Journals
- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
-
Electronic Research Announcements
-
Conference Publications
-
AIMS Mathematics
$\lambda=a_1 (n-2)(n-1)+\gamma_1 (m-2)(m-1)+\beta_1 (n-1)(m-1)+\delta_1 (n-1)+\epsilon_1 (m-1),$
where $n=1,2,...,N$, $m=1,2,...,M$ is a necessary and sufficient condition for the linear partial differential equation
$(a_1x^2+a_2x+a_3)u_{x x}+(\beta_1xy+\beta_2x+\beta_3y+\beta_4)u_{x y} $
$+(\gamma_1y^2+\gamma_2y+\gamma_3)u_{y y}+(\delta_1x+\delta_2)u_x+(\epsilon_1y+\epsilon_2)u_y=\lambda u, $
where $a_i$, $\beta_j$, $\gamma_i$, $\delta_s$, $\epsilon_s$, $i=1,2,3$, $j=1,2,3,4$, $s=1,2$ are real or complex constants, to have polynomial solutions of the form
$u(x,y)=\sum_{n=1}^N\sum_{m=1}^Mu_{n m}x^{n-1}y^{m-1}.$
The proof of this result is obtained using a functional analytic method which reduces the problem of polynomial solutions of such partial differential equations to an eigenvalue problem of a specific linear operator in an abstract Hilbert space. The main result of this paper generalizes previously obtained results by other researchers.
The problem of the existence of complex $\ell_1$ solutions of two difference equations with exponential nonlinearity is studied, one of which is nonautonomous. As a consequence, several information are obtained regarding the asymptotic stability of their equilibrium points, as well as the corresponding generating function and $z-$ transform of their solutions. The results, which are obtained using a general theorem based on a functional-analytic technique, provide also a rough estimate of the region of attraction of each equilibrium point for the autonomous case. When restricted to real solutions, the results are compared with other recently published results.
Year of publication
Related Authors
Related Keywords
[Back to Top]