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一类n阶拟线性奇异摄动边值问题的一致有效渐近展开 总被引:1,自引:0,他引:1
本文研究一类n阶拟线性奇异摄动边值问题:εy(n)=f(t,ε,y,…,y(n-2)y(n-1)+g(t,ε,y,…,y(n-2),pj(ε)y(j)(0,ε)-qj(ε)y(j+1)(0,ε)=αj(ε)(0≤j≤n-2),b1(ε)y(n-2)(1,ε)+b2(ε)y(n-1)(1,ε)=β(ε),其中ε>0为小参数.在较一般的条件之下,应用Banach/Picard不动点定理证明了摄动解的存在性及局部唯一性,并给出了摄动解直到n阶导函数的一致有效渐近展开式,推广和改进了已有的结果[1-5]. 相似文献
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刘其林 《高校应用数学学报(A辑)》1993,(3):231-238
本文研究一类非线性微分方程的非线性边值问题的奇摄动,应用边界层校正法构造出解的形式渐近展开式,并借助于上,下解及微分不等式理论研究解及其一阶导数的有关余项估计。 相似文献
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本文研究摄动边值问题dx/dt=f(x,y,t;ε),εdy/dt=g(x,y,t;ε),a1(ε)x(0,ε)+a2(ε)y(0,ε)=a(ε)b1(ε)x(1,ε)+εb2(ε)y(1,ε)=β(ε)这里x,f,β∈Em,y,g,a∈En,0<ε《1,a1(ε),a2(ε),b1(ε),b2(ε)为适当阶数的矩阵.在gy(t)是非奇异矩阵及其它的适当限制下,证明了解的存在唯一性,作出了解的n阶渐近近似式,并得出余项估计. 相似文献
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研究了在子区间上奇异摄动的一类半线性二阶微分方程边值问题,用边界层函数法构造出问题的形式渐近解,借助微分不等式理论证明了渐近解的一致有效性. 相似文献
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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,…,yn-2 Pj(ε)yj(0,ε)-qj(ε)yj+1(0,ε)=Aj(ε)(0≤j≤n-3)a1(ε)y(n-2)(0,ε)-a2(ε)yn-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。 相似文献
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研究不满足法向双曲条件的二阶半线性非自治奇摄动Dirichlet边值问题.首先,利用边界层函数法,构造了问题在两个区间端点的代数边界层,获得了形式渐近解;接着,利用上下解方法,证明了解的存在性、渐近解的一致有效性以及渐近解与精确解之间的误差估计.研究表明:通过对奇异摄动参数进行适当的尺度变换,一定条件下可处理任意退化的二阶半线性非自治奇摄动边值问题.最后,通过一个典型例子验证了理论结果的正确性. 相似文献
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奇摄动非线性边值问题 总被引:5,自引:0,他引:5
本文讨论了一类奇摄动非线性边值问题 .利用伸长变量和边界层校正法 ,得到了问题解的形式渐近展开式 .再用微分不等式理论 ,证明了解的一致有效性 相似文献
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§ 1 . Introduction AclassofintegraldifferentialequationsDirichletboundaryvalueproblemsforordinarydif ferentialequationandellipticequationarediscussedin [1]and [2 ]respectively .Andin [3]akindofnonlocalproblemsforsingularlyperturbedreactiondiffusionsystemsarestudied.Inthispaper,whatisworthpointingoutisaclassofnonlinearboundaryvalueproblemsdiscussed,applyingthemethodofcompositeexpandandthetheoryofdifferentialinequalities.εy″ =f(x ,y ,Tεy ,ε) y′ +g(x ,y ,Tεy,ε) ,0 相似文献
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研究了一类二阶线性椭圆型方程的奇摄动边值问题.利用边界层函数法构造出问题的零次形式近似,并应用椭圆型算子的最大值原理对问题的解作出渐近估计. 相似文献
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研究了一类奇摄动三阶非线性边值问题,在构造形式渐近解的基础上,用微分不等式理论证明了解的存在性,并得出了解的任意阶的一致有效展开式. 相似文献
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Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [1], is used to analytically calculate high-order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [2]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are shown to compare very favorably with two-term asymptotic results. Finally, some Sturm-Liouville operators with exponentially small spectral gap widths are studied. One such problem is applied to analyzing metastable internal layer motion for a certain forced Burgers equation. 相似文献
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The singularly perturbed boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem, the existence, uniqueness and asymptotic behavior of solution for the boundary value problems are studied. 相似文献
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The singularly perturbed boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem, the existence, uniqueness and asymptotic behavior of solution for the boundary value problems are studied. 相似文献
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本文讨论了拟线性椭圆型方程奇摄动Robin边值问题。在适当的条件下,利用不动点定理,研究了边值问题解的存在唯一性及其渐近性态。 相似文献
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We base on Taylor series expansions to construct the numerical method for solving singularly perturbed boundary value problems. We use the trapezoid method to approximate the integrals and obtain three‐term recurrence relationship. The efficiency of the proposed method is demonstrated by test problems. The numerical result is found in a good agreement with exact solution. 相似文献