TY - JOUR

T1 - Boundary elements and perturbation theory for certain classes of hyperbolic and parabolic problems

AU - Steven Gipson, G.

AU - Reible, D. D.

AU - Savant, S. A.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1988/9

Y1 - 1988/9

N2 - The boundary element method (BEM) has been applied to the solution of partial differential equations of the form ∇2φ = L + f{hook}ekt where L is the time derivative operator ∂/∂t or ∂/∂t, κ is a small magnitude complex quantity, and f{hook} is a known function of the spatial variables. This equation is quite useful in a variety of physical problems, including forced vibrations of membranes, acoustic waves, and damped membrane response. Reported herein is a special mathematical treatment in which the solution to Eq. (1) is obtained by combining perturbation theory with the boundary integral equation technique. A particularly elegant formulation develops in which the solution to Eq. (1) may be rewritten as the solution of a set of coupled Poisson equations. Each equation may be handled by boundary elements using established methods for solving the Poisson equation. The time dependence of the formulation is handled exactly and explicitly. The new technique is of extreme interest in the theory of time-dependent phenomena in that it couples the best features of boundary elements with the well-established mathematics of perturbation theory. The paper concludes with examples of the use of the technique to several engineering problems and a discussion of the method's viability for other types of problems.

AB - The boundary element method (BEM) has been applied to the solution of partial differential equations of the form ∇2φ = L + f{hook}ekt where L is the time derivative operator ∂/∂t or ∂/∂t, κ is a small magnitude complex quantity, and f{hook} is a known function of the spatial variables. This equation is quite useful in a variety of physical problems, including forced vibrations of membranes, acoustic waves, and damped membrane response. Reported herein is a special mathematical treatment in which the solution to Eq. (1) is obtained by combining perturbation theory with the boundary integral equation technique. A particularly elegant formulation develops in which the solution to Eq. (1) may be rewritten as the solution of a set of coupled Poisson equations. Each equation may be handled by boundary elements using established methods for solving the Poisson equation. The time dependence of the formulation is handled exactly and explicitly. The new technique is of extreme interest in the theory of time-dependent phenomena in that it couples the best features of boundary elements with the well-established mathematics of perturbation theory. The paper concludes with examples of the use of the technique to several engineering problems and a discussion of the method's viability for other types of problems.

UR - http://www.scopus.com/inward/record.url?scp=45549121185&partnerID=8YFLogxK

U2 - 10.1016/0264-682X(88)90003-2

DO - 10.1016/0264-682X(88)90003-2

M3 - Article

AN - SCOPUS:45549121185

VL - 5

SP - 133

EP - 139

JO - Engineering Analysis

JF - Engineering Analysis

SN - 0264-682X

IS - 3

ER -