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Topic: Difference equations

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	Difference Equations (Site not responding. Last check: 2007-10-31)
		Whereas the output of a continuous system with continuous inputs is described by differential equations, that of a discrete system is described by difference equations (DE's).
		A difference equation may be constructed as an approximation to samples from a continuous system described by a differential equation.
		However note that difference equations do not always arise this way and indeed this aspect of difference equations is not emphasized in this course.
	www.physics.uq.edu.au /people/jones/ph360/lectures/topic2/l2/node2.html (209 words)

	DIFFERENCE EQUATIONS \\ Applications and \\ Discrete Transforms Method
		Difference equations are often used to model ``an approximation'' of differential equations, an approach which underlies the development of many numerical methods.
		In the rest of the chapter we present some of the fundamental difference operators along with their basic properties and their inverses as ``sum'' operators, which are necessary for modeling difference equations as well as developing pairs for the basic discrete transforms.
		In addition to the main topic here which is linear difference equations of one variable, or ``ordinary linear difference equation,'' a clear introduction to difference equations of several variables, or ``partial difference equations'' is also presented, which is supported by a number of interesting examples.
	people.clarkson.edu /~jerria/book4.html (1647 words)

	Second-order difference equations (Site not responding. Last check: 2007-10-31)
		That is, any solution of the original equation is the sum of a some solution of this equation and a solution of the homogeneous equation.
		Equating coefficients, we have a = 1 and b = 0.
		The strategy for solving such an equation is very similar to the strategy for solving a second-order linear differential equations with constant coefficients.
	www.chass.utoronto.ca /~osborne/MathTutorial/SOD.HTM (881 words)

	Numerical Analysis of Applied Partial Differential Equations
		where (2) is the forward difference equation, (3) is the backward difference equation, and (4) is the central difference.
		The forward difference is the difference equation that would be used on the left boundary, when the function proceeds to the right.
		From this point, the difference approximations may be used to supply the replacement equations, which in turn are evaluated to reveal the unknowns.
	www.geocities.com /b_ward.rm/na.html (1809 words)

	39: Difference and functional equations
		Functional equations are those in which a function is sought which is to satisfy certain relations among its values at all points.
		When the focus of a functional equation is on continuity of functions and a domain is specified, this becomes a question of topology (in particular this sometimes becomes questions about the group of homeomorphism or diffeomorphisms of a set.
		Functions whose domains are integers are sequences, of course; thus a functional equation with this domain is essentially a recursion problem.
	www.math.niu.edu /~rusin/known-math/index/39-XX.html (583 words)

	Difference Equations and Its Applications
		stability of non-autonomous perturbed linear difference equations are derived.
		The differential equation is, in fact, a general dynamic equation containing delta-derivatives whose solution is defined on a measure chain.
		For a pair of eigenvalue problems for this dynamic equation, we first verify the existence of a smallest possible eigenvalue and then establish a comparison between the smallest eigenvalues of each eigenvalue problem.
	academic.udayton.edu /YoussefRaffoul/DifferenceEquations.htm (1150 words)

	Difference Equations: Solving Difference equations
		In some cases a difference equation in terms of a[n] may yield a solution for a[n] in terms of n alone.
		There are various techniques on how to derive the solution of such a difference equation, but we shall not cover them here.
		is a series of difference equations, including initial conditions, to be utilized for the purpose of solving for a[n] in terms of the variable n.
	www.mathcs.emory.edu /ccs/ccs110/webpages/moduleIII/modIIIsolvdiffeq.html (502 words)

	Difference Equations: John Hwang
		The homogeneous difference equations topic is covered in 5.5 while the non-homogeneous difference equations is covered in 5.7.
		Non-homogeneous difference equations are tougher to cover because there are varying types of problems.
		Tips: The most confusing part of non-homogeneous difference equations is Step 3 and all the algebra you have to do to find the general solution.
	coweb.math.gatech.edu:8888 /linear/732 (400 words)

	Digital Filter Design, Writing Difference Equations For Digital Filters, a Tutorial
		Difference equations are presented for 1st, 2nd, 3rd, and 4th order low pass and high pass filters, and 2nd, 4th and 6th order band-pass, band-stop and notch filters along with a resonance compensation (RES_COMP) filter.
		The low pass, high pass, bandpass and bandstop difference equations are obtained from the normalized butterworth continuous time filter descriptions given below.
		That is, to compensate for an inherent inaccuracy in the bi-linear transformation method that is a function of frequency and sample rate.
	www.apicsllc.com /apics/Sr_3/Sr_3.htm (477 words)

	Finding Closed-Form Solutions of Difference Equations by Symbolic Methods -- from Wolfram Library Archive
		Used as a collection of tools, the package can be employed to compute closed-form solutions of certain partial difference equations, to obtain recurrences for power-series coefficients of analytic functions, and to prove combinatorial identities.
		A proof that the generating function corresponding to the solution of a linear partial difference equation with constant coefficients with at most exponentially growing initial conditions is analytic.
		A proof of the theorem that the Galois group of a linear difference operator with polynomial coefficients over the difference ring of germs at infinity of sequences over a field is an algebraic matrix group.
	library.wolfram.com /infocenter/Articles/1425 (357 words)

	The Math Forum - Math Library - Difference Equations
		A short article designed to provide an introduction to functional equations, those in which a function is sought which is to satisfy certain relations among its values at all points.
		A special case involves difference equations, that is, equations comparing f(x) - f(x-1), for example, with some expression involving x and f(x).
		This journal presents papers on difference equations and the academic, analytical and engineering problems in which they arise.
	mathforum.org /library/topics/difference_eq (461 words)

	34: Ordinary differential equations
		Ordinary differential equations are equations to be solved in which the unknown element is a function, rather than a number, and in which the known information relates that function to its derivatives.
		Few such equations admit an explicit answer, but there is a wealth of qualitative information describing the solutions and their dependence on the defining equation.
		The solutions to many classic differential equations, particularly linear second-order differential equations, cannot be expressed in terms of the elementary functions but are themselves studied in 33: Special Functions.
	www.math.niu.edu /~rusin/known-math/index/34-XX.html (771 words)

	Difference Equations and Chaos in Mathematica -- from Wolfram Library Archive
		A difference equation (or map) of the form x_n-1 = f(x_n, x_n-1,...) which, together with some specified values or initial conditions defines a sequence {x_n}.
		Despite the seemingly simple form, difference equations have a variety of applications and can display a range of dynamics.
		Also, many of the approximations in numerical analysis (such as numerical solutions of differential equations) typically approximate continuous dynamical systems using discrete systems of difference equations.
	library.wolfram.com /infocenter/Articles/1032 (131 words)

	Difference Equations and Applications, Reza Kasravi and Mu-Ku Kang
		According to the book, any definition of the general term in a sequence in terms of one or more previous terms and initial cases is called a difference equation.
		Difference equations are being used in different fields by various companies everyday.
		Economic forecasting and business inventory which involve lots of difference equations are two good examples of common use of difference equations.
	coweb.math.gatech.edu:8888 /linear/731 (397 words)

	Discrete Dynamical Systems and Difference Equations
		Different ways of expressing the same idea often highlight different aspects -- something that at first appears to be very mysterious may suddenly seem clear when looked at from another perspective.
		is the most illuminating since the right hand side of this equation is a constant and thus we see immediately that this is an exponential model.
		is the most illuminating because we see immediately that the difference from one year to the next is constant.
	www.math.montana.edu /frankw/ccp/calculus/discdynm/ddsdfeqn/learn.htm (783 words)

	Difference Equations (Site not responding. Last check: 2007-10-31)
		Sometime in 1950 Professor Bohnenblust spoke at an afternoon seminar at Cal-Tech on “difference equations” which were evidently new to most of the audience.
		Bohnenblust contrasted difference equations with differential equations and presented them as a technique for arguing the existence of solutions to the differential equations and sometimes providing information about those solutions.
		Computers were not mentioned but it was clear to all that these results bore on the approximate solution of differential equations by difference equations with computers.
	www.cap-lore.com /stories/DiffEq.html (230 words)

	Eigenring and Reducibility of Difference Equations
		In analogy with the differential case, a concept of Liouvillian solutions of a difference equation is introduced, in relation to equations with solvable Galois group.
		In the differential case, a Liouvillian extension of a differential field is done by algebraic extensions and by the operations of exponentiation and integration of a function of the field.
		is the difference operator associated to the equation.
	algo.inria.fr /seminars/sem99-00/bomboy.html (1900 words)

	Z Transform of Difference Equations
		Since z transforming the convolution representation for digital filters was so fruitful, let's apply it now to the general difference equation, Eq.
		Repeating the general difference equation for LTI filters, we have (see Eq.
		Thus, taking the z transform of the general difference equation led to a new formula for the transfer function in terms of the difference equation coefficients.
	ccrma.stanford.edu /~jos/filters/Z_Transform_Difference_Equations.html (250 words)

	Egwald Mathematics — Nonlinear Dynamics
		The local stability of a nonlinear difference equation about a fixed point using linear stability analysis.
		In the continuous time version of the model, solution trajectories of the differential equation governing the growth of employment converge to a stable fixed point.
		In the discrete time version of the model, solution trajectories may follow periodic cycles and eventual exhibit chaotic behaviour as the parameter α of the model's difference equation increase.
	www.egwald.ca /nonlineardynamics (334 words)

	Recurrence Equation -- from Wolfram MathWorld
		A recurrence equation (also called a difference equation) is the discrete analog of a differential equation.
		The above equation is the discrete analog of the first-order ordinary differential equation
		The terms in a general recurrence sequence belong to a finitely generated ring over the integers, so it is impossible for every rational number to occur in any finitely generated recurrence sequence.
	mathworld.wolfram.com /RecurrenceEquation.html (480 words)

	The Solution of Macroeconomic Difference Equations
		From simulations, such as the above, some of the dynamics of an economy described by the difference equation specified.
		Thus the homogenous difference equation for the deviations from equilibrium (y
		For the previously specified simulation the analytical solution of the homogeneous difference equation for the deviations from the equilibrium can be found.
	www2.sjsu.edu /faculty/watkins/accel3.htm (205 words)

	Difference Equations (Site not responding. Last check: 2007-10-31)
		The above is an example of a difference equation.
		This equation can be used to model how this process changes over time by simply computing the value of X at each time step from the initial time, n = 0, up to any time required.
		As we shall see throughout the course, difference equations are extremely important in modeling biological processes.
	www2.sunysuffolk.edu /fultonj/difference_equations.htm (233 words)

	3 Material Balance Finite Difference Equations in One Dimension (Site not responding. Last check: 2007-10-31)
		To obtain a finite difference equation (FDE) for a partial differential equation, the continuous independent variables in the PDE (x and t in the previous examples) are restricted to a discrete grid of points, say
		One method begins with a PDE and then uses finite difference formulas obtained from Taylor series expansions to approximate the various derivatives in the PDE. This approach is usually favored when performing a mathematical analysis of the consistency of the FDE with the PDE and convergence of the solution of the FDE to the PDE.
		The second method for obtaining finite difference equations is based on a discrete conservation principle, similar to that introduced in Section 1.
	www.phy.ornl.gov /csep/CSEP/PDE2/NODE3.html (312 words)

	SECTION 1.6 Nonhomogeneous Linear Difference Equations (Site not responding. Last check: 2007-10-31)
		F, so that initially there is a large difference between the temperature of the soda and that of the refrigerator.
		The stability of a difference equation is of paramount concern in mathematics and its applications.
		Nonlinear difference equations, unstable equilibria, chaotic behavior, fractal patterns, sensitivity to initial conditions....
	www.webpearls.com /products/demos/pap/comap/chapter1/sec6/node1.html (2430 words)

	Nonlinear Population Dynamics - Publications
		Cushing, J.M. Nonlinear matrix equations as models for structured populations.
		Cushing, J.M. Periodically forced nonlinear systems of difference equations.
		Cushing, J.M. and Henson, S.M. Global dynamics of some periodically forced, monotone difference equations.
	caldera.calstatela.edu /nonlin/pubs.html (600 words)

	solving non linear difference equations
		This is a course on differential and difference equations with an emphasis on derivation and analysis of models of physical phenomena.
		The course will then introduce first order systems of ordinary differential and difference equations with constant coefficients.
		The topics covered in this course includes solving nonlinear equations; numerical linear algebra and solving systems of linear equations; approximation and interpolation using Lagrange polynomials, least square polynomials and splines; numerical differentiation and integration; solving ordinary differential equations; and simulation of stochastic processes.
	www.softmath.com /tutorials2/solving-non-linear-difference-equations.html (1097 words)

	Difference Equations with Hypergeometric Coefficients
		Let b be an element of k, and L be a linear ordinary difference operator with coefficients in k.
		A classical problem in the theory of difference equations is to compute all the solutions in k of the equation L(y)=b.
		He adapts the previous methods to difference equations with coefficients in an hypergeometric extension of C(n), and this gives an efficient algorithm to compute the rational solutions of such equations.
	algo.inria.fr /seminars/sem99-00/bronstein.html (1691 words)

	Difference Equations
		In the following papers, we have investigated the global stability of difference equations using different methods based on new discrete inequalities and monotonicity arguments.
		Sufficient conditions for the global exponential stability of nonautonomous higher order difference equations (with L. Berezansky and E. Braverman), Journal of Difference Equations and Applications, Vol.
		Convergence to equilibria in discrete population models (with H. El-Morshedy), Journal of Difference Equations and Applications, Vol.
	www.dma.uvigo.es /~eliz/Difference.html (299 words)

	Open Directory - Science: Math: Differential Equations
		Difference Method for Numerical Approximation to Applied Differential Equations.
		Differential Equations in Banach Algebras - Fuchsian Singularities of Linear Ordinary Differential Equations in Banach Algebras.
		Navier-Stokes Type Equations - Explicit solutions provided for this particular type of equation and their relations to the heat equation, Burger's equation, and Euler's equation.
	dmoz.org /Science/Math/Differential_Equations (706 words)

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