Hi.

■ Differential equations describe continuous systems. With these equations, rates of

change are defined in terms of other values in the system.

■ Difference equations are a discrete parallel to this where we use old values from the system to calculate new values.

■ The order of an equation is how many time step back you are looking at or what the

highest power derivative involved is.

■ Difference equations have an implicit time step in them. We keep track of

variables with integer subscripts and the difference between consecutive subscripts

is a time step.

■ You have to be careful when coding solutions to these systems to use the old

values in the calculation of all new values.

■ A very simple example of a difference equation is the logistic equation.

xt=a*xt-1(1-xt-1)

■ This deceptively simple equation holds a significant amount of complexity.

Depending on the value of “a” we get different types of behavior.

■ The best way to visualize this is actually with a picture plotting how the value of x evolves.

■ Fibonacci numbers are another form of difference equation used to model

population growth

■ If we have a difference equation with an order higher than 1 then keeping track of

previous values can be a difficult task. This can be simplified with the use of a circular

queue.

■ Basically, keep an array of previous states that holds as many states as the order of the equation (or one more if you do the calculation in the queue). You reference

states with an offset and overwrite old values as you go. Remembers mods

■ Most physical laws are defined in terms of differential equations or partial differential

equations. For this reason, being able to solve these is remarkably handy.

Unfortunately, they aren't as straightforward as difference equations.

■ To solve a differential equation, we basically convert it to a difference equation. That

conversion must be done with care though and how we do it depends no how accurate we need our method to be.

Also See http://im0.p.lodz.pl/~jbanasiak/Sem/aims.pdf

.

a fully worked out example :

Difference equations versus differential equations, a possible equivalence for the Rössler system ?

Christophe Letellier, Saber Elaydi, Luis A. Aguirre, Aziz Alaoui

Physica D 195 (2004) 29–49

Abstract

When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete. In numerical investigations a fourth-order Runge–Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rössler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rössler system as long as the time step is less than the threshold value associated with the Nyquist criterion.

available on line here

http://lmah.univ-lehavre.fr/~alaoui/publications.html

.

Hi.

■ Differential equations describe continuous systems. With these equations, rates of

change are defined in terms of other values in the system.

■ Difference equations are a discrete parallel to this where we use old values from the system to calculate new values.

■ The order of an equation is how many time step back you are looking at or what the

highest power derivative involved is.

■ Difference equations have an implicit time step in them. We keep track of

variables with integer subscripts and the difference between consecutive subscripts

is a time step.

■ You have to be careful when coding solutions to these systems to use the old

values in the calculation of all new values.

■ A very simple example of a difference equation is the logistic equation.

xt=a*xt-1(1-xt-1)

■ This deceptively simple equation holds a significant amount of complexity.

Depending on the value of “a” we get different types of behavior.

■ The best way to visualize this is actually with a picture plotting how the value of x evolves.

■ Fibonacci numbers are another form of difference equation used to model

population growth

■ If we have a difference equation with an order higher than 1 then keeping track of

previous values can be a difficult task. This can be simplified with the use of a circular

queue.

■ Basically, keep an array of previous states that holds as many states as the order of the equation (or one more if you do the calculation in the queue). You reference

states with an offset and overwrite old values as you go. Remembers mods

■ Most physical laws are defined in terms of differential equations or partial differential

equations. For this reason, being able to solve these is remarkably handy.

Unfortunately, they aren't as straightforward as difference equations.

■ To solve a differential equation, we basically convert it to a difference equation. That

conversion must be done with care though and how we do it depends no how accurate we need our method to be.

Also See http://im0.p.lodz.pl/~jbanasiak/Sem/aims.pdf

Thanks to all of you

Differential equation is a mathematical equation that relates some function and its certain derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions and the set of functions that satisfy the equation. Simplest differential equations are solvable by explicit formulas, some properties of solutions of a given differential equation may be estimated without finding their exact form. If a self contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Difference equations are a numerical methods approach to getting approximate solutions to differential equations. You break the continuity of the domain up into discrete steps or some discrete 'mesh' (in the case of PDEqs) and use numerical techniques so that it can be calculated using a computer algorithm.

Differential equation is a Mathematical tool to describe continuous systems (mathematical model). Whereas, Difference equation is also another mathematical tool that deals with discrete system.