scholarly journals A Comparative Study on the Stability of Laplace-Adomian Algorithm and Numerical Methods in Generalized Pantograph Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Sabir Widatalla
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Comparative Study ◽  
Approximate Solutions ◽  
Stability And Convergence ◽  
Multiple Delays ◽  
The Stability

The main objective of this paper is to examine the stability and convergence of the Laplace-Adomian algorithm to approximate solutions of the pantograph-type differential equations with multiple delays. This is done by comparatively investigating it with other methods.

scholarly journals Implicit difference schemes for quasilinear parabolic functional equations

2012 ◽  
Vol 45 (4) ◽  
Author(s):  
Milena Matusik
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Difference Schemes ◽  
New Class ◽  
Implicit Difference Schemes ◽  
Functional Differential ◽  
The Stability ◽  
Quasilinear Parabolic

AbstractWe present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

Stability and Convergence of Two New Implicit Numerical Methods for the Fractional Cable Equation

2009 ◽  
Author(s):  
F. Liu ◽  
Q. Yang ◽  
I. Turner
Keyword(s):  
Numerical Methods ◽  
Partial Derivative ◽  
Stability And Convergence ◽  
Cable Equation ◽  
Neuronal Dynamics ◽  
Neuronal Dendrites ◽  
Space Step ◽  
The Stability ◽  
Fundamental Equations ◽  
Fractional Cable Equation

The cable equation is one the most fundamental equations for modeling neuronal dynamics. Cable equations with fractional order temporal operators have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper we consider the following fractional cable equation involving two fractional temporal derivatives: ∂u(x,t)∂t=0Dt1−γ1κ∂2u(x,t)∂x2−μ02Dt1−γ2u(x,t)+f(x,t), where 0 < γ1,γ2 < 1, κ > 0, and μ02 are constants, and 0Dt1−γu(x,t) is the Rieman-Liouville fractional partial derivative of order 1 − γ. Two new implicit numerical methods with convergence order O(τ + h2) and O(τ2 + h2) for the fractional cable equation are proposed respectively, where τ and h are the time and space step sizes. The stability and convergence of these methods are investigated using the energy method. Finally, numerical results are given to demonstrate the effectiveness of both implicit numerical methods. These techniques can also be applied to solve other types of anomalous subdiffusion problems.

Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

scholarly journals A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

2021 ◽  
pp. 58-69
Author(s):  
Lawrence Osa Adoghe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Stiff Systems ◽  
Convergence Properties ◽  
Main Method ◽  
The Stability ◽  
Third Derivative

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

scholarly journals New Numerical Methods for Solving Differential Equations

2019 ◽  
Vol 16 ◽  
pp. 8384-8390 ◽  
Author(s):  
Osama. Y. Ababneh
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Theta Method

In this paper, we present new numerical methods to solve ordinary differential equations in both linear and nonlinear cases. we apply Daftardar-Gejji technique on theta-method to derive anew family of numerical method. It is shown that the method may be formulated in an equivalent way as a RungeKutta method. The stability of the methods is analyzed.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
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.

scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

scholarly journals 3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
J. O. Ehigie ◽  
S. A. Okunuga ◽  
A. B. Sofoluwe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Absolute Stability ◽  
Second Order ◽  
Stability And Convergence ◽  
Block Methods ◽  
Good Region ◽  
The Stability ◽  
The Individual ◽  
Region Of Absolute Stability

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form . The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

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