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A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations. (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:

$-\epsilon {y}^{\left(iv\right)}\left(x\right)+b\left(x\right){y}^{\text{'}\text{'}}\left(x\right)-c\left(x\right)y\left(x\right)=-f\left(x\right),\phantom{\rule{1.em}{0ex}}x\in D,$
$y\left(0\right)=p,\phantom{\rule{4pt}{0ex}}y\left(1\right)=q,\phantom{\rule{4pt}{0ex}}{y}^{\text{'}\text{'}}\left(0\right)=-r,\phantom{\rule{4pt}{0ex}}{y}^{\text{'}\text{'}}\left(1\right)=-s,$

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:

$b\left(x\right)\ge \beta >0,$
$0\ge c\left(x\right)\ge -\gamma ,\phantom{\rule{1.em}{0ex}}\gamma >0,$
$\beta -2\gamma \ge \eta >0\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\eta ,$
$D=\left(0,1\right),\phantom{\rule{1.em}{0ex}}\overline{D}=\left[0,1\right],\phantom{\rule{1.em}{0ex}}y\in \phantom{\rule{4.pt}{0ex}}{\text{C}}^{4}\left(D\right)\cap \phantom{\rule{4.pt}{0ex}}{\text{C}}^{2}\left(\overline{D}\right)·$

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)