Behavior of some higher order nonlinear rational partial difference equations | ||
| Journal of the Egyptian Mathematical Society | ||
| Volume 24, Issue 4, 2016, Pages 532-537 PDF (347.18 K) | ||
| DOI: 10.1016/j.joems.2016.03.004 | ||
| Author | ||
| Tarek F. Ibrahim | ||
| Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt | ||
| Abstract | ||
| 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 | ||
| Keywords | ||
| Partial difference equations; Solutions; Double mathematical induction | ||
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