Applications of the Maximum Principle to Singular Perturbation Problems

Dorr, Fred W.; Parter, Seymour V.; Shampine, L. F.

doi:10.1137/1015002

Cited by 76 publications

(33 citation statements)

References 14 publications

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“…e-»o^v \uR(t), t0<t<b, and lim /(/, e) -u'L(t), a< t <t0-8, e^o* 7 v' "' [ M^(f), t0 + 8 < t < b, for each fixed 5 > 0 independent of e. 3 The results in [4] are valid under mild (i.e., C(1)) smoothness restrictions on the function/ = /(/,>',>'')• Later writers have almost exclusively concerned themselves with constructing complete asymptotic expansions of the solution of (1.1), (1.2) under basically the same assumptions as in the original formulation of Haber and Levinson. The only additional requirement is that / have sufficiently many derivatives.…”

mentioning

confidence: 99%

“…, for t in (t0 -8, t0 + 5), where 8 > 0 is a small constant; in addition, |/(i,j>,y)l -» oo, as \y'\-+co,for(t,y)in<Sl; (3) there exists a positive constant k such that fy. (t, uL(t), u'L(t)) > k > 0, t0-8 < t < t0, andfy.…”

mentioning

confidence: 99%

“…and it follows from the maximum principle (cf., e.g., [3]) that z'(t, e) = O (e), a < t < t0 -8. On (t0 -8, tQ) z satisfies ez" =fy{t, e}Lz' + O (e), t0 -8 < t < t0,…”

mentioning

confidence: 99%

“…for t in (tx -8, tx + 8), and d(t,e) = 0(\u'M(t2) -u'R(t2)\), for t in (t2 -8, t2 + 8); here 8 > 0 is a small constant; in addition, \f(t,y,y')\ -> oo as \y'\ -> oo, for (t,y) in <$ ; (3) there exists a positive constant k such that…”

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confidence: 99%

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Singularly Perturbed Boundary Value Problems with Angular Limiting Solutions

Howes¹

1978

Transactions of the American Mathematical Society

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Abstract. A basic result of Haber and Levinson which describes the behavior of solutions of ey" = f(t,y,yr), a < t < b, y(a, e), y(b, e) prescribed, in the presence of a reduced solution with corners is modified to treat related classes of problems. Under various stability assumptions, solutions are shown to remain, for small e > 0, in a o(l)-neighborhood of an angular reduced solution with the possible exception of narrow layers near the boundaries in some cases. Each aspect of the theory developed here is illustrated by several examples.

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confidence: 99%

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confidence: 99%

“…and it follows from the maximum principle (cf., e.g., [3]) that z'(t, e) = O (e), a < t < t0 -8. On (t0 -8, tQ) z satisfies ez" =fy{t, e}Lz' + O (e), t0 -8 < t < t0,…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Singularly Perturbed Boundary Value Problems with Angular Limiting Solutions

Howes¹

1978

Transactions of the American Mathematical Society

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“…The driving term on the right-hand side is simply the error due to the linear interpolation and can be bounded by a standard result of approximation theory. For a Dirichlet problem, we can use the maximum principle (Dorr, Parter and Shampine [16], Protter and Weinberger [35]) to obtain the following a posteriori error bound. homogeneous Dirichlet boundary condition.…”

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confidence: 99%

A Hybrid Asymptotic-Finite Element Method for Stiff Two-Point Boundary Value Problems

Chin¹,

Krasny²

1983

SIAM J. Sci. and Stat. Comput.

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Abstract. An accurate and efficient numerical method has been developed for a nonlinear stiff second order two-point boundary value problem. The scheme combines asymptotic methods with the usual solution techniques for two-point boundary value problems. A new modification of Newton's method or quasilinearization is used to reduce the nonlinear problem to a sequence of linear problems. The resultant linear problem is solved by patching local solutions at the knots or equivalently by projecting onto an affine subset constructed from asymptotic expansions. In this way, boundary layers are naturally incorporated into the approximation. An adaptive mesh is employed to achieve an error of O(1/N2)+O(/e). Here, N is the number of intervals and e << is the singular perturbation parameter. Numerical computations are presented.

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Singularly Perturbed Nonlinear Boundary‐Value Problems Whose Reduced Equations Have Singular Points

Howes

1977

Stud Appl Math

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Using results on implicit first‐order differential equations, the existence and the asymptotic behavior of solutions of singularly perturbed second‐order boundary‐value problems whose reduced equations have singular points are studied. The variety of possible nonuniform behavior is investigated by means of a stability analysis of the reduced solutions and is illustrated with several model problems which are canonical in terms of the first‐order singular‐point theory.

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Applications of the Maximum Principle to Singular Perturbation Problems

Cited by 76 publications

References 14 publications

Singularly Perturbed Boundary Value Problems with Angular Limiting Solutions

Singularly Perturbed Boundary Value Problems with Angular Limiting Solutions

A Hybrid Asymptotic-Finite Element Method for Stiff Two-Point Boundary Value Problems

Singularly Perturbed Nonlinear Boundary‐Value Problems Whose Reduced Equations Have Singular Points

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