Journal of Mathematical Sciences: Advances and ApplicationsAuthor's: Mircea Ion Cîrnu
Pages: [99] - [113]
Received Date: June 9, 2014; Revised June 25, 2014
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A differential recurrence equation consists of a recurrent sequence of differential equations, from which a sequence of unknown functions must be determined. In this paper, we will present several methods for solving two nonlinear (quadratic) first-order homogeneous differential recurrence equations with discrete auto-convolution of the unknown functions or their derivatives. We use here three types of proofs: The first by mathematical induction, the second based on generating function method, and the third by a substitution which reduces the differential recurrence equation to the corresponding algebraic recurrence equation. We will present these methods on the simplest differential recurrence equations with discrete auto-convolution. For the first equation, we will determine the solutions that are in geometric progression, while the second is solved without any supplementary condition. Finally, we present two differential recurrence equations with combinatorial auto-convolution that are reduced to the first ones by substitutions, and some applications of the results from this paper to the discrete linear time-invariant physical systems theory are also presented.
discrete auto-convolution, combinatorial auto-convolution, algebraic and differential recurrence equations, discrete linear time-invariant physical systems, impulse response function.
