New Spectral Solutions for High Odd-Order Boundary Value Problems via Generalized Jacobi Polynomials

2017 ◽  
Vol 40 (4) ◽  
pp. 1393-1412
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Generalized Jacobi Polynomials ◽  
Odd Order

scholarly journals Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Third Order ◽  
Generalized Jacobi Polynomials ◽  
Nonlinear Algebraic Equations ◽  
Derivatives Of

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

scholarly journals On Triple Series Equations Involving Series of Jacobi Polynomials

1967 ◽  
Vol 15 (3) ◽  
pp. 221-231 ◽  
Author(s):  
K. N. Srivastava
Keyword(s):  
Boundary Value Problems ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problems ◽  
Dual Series Equations

Recently Collins (2) has studied triple series equations involving series of Legendre polynomials. These equations arise in the study of mixed boundary value problems and can be regarded as extensions of the dual series equations considered by Collins in (1).

scholarly journals Spectral approximations to the fractional integral and derivative

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Changpin Li ◽  
Fanhai Zeng ◽  
Fawang Liu
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Caputo Derivative ◽  
Initial Value Problems ◽  
Fractional Integral ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Initial Value ◽  
Numerical Examples ◽  
Spectral Approximations

AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

scholarly journals On Lyapunov-type inequalities for odd order boundary value problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mustafa Fahri Aktaş ◽  
Devrim Çakmak ◽  
Abdullah Ahmetoğlu
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Odd Order
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