*(English)*Zbl 1205.65216

This paper is concerned with the approximate solution of second order differential equations ${u}^{\text{'}\text{'}}\left(t\right)+\sigma {u}^{\text{'}}\left(t\right)+f(t,u\left(t\right))=0,$ $t\in [0,1],$ where $\sigma $ is a non zero constant and $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ 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 $H(u,p)\equiv {u}^{\text{'}\text{'}}+\sigma {u}^{\text{'}}+pf(t,u)=0$, 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 $u={u}_{0}\left(t\right)$ 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 $H(u,p)=0$ 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 $(m+1)$-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 $[0,1]\xb7$

##### 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.) |