Electronic Journal of Differential Equations,
Vol. 2011(2011), No. 02, pp. 1-13.
Title: Initial-value problems for first-order differential recurrence
equations with auto-convolution
Author: Mircea Cirnu (Univ. Politehnica of Bucharest, Romania)
Abstract:
A differential recurrence equation consists of a sequence of
differential equations, from which must be determined by
recurrence a sequence of unknown functions. In this article, we
solve two initial-value problems for some new types of nonlinear
(quadratic) first order homogeneous differential recurrence
equations, namely with discrete auto-convolution and with
combinatorial auto-convolution of the unknown functions. In
both problems, all initial values form a geometric progression,
but in the second problem the first initial value is exempted
and has a prescribed form.
Some preliminary results showing the importance of the
initial conditions are obtained by reducing the differential
recurrence equations to algebraic type.
Final results about solving the considered initial
value problems, are shown by mathematical induction. However,
they can also be shown by changing the unknown
functions, or by the generating function method.
So in a remark, we give a proof of the first theorem by the
generating function method.
Submitted September 10, 2010. Published January 04, 2011.
Math Subject Classifications: 11B37, 34A12.
Key Words: Differential recurrence equations; discrete auto-convolution;
algebraic recurrence equations.