scholarly journals Numerical Approach of High-Order Linear Delay Difference Equations with Variable Coefficients in Terms of Laguerre Polynomials

2011 ◽  
Vol 16 (1) ◽  
pp. 267-278 ◽  
Author(s):  
Burcu Gürbüz ◽  
Mustafa Gülsu ◽  
Mehmet Sezer
Keyword(s):  
Difference Equations ◽  
Laguerre Polynomials ◽  
Numerical Approach ◽  
Variable Coefficients ◽  
High Order ◽  
Order Linear ◽  
Linear Delay ◽  
Delay Difference Equations ◽  
Delay Difference

A Numerical Approach for Solving High-Order Linear Delay Volterra Integro-Differential Equations

2018 ◽  
Vol 15 (05) ◽  
pp. 1850042 ◽  
Author(s):  
Şuayip Yüzbaşı ◽  
Murat Karaçayır
Keyword(s):  
Differential Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Numerical Approach ◽  
High Order ◽  
Inner Product ◽  
Present Scheme ◽  
Order Linear ◽  
Linear Delay ◽  
Residual Correction

In this study, a numerical method is proposed to solve high-order linear Volterra delay integro-differential equations. In this approach, we assume that the exact solution can be expressed as a power series, which we truncate after the [Formula: see text]-st term so that it becomes a polynomial of degree [Formula: see text]. Substituting the unknown function, its derivatives and the integrals by their matrix counterparts, we obtain a vector equivalent of the equation in question. Applying inner product to this vector with a set of monomials, we are left with a linear algebraic equation system of [Formula: see text] unknowns. The approximate solution of the problem is then computed from the solution of the resulting linear system. In addition, the technique of residual correction, whose aim is to increase the accuracy of the approximate solutions by estimating the error of those solutions, is discussed briefly. Both the method and this technique are illustrated with several examples. Finally, comparison of the present scheme with other methods is made wherever possible.

scholarly journals Nontrivial periodic solutions to delay difference equations via Morse theory

2018 ◽  
Vol 16 (1) ◽  
pp. 885-896 ◽  
Author(s):  
Yuhua Long ◽  
Haiping Shi ◽  
Xiaoqing Deng
Keyword(s):  
Periodic Solutions ◽  
Difference Equations ◽  
Morse Theory ◽  
Sufficient Conditions ◽  
Asymptotically Linear ◽  
Linear Delay ◽  
Nontrivial Periodic Solutions ◽  
Delay Difference Equations ◽  
Delay Difference

AbstractIn this paper some sufficient conditions are obtained to guarantee the existence of nontrivial 4T + 2 periodic solutions of asymptotically linear delay difference equations. The approach used is based on Morse theory.

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