scholarly journals Developing a Series Solution Method of -Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Hsuan-Ku Liu
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Series Solution ◽  
Solution Method ◽  
Generalized Polynomials ◽  
Order Difference ◽  
Multiplication Rule ◽  
Second Order Difference Equation ◽  
Order Difference Equation

The series solution is widely applied to differential equations on but is not found in -differential equations. Applying the Taylor and multiplication rule of two generalized polynomials, we develop a series solution of linear homogeneous -difference equations. As an example, the series solution method is used to find a series solution of the second-order -difference equation of Hermite’s type.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

scholarly journals Solution of homogeneous linear difference equations

1980 ◽  
Vol 21 (3) ◽  
pp. 293-296 ◽  
Author(s):  
J. D. Love
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Continued Fraction ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Simple Product ◽  
Homogeneous Linear

AbstractWhen the first two elements of a sequence satisfying a second order difference equation are prescribed, the remaining elements are evaluated from a continued fraction and a simple product.

scholarly journals Growth of Meromorphic Solutions of Some -Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Guowei Zhang
Keyword(s):  
Difference Equation ◽  
Fixed Points ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Order Difference ◽  
Complex Difference ◽  
Complex Difference Equations ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

We estimate the growth of the meromorphic solutions of some complex -difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order -difference equation. We also obtain a theorem about the -difference equation mixing with difference.

scholarly journals TRANSFORMATION OF THE LINEAR DIFFERENCE EQUATION INTO A SYSTEM OF THE FIRST ORDER DIFFERENCE EQUATIONS

2019 ◽  
pp. 76-80
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Second Order ◽  
Dimensional System ◽  
Order Difference ◽  
Order Linear ◽  
First Order ◽  
Second Order Difference Equation ◽  
Order Difference Equation

The transformation of the N-th-order linear difference equation into a system of the first order difference equations is presented. The proposed transformation opens possibility to obtain new forms of the N-dimensional system of the first order equations that can be useful for the analysis of solutions of the N-th-order difference equations. In particular for the third-order linear difference equation the nonlinear second-order difference equation that plays the same role as the Riccati equation for second-order linear difference equation is obtained. The new form of the Ndimensional system of first order equations can also be used to find the WKB solutions of the linear difference equation with coefficients that vary slowly with index.

scholarly journals Sequences of positive homoclinic solutions to difference equations with variable exponent

2020 ◽  
Vol 70 (2) ◽  
pp. 417-430
Author(s):  
Robert Stegliński ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Variational Principle ◽  
Difference Equation ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Variable Exponent ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

Abstract We study the existence of infinitely many positive homoclinic solutions to a second-order difference equation on integers with pk-Laplacian. To achieve our goal we use the critical point theory and the general variational principle of Ricceri.

scholarly journals On The Solutions of Some Difference Equations Systems and Analytical Properties

2018 ◽  
Vol 22 ◽  
pp. 01050
Author(s):  
Münevver Tuz
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Equation System ◽  
Asymptotic Behavior Of Solutions ◽  
Asymptotic Results ◽  
Order Difference ◽  
Positive Balance ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Given

In this study, we investigated the global asymptotic behaviors of their solutions by taking the second-order difference equation system. According to the given conditions, we obtained some asymptotic results for the positive balance of the system. We have also worked on q-fast changing functions. Such functions form the class of q-Caramate functions. We have applied q-Caramate functions to linear q-difference equations and We have also learned about the asymptotic behavior of solutions. In addition, we have studied the problem of initial and boundary value for the q-difference equation

scholarly journals Necessary and sufficient conditions for oscillation of second order neutral difference equations

2003 ◽  
Vol 34 (2) ◽  
pp. 137-146 ◽  
Author(s):  
E. Thandapani ◽  
K. Mahalingam
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Real Sequence ◽  
Order Difference ◽  
Neutral Difference Equations ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Necessary And Sufficient

Consider the second order difference equation of the form$\Delta^2(y\n-py_{n-1-k})+q_nf(y_{n-\ell})=0,\quad n=1,2,3,\ldots  \hskip 1.9cm\hbox{(E)}$where $ \{q_n\}$ is a nonnegative real sequence, $ f:{\Bbb R}\rightarrow {\Bbb R}$ is continuous such that $ uf(u)>0$ for $ u\not= 0$, $ 0\le p

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

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