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New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions. (English) Zbl 1205.65216

This paper is concerned with the approximate solution of second order differential equations ${u}^{\text{'}\text{'}}\left(t\right)+\sigma {u}^{\text{'}}\left(t\right)+f\left(t,u\left(t\right)\right)=0,$ $t\in \left[0,1\right],$ where $\sigma$ is a non zero constant and $f:\left[0,1\right]×ℝ\to ℝ$ is a sufficiently smooth function, supplemented with linear integral boundary conditions of type: $u\left(0\right)-{\mu }_{1}{u}^{\text{'}}\left(0\right)={\int }_{0}^{1}{h}_{1}\left(s\right)u\left(s\right)ds,$ $u\left(1\right)+{\mu }_{2}{u}^{\text{'}}\left(1\right)={\int }_{0}^{1}{h}_{2}\left(s\right)u\left(s\right)ds,$ with positive constants ${\mu }_{j}$ and given smooth functions ${h}_{j}\left(t\right)$.

The proposed approach starts establishing an homotopy defined by family of differential equations $H\left(u,p\right)\equiv {u}^{\text{'}\text{'}}+\sigma {u}^{\text{'}}+pf\left(t,u\right)=0$, with the parameter $p\in \left[0,1\right]$ so that for $p=0$ gives a linear equation such that with the boundary conditions has a unique solution $u={u}_{0}\left(t\right)$ 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 $H\left(u,p\right)=0$ can be written as an asymptotic series $u={u}_{0}+p{u}_{1}+\cdots$ 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 $\left(m+1\right)$-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 $\left[0,1\right]·$

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B15 Nonlinear boundary value problems for ODE 46E22 Hilbert spaces with reproducing kernels 34B30 Special ODE (Mathieu, Hill, Bessel, etc.)