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Linear integral equations and nonlinear difference-difference equations
Physica D: Nonlinear Phenomena, 1984We present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Bäcklund transformations.
Quispel, G. R. W. +3 more
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2014
In this chapter we present some techniques to solve linear difference equations with one or more steps, modeling the dynamics of scalar quantities. Nonlinear models and the vector-valued case will be addressed respectively in Chaps. 3– 4 and 5– 6.
Ernesto Salinelli, Franco Tomarelli
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In this chapter we present some techniques to solve linear difference equations with one or more steps, modeling the dynamics of scalar quantities. Nonlinear models and the vector-valued case will be addressed respectively in Chaps. 3– 4 and 5– 6.
Ernesto Salinelli, Franco Tomarelli
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2000
In this Chapter we shall consider such systems of equations where each variable has a time index £ = 0,1,2,..., and variables of different time-periods are connected in a nontrivial way. To avoid complicated situations, we assume that there is one variable at the left hand side (to be abbreviated as L.H.S) of each equation, its time index is at least ...
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In this Chapter we shall consider such systems of equations where each variable has a time index £ = 0,1,2,..., and variables of different time-periods are connected in a nontrivial way. To avoid complicated situations, we assume that there is one variable at the left hand side (to be abbreviated as L.H.S) of each equation, its time index is at least ...
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Dynamics of a non-linear difference equation
Applied Mathematics and Computation, 2006The authors consider the dynamics of the difference equation \[ y_{n+1}=\frac{y_n + p y_{n-k}}{y_n+q},\quad n=0,1,2\dots \] where the initial conditions \(y_{-k},\dots,y_{-1}, y_0\) are non-negative, \(k\in \mathbb{N}\), and the parameters \(p\) and \(q\) are non-negative.
Reza Mazrooei-Sebdani +1 more
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2010
Already in an analysis of nonlinear systems, linear problems frequently occur in form of variational equations (cf. Corollary 2.3.11) when linearizing along a given reference solution. Provided this solution does lack a specific time-dependence (e.g., (almost) periodicity, or being convergent), then the resulting variational equations are nonautonomous
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Already in an analysis of nonlinear systems, linear problems frequently occur in form of variational equations (cf. Corollary 2.3.11) when linearizing along a given reference solution. Provided this solution does lack a specific time-dependence (e.g., (almost) periodicity, or being convergent), then the resulting variational equations are nonautonomous
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The Theory of Linear G-Difference Equations
Acta Applicandae Mathematica, 1999zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Jakobsen, Per K., Lychagin, Valentin V.
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Perturbation of a linear difference equation
Zbornik radova Prirodno-matematičkog fakulteta u Novom Sadu, serija Matematika, 1986The author proves a generalization of Weyl's alternative for perturbed linear difference equations. Let \(Dy\equiv y(m+n)+p_{n-1}(m)y(n+m- 1)+...+p_ 0(m)y(m)\) and \(D_{L^ y}\equiv Dy+q_{n-1}(m)y(m+n- 1)+...+q_ 0(m)y(m).\) Suppose \(\prod^{S}_{m=0}| p_ 0(m)| \geq M>0\) and \(q_ i\in l^ k\) where \(1\leq k\leq \infty\) for \(1\leq p\leq 2\) and \(1\leq ...
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1998
This brief section treats linear difference equations, which play a role in the theory of discrete-time linear systems similar to that played by linear differential equations in the theory of continuous-time linear systems. The existence and uniqueness theorems for difference equations are much easier than those for differential equations; as we shall ...
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This brief section treats linear difference equations, which play a role in the theory of discrete-time linear systems similar to that played by linear differential equations in the theory of continuous-time linear systems. The existence and uniqueness theorems for difference equations are much easier than those for differential equations; as we shall ...
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