Solvability of boundary value problems for a class of partial difference equations on the combinatorial domain
- Stevo Stević^{1, 2}Email author
Received: 17 June 2016
Accepted: 5 October 2016
Published: 18 October 2016
Abstract
Keywords
partial difference equation boundary value problem equation solvable in closed form the method of half-linesMSC
39A14 05A101 Introduction
Let \(\mathbb{N}\) denote the set of natural numbers and \(\mathbb{N} _{0}=\mathbb{N}\cup \{0\}\). Let \(k,l\in \mathbb{N}_{0}\) be such that \(k< l\), then the notation \(j=\overline{k,l}\) means \(k\le j\le l\). In the rest of this section we give some motivation for the study, as well as notions that will be used in the rest of the paper.
Motivated by [29] and numerous recent applications of equation (8) (for example, those in [9, 10, 13–17, 19, 20]), here we show that there is a closed form formula for solutions to an extension of equation (9) on domain \(A\setminus \{(0,0)\}\) in terms of given boundary values \(d_{n,0}\) and \(d_{n,n}\), \(n\in \mathbb{N}\).
2 Main results
We show how general solution to (9) on set \(A\setminus \{(0,0) \}\) can be found by using a method in [29], which we call the method of half-lines. Namely, the domain is divided into some half-lines, and (9) is regarded on each line as an equation of type (8). It is solved, and based on the obtained formulas one gets the general solution. However, the method cannot be applied directly, so it needs some modifications.
Now we are in a position to state and prove our first result based on the above consideration.
Theorem 1
Proof
By choosing \(l=n-k\) in (17), using the conditions in (18), and some calculations, formula (19) is obtained. □
Remark 1
Corollary 1
Remark 2
Note that under the conditions of Corollary 1 equation (22) gives a closed form formula for such solutions due to equation (6).
Theorem 2
Now we prove the following lemma.
Lemma 1
Proof
Hence, the following result holds.
Theorem 3
Declarations
Acknowledgements
The author would like to express his sincere thanks to the referee of the paper for her/his comments and for suggesting some references on the solvability of partial difference equations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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