Electron. J. Differential Equations, Vol. 2017 (2017), No. 17, pp. 1-10.

Solvability of boundary-value problems for a linear partial difference equation

Stevo Stevic

Abstract:
In this article we consider the two-dimensional boundary-value problem
$$\displaylines{ d_{m,n}=d_{m-1,n}+f_nd_{m-1,n-1},\quad 1\le n<m,\cr d_{m,0}=a_m,\quad d_{m,m}=b_m,\quad m\in\mathbb{N}, }$$
where $a_m$, $b_m$, $m\in\mathbb{N}$ and $f_n$, $n\in\mathbb{N}$, are complex sequences. Employing recently introduced method of half-lines, it is shown that the boundary-value problem is solvable, by finding an explicit formula for its solution on the domain, the, so called, combinatorial domain. The problem is solved for each complex sequence $f_n$, $n\in\mathbb{N}$, that is, even if some of its members are equal to zero. The main result here extends a recent result in the topic.

Submitted October 23, 2016. Published January 14, 2017.
Math Subject Classifications: 39A14, 39A06.
Key Words: Partial difference equation; solvable difference equation; method of half-lines; combinatorial domain.

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Stevo Stevic
Mathematical Institute of the Serbian Academy of Sciences
Knez Mihailova 36/III, 11000 Beograd, Serbia
email: sstevic@ptt.rs

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