scholarly journals The step-type contrast structure for a second order semi-linear singularly perturbed differential-difference equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mei Xu ◽  
Bingxian Wang
Keyword(s):  
Asymptotic Expansion ◽  
Difference Equation ◽  
Boundary Function ◽  
Second Order ◽  
Singularly Perturbed ◽  
Contrast Structure ◽  
Structure Solution ◽  
Differential Difference Equation ◽  
Step Type

Abstract The step-type contrast structure for a second order semi-linear singularly perturbed differential-difference equation is studied. Using the methods of boundary function and fractional steps, we construct the formula asymptotic expansion of the problem. At the same time, based on sewing techniques, the existence of the step-type contrast structure solution and the uniform validity of the asymptotic expansion are proved.

Spline approximation method for singularly perturbed differential-difference equation on nonuniform grids

2020 ◽  
pp. 2150005
Author(s):  
P. Mushahary ◽  
S. R. Sahu ◽  
J. Mohapatra
Keyword(s):  
Difference Equation ◽  
Spline Approximation ◽  
Perturbation Parameter ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Nonuniform Grids ◽  
Fine Mesh ◽  
Differential Difference Equation ◽  
Perturbed Differential Equation

In this paper, a second-order singularly perturbed differential-difference equation involving mixed shifts is considered. At first, through Taylor series approximation, the original model is reduced to an equivalent singularly perturbed differential equation. Then, the model is treated by using the hybrid finite difference scheme on different types of layer adapted meshes like Shishkin mesh, Bakhvalov–Shishkin mesh and Vulanović mesh. Here, the hybrid scheme consists of a cubic spline approximation in the fine mesh region and a midpoint upwind scheme in the coarse mesh region. The error analysis is carried out and it is shown that the proposed scheme is of second-order convergence irrespective of the perturbation parameter. To display the efficacy and accuracy of the proposed scheme, some numerical experiments are presented which support the theoretical results.

A differential equation for varying Krall-type orthogonal polynomials

2019 ◽  
Vol 09 (01) ◽  
pp. 2040002
Author(s):  
Galina Filipuk ◽  
Juan F. Mañas-Mañas
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Second Order ◽  
Inner Product ◽  
Ladder Operators ◽  
Differential Difference Equation

In this contribution, we consider varying Krall-type polynomials which are orthogonal with respect to a varying discrete Krall-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as to find a second-order differential-difference equation that these polynomials satisfy. We generalize some results that appeared recently in the literature.

scholarly journals Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Guoqiang Dang ◽  
Jinhua Cai
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
System Of Equations ◽  
Entire Solutions ◽  
Differential Difference Equation ◽  
Necessary And Sufficient

In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation f″z2+△ckfz2=1 and the system of equations f1″z2+△ckf2z2=1 and f2″z2+△ckf1z2=1 have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

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