scholarly journals Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations

2000 ◽  
Vol 4 (1) ◽  
pp. 21-28
Author(s):  
Alicia Serfaty de Markus
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Equation ◽  
Nonlinear Equations ◽  
Chaotic Dynamics ◽  
Nonlinear Models ◽  
Lorenz Equations ◽  
Van Der Pol ◽  
Complex Difference Equation ◽  
Numerical Instabilities

For computational purposes, a numerical algorithm maps a differential equation into an often complex difference equation whose structure and stability depends on the scheme used. When considering nonlinear models, standard and nonstandard integration routines can act invasively and numerical chaotic instabilities may arise. However, because nonstandard schemes offer a direct and generally simpler finite-difference representations, in this work nonstandard constructions were tested over three different systems: a photoconductor model, the Lorenz equations and the Van der Pol equations. Results showed that although some nonstandard constructions created a chaotic dynamics of their own, there was found a construction in every case that greatly reduced or successfully removed numerical chaotic instabilities. These improvements represent a valuable development to incorporate into more sophisticated algorithms.

Efficient Family of Sixth-Order Methods for Nonlinear Models with Its Dynamics

2019 ◽  
Vol 16 (05) ◽  
pp. 1840008
Author(s):  
Ramandeep Behl ◽  
Prashanth Maroju ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Complex Dynamics ◽  
Nonlinear Models ◽  
Computational Cost ◽  
Sixth Order ◽  
System Of Nonlinear Equations ◽  
Van Der Pol ◽  
Numerical Work ◽  
The Family

In this study, we design a new efficient family of sixth-order iterative methods for solving scalar as well as system of nonlinear equations. The main beauty of the proposed family is that we have to calculate only one inverse of the Jacobian matrix in the case of nonlinear system which reduces the computational cost. The convergence properties are fully investigated along with two main theorems describing their order of convergence. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. From this study, we intend to have more information about these methods in order to detect those with best stability properties. In addition, we also presented a numerical work which confirms the order of convergence of the proposed family is well deduced for scalar, as well as system of nonlinear equations. Further, we have also shown the implementation of the proposed techniques on real world problems like Van der Pol equation, Hammerstein integral equation, etc.

scholarly journals PENYELESAIAN NUMERIK MODEL PREDATOR-PREY DENGAN SKEMA BEDA HINGGA TAK-STANDAR

2019 ◽  
Vol 2 (1) ◽  
pp. 6
Author(s):  
Rina Reorita ◽  
Renny Renny
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Equation ◽  
Difference Scheme ◽  
Continuous System ◽  
Qualitative Property ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Linear Differential ◽  
Model Predator

Interaction between predator and prey can be represented as a system of non-linear differential equation which is difficult to be solved analytically. In this research, a predator-prey model with an addition of harvesting factor is discretized into a system of difference equation using non-standard finite difference scheme. The analysis result shows that the developed scheme has qualitative property which is consistent to the continuous system.

Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations

1993 ◽  
Vol 60 (1) ◽  
pp. 167-174 ◽  
Author(s):  
N. S. Abhyankar ◽  
E. K. Hall ◽  
S. V. Hanagud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Chaotic Dynamics ◽  
Partial Differential ◽  
Galerkin Approximations ◽  
The Finite Difference Method

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

On the non-linear difference equation Δxn = kΦ (xn)

1932 ◽  
Vol 28 (2) ◽  
pp. 234-243 ◽  
Author(s):  
J. B. S. Haldane
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Finite Difference ◽  
Difference Equation ◽  
Linear Difference Equation ◽  
Finite Difference Equation ◽  
Non Linear ◽  
Image Position

1. Among the equations arising in the theory of natural selection is the finite difference equationwhere k is a constant which may have any value between 1 and − ∞ inclusive, but is often small. Its solution is discussed in the accompanying paper(1). It is a particular case of the equationwhere Φ(x) is a known one-valued function of x. When k is small this may obviously be solved approximately by treating it as a differential equationwhence

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals p-adic confluence of q-difference equations

2008 ◽  
Vol 144 (4) ◽  
pp. 867-919 ◽  
Author(s):  
Andrea Pulita
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Root Of Unity

AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

Numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument

2017 ◽  
Vol 23 (2) ◽  
Author(s):  
Muhad H. Abregov ◽  
Vladimir Z. Kanchukoev ◽  
Maryana A. Shardanova
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Deviating Argument ◽  
Second Order Differential Equation

AbstractThis work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.

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