The Buchstab’s function and the operational Tau Method

2000 ◽  
Vol 7 (3) ◽  
pp. 673-683 ◽  
Author(s):  
M. Hosseini Aliabadi
Keyword(s):  
Tau Method ◽  
Operational Tau Method

Free response of a continuous vibrational system with attachments and/or discontinuities using segmented operational Tau method

2017 ◽  
Vol 24 (11) ◽  
pp. 2120-2133
Author(s):  
H. Akbarzadeh ◽  
Morteza H. Sadeghi ◽  
F. Talati ◽  
S. Shahmorad
Keyword(s):  
Finite Element Method ◽  
Analytical Techniques ◽  
Constrained Systems ◽  
Mode Shapes ◽  
Tau Method ◽  
The Finite Element Method ◽  
Free Response ◽  
Complicated Case ◽  
Continuity Equations ◽  
Operational Tau Method

It has been widely known that for complicated beam-like structures with various types of attachments and/or discontinuities analytical techniques are not always applicable. In this paper, a very efficient numerical method based on the Tau method is proposed to tackle the mentioned problem. A general form of the linear vibrational eigen-equation, based on the Euler–Bernoulli bending theory, together with its boundary conditions and continuity equations is considered. The problem is then formulated using a segmented form of the operational Tau method which is called the segmented operational Tau method. To investigate the reliability and accuracy of the proposed method some vibrational problems are solved and compared with the analytical solutions providing the exact frequencies and mode shapes. For a complicated case of a non-uniform beam with various types of attachments, since there was no analytical solution, results are validated with the finite element method. This paper has provided a platform for solving free vibrational problems of many complicated constrained systems through a very simple, highly accurate technique.

Development of the Tau Method for the Numerical Solution of Two-dimensional Linear Volterra Integro-differential Equations

2009 ◽  
Vol 9 (4) ◽  
pp. 421-435 ◽  
Author(s):  
A. Tari ◽  
M. Y. Rahimi ◽  
S. Shahmorad ◽  
F. Talati
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Second Order ◽  
Tau Method ◽  
Two Dimensional ◽  
Operational Tau Method ◽  
Standard Base

AbstractIn this paper, we consider a general form of two-dimensional linear Volterra integro-differential equations(TDLVIDE) of the second order with some sup- plementary conditions and develop the operational Tau method with standard base for obtaining a numerical solution.

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.

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