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Linear approximation and asymptotic expansion associated with the system of nonlinear functional equations

Ngoc L. T. P., Dung H. T. H., Danh P. H., Long N. T.

Demonstratio Mathematica

2014

Vol. 47, nr 1

103--124

This paper is devoted to the study of the following perturbed system of nonlinear functional equations (…) , where ε is a small parameter, aijk; bijk are the given real constants, Rijk; Sijk; Xijk : (…) are the given continuous functions and (…) are unknown functions. First, by using the Banach fixed point theorem, we find sufficient conditions for the unique existence and stability of a solution of (E). Next, in the case of (…) ; we investigate the quadratic convergence of (E). Finally, in the case of (…) and ε sufficiently small, we establish an asymptotic expansion of the solution of (E) up to order N + 1 in ε. In order to illustrate the results obtained, some examples are also given.

High-order iterative methods for a nonlinear Kirchhoff wave equation

Ngoc L. T. P., Truong L. X., Long N. T.

Demonstratio Mathematica

2010

Vol. 43, nr 3

605—634

In this paper we consider the following nonlinear wave equation (1) (…) where μ, f, ũ0, ũ1 are given functions satisfying conditions specified later. In Eq. (1)1, the nonlinear term μ(…) depends on the integral (…) dx. In this paper we associate with equation (1)1 a recurrent sequence {um} defined by (2) (…), 0< x <1, 0< t < T, with um satisfying (1)2,3. The first term u0 is chosen as u0≡0. If f∈CN([0,1]×R+×R), we prove that the sequence {um} converges at a rate of order N to a unique weak solution of problem (1).