doi:10.1016/j.na.2003.10.001 
Copyright © 2003 Elsevier Ltd. All rights reserved.
Approximating crossed symmetric solutions of nonlinear dynamic equations via quasilinearization
P. W. Eloe
, 
and Q. Sheng
Department of Mathematics, University of Dayton, 300 College Park, Dayton, OH 45469-2316, USA
Available online 21 November 2003.
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Abstract
Crossed symmetric solutions of nonlinear boundary value dynamic problems play an important role in many applications, in particular in adaptive algorithm designs. This article is devoted to the continuation of our investigation on second-order nonlinear companion dynamic boundary value problems on time scales. Monotonically convergent upper and lower solutions of the problems and their quasilinear approximations are investigated. It is shown that, under proper smoothness constraints, the iterative sequences constructed not only converge to the analytic solutions of the desired companion problems monotonically, but also preserve important crossed symmetry properties. The quasilinearization offers an efficient way in the solution approximation. Computational examples are given to illustrate our results.
Author Keywords: Quasilinearization; Upper and lower solutions; Crossed symmetry; Dynamic equation's on time scales; Δ and 
derivatives
Mathematical subject codes: 34B10; 34B15; 39A10; 65M06
Fig. 1. Top: circled curve is for u, triangulated curve is for v. Dotted curve is for w, the solution of the differential equation problem (5.3). Bottom: function u−v. The crossed symmetry of solutions u and v is clearly demonstrated.
Fig. 2. Top: lower solutions. Circled curve is for α
0, triangulated curve is for α
1. Bottom: upper solutions. Circled curve is for β
0, triangulated curve is for β
1. Dotted curves are for the solution
u of the forward dynamic problem (
(4.1) and
(4.2)).
Fig. 3. Top: error of the lower solutions. Solid curve is for
u−α
0, triangulated curve is for
u−α
1. Bottom: error of the upper solutions. Solid curve is for β
0−
u, triangulated curve is for β
1−
u.
u is the solution of the forward dynamic problem (
(4.1) and
(4.2)). Logarithmic scales are used in the vertical direction to see better details of the profiles.
Fig. 4. Top: lower solutions. Circled curve is for φ
0 triangulated curve is for φ
1. Bottom: upper solutions. Circled curve is for ψ
0, triangulated curve is for ψ
1. Dotted curves are for the solution
v of the forward dynamic problem (
(4.3) and
(4.4)).
Fig. 5. Top: error of the lower solutions. Solid curve is for
v−φ
0, triangulated curve is for
v−φ
1. Bottom: error of the upper solutions. Solid curve is for ψ
0−
v, triangulated curve is for ψ
1−
v.
v is the solution of the backward dynamic problem (
(4.3) and
(4.4)). Logarithmic scales are used in the vertical direction.
Fig. 6. Top: crossed symmetry of the numerical solutions α
n, φ
n. Circled curves are for α
n,
n=0,1, triangulated curves are for φ
n,
n=0,1. Bottom: crossed symmetry of the numerical solutions α
n, ψ
n. Circled curves are for α
n,
n=0,1, triangulated curves are for ψ
n,
n=0,1. Dotted curves are for
w, the solution of the differential equation problem (
5.3).
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