Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1205.65216
Geng, Fazhan; Cui, Minggen
New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions. (English)
[J] J. Comput. Appl. Math. 233, No. 2, 165-172 (2009). ISSN 0377-0427

This paper is concerned with the approximate solution of second order differential equations $ u''(t) + \sigma u'(t) + f(t, u(t)) = 0,$ $t \in [0,1],$ where $ \sigma $ is a non zero constant and $ f: [0,1] \times {\Bbb R} \to {\Bbb R} $ is a sufficiently smooth function, supplemented with linear integral boundary conditions of type: $ u(0) - \mu_1 u'(0) = \int_0^1 h_1(s) u(s) ds,$ $ u(1) + \mu_2 u'(1) = \int_0^1 h_2(s) u(s) ds, $ with positive constants $ \mu_j$ and given smooth functions $ h_j(t)$. The proposed approach starts establishing an homotopy defined by family of differential equations $ H(u,p) \equiv u'' + \sigma u' + p f(t,u) = 0$, with the parameter $ p \in [0,1]$ so that for $ p=0$ gives a linear equation such that with the boundary conditions has a unique solution $ u = u_0(t)$ easily computed and for the parameter value $ p=1$ is the desired solution of the non linear problem. Now by using $ p$ as a small parameter the solution of $ H(u,p)=0$ can be written as an asymptotic series $ u=u_0+ p u_1 + \dots $ where the successive $ u_j$ can be computed recursively as a solution linear problems and then the solution for $ p=1$ is approximated by the $(m+1)$-sum $ u = \sum_{j=0}^m u_j$. For solving each linear boundary value problem of $ u_j$ the authors propose a reproducing kernel Hilbert space method. Two numerical experiments are presented to show the behaviour of the method depending on the terms $m$ of the series and the number of grid points in the interval $[0,1].$
[Manuel Calvo (Zaragoza)]

MSC 2000:: *65L10 Boundary value problems for ODE (numerical methods)
34B15 Nonlinear boundary value problems of ODE
46E22 Hilbert spaces with reproducing kernels
34B30 Special ODE

Keywords: second order ordinary differential equations; integral boundary conditions; perturbation methods; reproducing kernel Hilbert space method (RKHSM); numerical experiments; homotopy perturbation method (HPM)

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]