Nonlinear Stability and D-Convergence of Additive Runge-Kutta Methods for Multidelay-Integro-Differential Equations

This paper is devoted to the stability and convergence analysis of the Additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of multidelay-integro-differential equations (MDIDEs). GDN-stability and D-convergence are introduced and proved. It is shown that strongly algebraically stability gives D-convergence, DA- DAS- and ASI-stability give GDN-stability. A numerical example is given to illustrate the theoretical results.

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Some Stability and Convergence of Additive Runge-Kutta Methods for Delay Differential Equations with Many Delays

Journal of Applied Mathematics ◽

10.1155/2012/456814 ◽

2012 ◽

Vol 2012 ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Haiyan Yuan ◽

Jingjun Zhao ◽

Yang Xu

Keyword(s):

Differential Equation ◽

Numerical Solution ◽

Convergence Analysis ◽

Delay Differential Equation ◽

Delay Differential Equations ◽

Stability And Convergence ◽

Runge Kutta ◽

Lagrangian Interpolation ◽

The Stability ◽

Delay Differential

This paper is devoted to the stability and convergence analysis of the additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of a delay differential equation with many delays. GDN stability and D-Convergence are introduced and proved. It is shown that strongly algebraically stability gives D-Convergence DA, DAS, and ASI stability give GDN stability. Some examples are given in the end of this paper which confirms our results.

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Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Neutral Delay Differential Equations

Mathematical Problems in Engineering ◽

10.1155/2013/683137 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Haiyan Yuan ◽

Cheng Song ◽

Peichen Wang

Keyword(s):

Differential Equations ◽

Nonlinear Stability ◽

Delay Differential Equations ◽

Stability And Convergence ◽

Neutral Delay ◽

Neutral Delay Differential Equations ◽

Restricted Type ◽

Runge Kutta ◽

The Stability ◽

Delay Differential

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with the Lagrange interpolation of the numerical solution for nonlinear neutral delay differential equations. Nonlinear stability and D-convergence are introduced and proved. We discuss theGR(l)-stability,GAR(l)-stability, and the weakGAR(l)-stability on the basis of(k,l)-algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

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Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

Abstract and Applied Analysis ◽

10.1155/2013/679075 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13

Author(s):

Haiyan Yuan ◽

Cheng Song

Keyword(s):

Differential Equations ◽

Quadrature Formula ◽

Nonlinear Stability ◽

Stability And Convergence ◽

Asymptotically Stable ◽

Kutta Method ◽

Runge Kutta Method ◽

Stage Order ◽

Runge Kutta ◽

The Stability

This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of(k,l)-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a(k,l)-algebraically stable two-step Runge-Kutta method with0<k<1is proved. For the convergence, the concepts ofD-convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order isp, then the method with compound quadrature formula isD-convergent of order at leastmin{p,ν}, whereνdepends on the compound quadrature formula.

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A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

Baghdad Science Journal ◽

10.21123/bsj.2020.17.1.0166 ◽

2020 ◽

Vol 17 (1) ◽

pp. 0166

Author(s):

Hussain Et al.

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Ordinary Differential Equations ◽

Numerical Results ◽

New Method ◽

Kutta Method ◽

First Order ◽

Runge Kutta Method ◽

Runge Kutta ◽

The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

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A Class of Hybrid Multistep Block Methods with A–Stability for the Numerical Solution of Stiff Ordinary Differential Equations

Mathematics ◽

10.3390/math8060914 ◽

2020 ◽

Vol 8 (6) ◽

pp. 914

Author(s):

Zarina Bibi Ibrahim ◽

Amiratul Ashikin Nasarudin

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Ordinary Differential Equations ◽

Stability And Convergence ◽

Stiff Differential Equations ◽

Stiff Ordinary Differential Equations ◽

Block Methods ◽

Backward Differentiation Formulas ◽

The Stability ◽

Differentiation Formulas

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

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Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Fractional Order by Using Adomian Decomposition Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.635-637.1582 ◽

2014 ◽

Vol 635-637 ◽

pp. 1582-1585

Author(s):

Li Feng Wang ◽

Yun Peng Ma ◽

Yong Qiang Yang

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Convergence Analysis ◽

Fractional Order ◽

Decomposition Method ◽

Series Solution ◽

Adomian Decomposition Method ◽

Computational Method ◽

Numerical Example ◽

Adomian Decomposition

In this work we present a computational method for for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on Adomian Decom-position Method. Convergence analysis is dependable enough to estimate the maximum absolute truncated error of the Adomian series solution. Numerical example is included to demonstrate the validity and applicability of the method.

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