New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials

2012 ◽  
Vol 64 (3) ◽  
pp. 373-394 ◽  
Author(s):  
E. H. Doha ◽  
W. M. Abd-Elhameed ◽  
A. H. Bhrawy
Keyword(s):  
Differential Equations ◽  
Jacobi Polynomials ◽  
Direct Solution ◽  
Generalized Jacobi Polynomials ◽  
Even 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.

A nonlocal problem for a differential operator of even order with involution

2020 ◽  
Vol 26 (2) ◽  
pp. 297-307
Author(s):  
Petro I. Kalenyuk ◽  
Yaroslav O. Baranetskij ◽  
Lubov I. Kolyasa
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

scholarly journals Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 285
Author(s):  
Saad Althobati ◽  
Jehad Alzabut ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Order Equation ◽  
Second Order ◽  
Riccati Technique ◽  
Neutral Equations ◽  
Neutral Differential Equations ◽  
Main Equation ◽  
Non Linear ◽  
Even Order

The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

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