Affiliation:
1. School of Arts and Sciences, Suqian University , Suqian , Jiangsu 223800 , P. R. China
2. School of Mathematics and Computer Science, Shangrao Normal University , Shangrao Jiangxi , 334001 , P. R. China
3. School of Mathematics, Qilu Normal University , Jinan , Shandong, 250200 , P. R. China
Abstract
Abstract
This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type
u
(
z
+
c
)
[
α
1
u
(
z
)
+
β
1
u
z
1
+
γ
1
u
z
2
+
α
2
v
(
z
)
+
β
2
v
z
1
+
γ
2
v
z
2
]
=
1
,
v
(
z
+
c
)
[
α
1
v
(
z
)
+
β
1
v
z
1
+
γ
1
v
z
2
+
α
2
u
(
z
)
+
β
2
u
z
1
+
γ
2
u
z
2
]
=
1
,
\left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right.
where
c
=
(
c
1
,
c
2
)
∈
C
2
c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2}
,
α
j
,
β
j
,
γ
j
∈
C
,
j
=
1
,
2
{\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2
. Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.