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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)

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