一类高阶非线性边值问题的奇异摄动

Singular Perturbations for a Class of Boundary Value Problems of Higher Order Nonlinear Differential Equations

摘要: 本文应用高阶微分不等式技巧和边界层校正法研究一类高阶非线性方程混合边值问题: e²yⁿ=f(t,e,y,…,y^n-2 Pj(ε)y^j(0,ε)-qj(ε)y^j+1(0,ε)=Aj(ε)(0≤j≤n-3)a₁(ε)y^(n-2)(0,ε)-a₂(ε)y^n-1(0,ε)=B(ε)b₁(ε)y^(n-2)(1,ε)十b₂(ε)y^(n-1)(1,ε)=C(ε)的奇异摄动。在较一般的条件下,证明了摄动解的存在性,并得到了摄动解直到n阶导函数的一致有效渐近展开式,从而推广和改进了前人的结果。
- 非线性边值问题 /
- 奇异摄动 /
- 一致有效渐近展开 /
- 高阶微分不等式 /
- 边界层校正项
Abstract: In this paper, it has been studied that the singular perturbations for the higherorder nonlinear boundary value problem of the formε²y(n)=f(t, ε, y.…, y^(n-2))pj(ε)y⁽¹⁾(0, ε)-qj(ε)y^(j+1)(0.ε)=Aj(ε)(0≤j≤n-3)a₁(ε)y^(n-2)(0.ε)-a₂(ε)y^(n-1)(0, ε)=B(ε)b₁(ε)y^(n-2)(1, ε)+b₂(ε)y^(n-1),(1. ε)=C(ε)by the method of higher order differential inequalities and boundary layer corrections.Under some mild conditions, the existence of the perturbed solution is proved and itsuniformly efficient asymptotic expansions up to its n-th order derivative function aregiven out. Hence, the existing results are extended and improved.
- nonlinear boundary value problem /
- singular perturbation /
- uniformly efficient asymptotic expansion /
- higher orderdifferential inequalities /
- boundary layer correction

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