Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials

2013 ◽  
Vol 36 (1) ◽  
pp. 15-38 ◽  
Author(s):  
W.M. Abd-Elhameed ◽  
E.H. Doha ◽  
Y.H. Youssri
Keyword(s):  
Differential Equations ◽  
Jacobi Polynomials ◽  
Galerkin Methods ◽  
Generalized Jacobi Polynomials ◽  
Fifth Order

scholarly journals An Accurate Spectral Galerkin Method for Solving Multiterm Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
A. H. Bhrawy ◽  
A. S. Alofi
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Differential Equations ◽  
Jacobi Polynomials ◽  
Initial Conditions ◽  
Solution Technique ◽  
Caputo Fractional Derivatives ◽  
Polynomial Degree ◽  
Generalized Jacobi Polynomials ◽  
Fractional Differential

This paper reports a new formula expressing the Caputo fractional derivatives for any order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. A direct solution technique is presented for solving multiterm fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using spectral shifted generalized Jacobi Galerkin method. The homogeneous initial conditions are satisfied exactly by using a class of shifted generalized Jacobi polynomials as a polynomial basis of the truncated expansion for the approximate solution. The approximation of the spatial Caputo fractional order derivatives is expanded in terms of a class of shifted generalized Jacobi polynomialsJnα,−β(x)withx∈(0,1), andnis the polynomial degree. Several numerical examples with comparisons with the exact solutions are given to confirm the reliability of the proposed method for multiterm FDEs.

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