Applied Mathematics and Mechanics

, Volume 17, Issue 12, pp 1193–1201

# Singular perturbations for a class of boundary value problems of higher order nonlinear differential equations

• Shi Yuming
• Liu Guangxu
Article

## Abstract

In this paper, it has been studied that the singular perturbations for the higher order nonlinear boundary value problem of the form by the method of higher order differential inequalities and boundary layer correction. Under some mild conditions, the existence of the perturbed solution is proved and its uniformly efficient asymptotic expansions up to its n-th order derivative function are given out. Hence, the existing results are extended and improved.

$$\begin{gathered} \varepsilon ^{\text{2}} y^{(n)} = f(t, \varepsilon , y, \cdot \cdot \cdot , y^{(n + 2)} ) \hfill \\ p_j (\varepsilon )y^{(j)} (0, e) - q_j (\varepsilon )y^{(i + 1)} (0, \varepsilon ) = A_j (\varepsilon ) (0 \leqslant j \leqslant n - 3) \hfill \\ a_1 (\varepsilon )y^{(n - 2)} (0, \varepsilon ) - a_2 (\varepsilon )y^{(n - 1)} (0, \varepsilon ) = B(\varepsilon ) \hfill \\ b_1 (\varepsilon )y^{(n - 2)} (1, \varepsilon ) - b_2 (\varepsilon )y^{(n - 1)} (1, \varepsilon ) = C(\varepsilon ) \hfill \\ \end{gathered}$$
by the method of higher order differential inequalities and boundry layer corrections. Under some mild conditions, the existense of the perturbed solution is proved and its uniformly efficient asymptotic expansions up to its n-th order derivative function are given out. Hence, the existing results are extended and improved.

### Key words

nonlinear boundary value problem singular perturbation uniformly efficient asymptotic expansion higher order differential inequalities boundary layer correction

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