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In this paper we give the closed form expressions of some higher order nonlinear rational partial difference equations in the form Xn,m=Xn−r,m−rΨ+∏i=1rXn−i,m−i where n,m∈N and the initial values Xn, t, Xt,m−r are real numbers with t∈{0,−1,−2,…,−r+1} such that ∏j=0r−1Xj−r+1,i+j−r+1≠−Ψ and ∏j=0r−1Xi+j−r+2,j−r+1≠−Ψ, i∈N0. 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.