Multi-dimensional initial-boundary value problems with strong nonlinearities

Howes, F. A.

doi:10.1007/bf00276861

Cited by 16 publications

(11 citation statements)

References 17 publications

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“…In this section we present several of the rules that we have defined for the semilinear Dirichlet problem given by Equations (11)(12)(13). The rules consist of a comment section, a preconditions section and an actions section.…”

Section: Reducing the Theory To Rulesmentioning

confidence: 99%

“…Results from nonlinear singular perturbation theory provide the means by which the asymptotic behavior of certain classes of nonlinear boundary value problems can be characterized in their solution space. While results are available for boundary and initial-boundary value problems based on nonlinear partial differential equations, as well as systems of certain classes of nonlinear ordinary differential equations [13,14], we will only discuss results for boundary value problems based on scalar, singularly perturbed, second order, nonlinear ordinary differential equations since this portion of the theory is well known and fairly complete [1]. Equations of this type have the following general formula,…”

Section: The Nonlinear Boundary Value Problemmentioning

confidence: 99%

“…These requirements, if satisfied, yield parameters that are used in the definition of the bounding functions. For example, consider one form of the definition of (~) -stability for the problem defined by Equations (11)(12)(13). We assume that the function h(x, y) has the appropriate number of partial derivatives with respect to y, and that q ~> 0 and n /> 2, where both n and q are integers.…”

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confidence: 99%

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Automatically identifying the asymptotic behavior of nonlinear singularly perturbed boundary value problems

Russo

Peskin

1992

J Autom Reasoning

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Abstract. We describe a computational procedure designed to automatically analyze the behavior of certain general classes of nonhnear singular perturbation problems by applying the combined results of a body of theory that proves the existence of solutions for these problems. We have also created a computer program that implements the computational procedure. The core mathematical knowledge contained in our program is composed of rules that embody the results of mathematical theorems from nonhnear singular perturbation theory The principle method of proof used in the mathematical theory yields an estimate of a soiutlon by constructing sharp bounding functions that define a region in which the solution exists uniquely As a result, a successful application of our program produces an approximation of a solution as a side effect. In addition, the mathematical theory can be used to show the existence of multiple solutions for a nonlinear singularly perturbed boundary value problem. This feature is also reflected in the results obtained from our program. The ability to construct such a program depends critically on the successful coupling of a non-deterministic programming technique called path-finding with the capabilities of a computer algebra system

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Section: Reducing the Theory To Rulesmentioning

confidence: 99%

Section: The Nonlinear Boundary Value Problemmentioning

confidence: 99%

mentioning

confidence: 99%

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Automatically identifying the asymptotic behavior of nonlinear singularly perturbed boundary value problems

Russo

Peskin

1992

J Autom Reasoning

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“…CCC 0749-159)(/96/04044 [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] passive scalar in which they studied the initial-value problem where $~(z,y,t) is the streamfunction of the flow, 8 is the passive scalar, and k is the diffusivity. In a steady plane shear-flow, the tracer equation becomes Or + UO,.…”

Section: Introductionmentioning

confidence: 99%

An asymptotic-numerical method for a time-dependent singularly perturbed system with turning points

Shao

Chin

1996

Numer. Methods Partial Differential Eq.

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“…The analysis is performed in the style of Howes [2,3,4], It begins with a multiple-scales asymptotic analysis. This provides the appropriate local scalings and indicates candidate forms for a bounding function.…”

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Shock-layer bounds for a singularly perturbed equation

Scroggs¹

1995

Quart. Appl. Math.

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Abstract. The size of the shock-layer governed by a conservation law is studied. The conservation law is a parabolic reaction-convection-diffusion equation with a small parameter multiplying the diffusion term and convex flux. Rigorous upper and lower bounding functions for the solution of the conservation law are established based on maximum-principle arguments. The bounding functions demonstrate that the size of the shock-layer is proportional to the parameter multiplying the diffusion term.

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Multi-dimensional initial-boundary value problems with strong nonlinearities

Cited by 16 publications

References 17 publications

Automatically identifying the asymptotic behavior of nonlinear singularly perturbed boundary value problems

Automatically identifying the asymptotic behavior of nonlinear singularly perturbed boundary value problems

An asymptotic-numerical method for a time-dependent singularly perturbed system with turning points

Shock-layer bounds for a singularly perturbed equation

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