A singular perturbation problem for the Lavrent'ev-Bitsadze equation

In this note we consider a singular perturbation problem for the equationwhere K(y) = sgn y and. Ε is a small (positive) parameter. This equation for ε≠O is elliptic for y<0 and hyperbolic for y>0. Many of the results carry over to more difficult and interesting problems for equations of mixed type. The particularly simple model treated here permits the elimination of some complications in the analysis involving singular integral equations while preserving the main qualitative features of more general cases. For a special Tricomi-like problem for (1.1) we construct asymptotic expansions in ε, including boundary layer corrections, of the solution. A proof of uniform asymptotic validity of the lowest order terms is given.

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Asymptotic behaviour of solutions of hyperbolic conservation laws ut + (um)x = 0 as m → ∞ with inconsistent initial values

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023957 ◽

1989 ◽

Vol 113 (1-2) ◽

pp. 61-71 ◽

Cited By ~ 3

Author(s):

Xiangsheng Xu

Keyword(s):

Singular Perturbation ◽

Asymptotic Behaviour ◽

Conservation Laws ◽

Positive Measure ◽

Hyperbolic Conservation Laws ◽

Singular Perturbation Problem ◽

Stationary Equation ◽

Hyperbolic Conservation ◽

Perturbation Problem ◽

Image Position

SynopsisWe study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

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A singular perturbation problem with a turning point

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700046888 ◽

1971 ◽

Vol 5 (1) ◽

pp. 61-73 ◽

Cited By ~ 39

Author(s):

A.M. Watts

Keyword(s):

Singular Perturbation ◽

Exact Solutions ◽

Turning Point ◽

Singular Perturbation Problem ◽

Positive Parameter ◽

Perturbation Problem ◽

Small Positive Parameter ◽

The Difference ◽

Asymptotic Forms

We consider the equationεy″ + p(x)y′ + q(x)y = 0, where ε is a small positive parameter and p vanishes in the interval. Two asymptotic forms of solution are obtained and a rigorous estimate is made of the difference between the exact solutions and the asymptotic forms.

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The Schrödinger equation as a singular perturbation problem

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500010982 ◽

1978 ◽

Vol 82 (1-2) ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

E. M. de Jager ◽

T. Küpper

Keyword(s):

Schrödinger Equation ◽

Singular Perturbation ◽

Schrodinger Equation ◽

Eigenvalue Problems ◽

Singular Perturbation Problem ◽

Perturbation Problem ◽

First Approximation ◽

Image Position

SynopsisComparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problemsandwith φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.

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