tau Method ◽  
Fractional Telegraph Equation

A Convergence Result For The Tau Method

1990 ◽  
Vol 23 (4) ◽  
Author(s):  
H.-G. Roos ◽  
E. Pfeifer
Keyword(s):  
Convergence Result ◽  
Tau Method
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