*(English)*Zbl 1025.65044

Authors’ abstract: “Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative are considered. A numerical method is suggested to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable boundary conditions. Then the domain of definition of the differential equation (a closed interval) is divided into three non-overlapping sub-intervals, which we call them inner regions (boundary layers) and outer region. Then the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then nonlinear equations. To solve nonlinear equations, Newton’s method of quasi-linearization is applied. The present method is demonstrated by providing examples. The method is easy to implement and suitable for parallel computing”.

The method solves equations of the following form:

where $\epsilon $ is a small positive parameter and $b\left(x\right)$, $c\left(x\right)$ and $f\left(x\right)$ are sufficiently smooth functions satisfying the following conditions:

The sign of the function $f$ and the constants $p$, $q$, $r$ and $s$ does not seem to be arbitrary, because of the presence of the minus sign in the equations, but the authors do not clarify this point in the Introduction.

##### MSC:

65L10 | Boundary value problems for ODE (numerical methods) |

34B05 | Linear boundary value problems for ODE |

34E15 | Asymptotic singular perturbations, general theory (ODE) |

65Y05 | Parallel computation (numerical methods) |

34B15 | Nonlinear boundary value problems for ODE |

65L12 | Finite difference methods for ODE (numerical methods) |