In this paper we give the closed form expressions of some higher order nonlinear rational
partial difference equations in the form
X n,m =
X n −r,m −r
 +
 r
i=1 X n −i,m −i
where n, m ∈ N and the initial values X n , t , X t,m −r are real numbers with t ∈ { 0 , −1 , −2 , . . . , −r + 1 }
such that
 r −1
j=0 X j −r +1 ,i+ j −r +1  = − and
 r −1
j=0 X i+ j −r +2 , j −r +1  = −, i ∈ N 0 .
We will use a new method to prove the results by using what we call ‘piecewise double mathematical
induction' which we introduce here for the first time as a generalization of many types of mathematical
induction. As a direct consequences, we investigate and conclude the explicit solutions of some higher
order ordinary difference equations