scholarly journals A unified boundary integral equation method for a class of second order elliptic boundary value problems

1984 ◽  
Vol 25 (4) ◽  
pp. 501-517 ◽  
Author(s):  
M. Rezayat ◽  
F. J. Rizzo ◽  
D. J. Shippy
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Second Order ◽  
Elliptic Type ◽  
Integral Equation Formulation

AbstractA generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.

On mixed boundary value problems for the Helmholtz equation

1977 ◽  
Vol 77 (1-2) ◽  
pp. 65-77 ◽  
Author(s):  
R. Kress ◽  
G. F. Roach
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Theoretic Approach ◽  
Mixed Boundary Value Problems

SynopsisExistence and uniqueness theorems are obtained for a class of mixed boundary value problems associated with the three-dimensional Helmholtz equation. In this context the boundary of the region of interest is assumed to consist of the union of a finite number of disjoint, closed, bounded Lyapunov surfaces on some of which are imposed Dirichlet conditions whilst Neumann conditions are imposed on the remainder. An integral equation method is adopted throughout. The required boundary integral equations are generated by a modified layer theoretic approach which extends the work of Brakhage and Werner [1] and Leis [2, 3].

scholarly journals The Numerical Analysis of the Flow on the Smooth and Nonsmooth Boundaries by IBEM/DBIEM

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Gang Xu ◽  
Guangwei Zhao ◽  
Jing Chen ◽  
Shuqi Wang ◽  
Weichao Shi
Keyword(s):  
Boundary Value ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Tangential Velocity ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Surface Shape ◽  
Normal Vector ◽  
Computational Domain ◽  
Nonsmooth Boundary

The value of the tangential velocity on the Boundary Value Problem (BVP) is inaccurate when comparing the results with analytical solutions by Indirect Boundary Element Method (IBEM), especially at the intersection region where the normal vector is changing rapidly (named nonsmooth boundary). In this study, the singularity of the BVP, which is directly arranged in the center of the surface of the fluid computing domain, is moved outside the computational domain by using the Desingularized Boundary Integral Equation Method (DBIEM). In order to analyze the accuracy of the IBEM/DBIEM and validate the above-mentioned problem, three-dimensional uniform flow over a sphere has been presented. The convergent study of the presented model has been investigated, including desingularized distance in the DBIEM. Then, the numerical results were compared with the analytical solution. It was found that the accuracy of velocity distribution in the flow field has been greatly improved at the intersection region, which has suddenly changed the boundary surface shape of the fluid domain. The conclusions can guide the study on the flow over nonsmooth boundaries by using boundary value method.

scholarly journals Forward

2007 ◽  
Vol 1 (1) ◽  
Author(s):  
Thomas J. Rudolphi
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Equation Method ◽  
Equation Approach ◽  
Equation Formulation ◽  
Integral Equation Formulation ◽  
Potential Problems ◽  
Integral Equation Approach

<br /><br /> <table width="530" border="0" cellspacing="0" cellpadding="0"> <tr> <td align="left" valign="top"> <a name="abstract"></a> <span class="subtitle" style="font-weight:bold">Abstract</span><br /> <p><img src="http://ejbe.libraries.rutgers.edu/files/rizzo.gif" align=left HSPACE=20>This is the first of two special issues of the Electronic Journal of Boundary Elements dedicated to Frank Rizzo. To say that Frank Rizzo played an important role in the development of what he referred to as “boundary integral equations� would not give much credit to where much credit is due. While it could be argued that the use of integral equations to formulate and form a computational basis of many of the problems of applied mathematics and engineering would probably have been inevitably developed, it was Frank’s seminal work on using the integral equation approach to classical elastostatics that set a whole new research area into motion. His dissertation (which we thought would be of interest to include in this issue) topic, as suggested by his mentor Marvin Stippes at the University of Illinois, and subsequently so well documented in the oft-cited paper “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics�, Quarterly of Applied Mechanics, 1967, represented the quantum step in the use of integral equations for classical scalar potential problems to the vector potential problems of practical engineering significance. The theoretical basis for this development was Betti’s reciprocal work theorem with the fundamental (response to a point force) solution of the equations of elastostatics, but it was Frank Rizzo who actually breathed the new life into this classical mathematics. A nontrivial contribution of Frank’s original work was to not only to achieve the singular integral equation formulation, but also the systematic methodology of reducing the elegant integral equation formulation to well conditioned, linear algebraic equations by proper analytical integration of the singular terms. Those combined theoretical and practical developments by Frank set into motion a whole new and modern approach to numerically solving partial differential equations, at least of the elliptic type. With Frank’s hard work and the recognition of its elegance and potential by several of his early disciples, the integral equation method blossomed into a powerful and practical computational methodology that would eventually be called “boundary elements�. Amongst the early disciples of the integral equation method, several of which contributed significantly to advancing the methodology to a sophisticated and now mature state, are the authors of this issue and its sequel dedicated to Frank. It is undoubtedly fair to say that most of these authors were, at one time or even continuously, colleagues and personal friends of Frank Rizzo. Frank’s contributions to the boundary integral equation method spanned nearly four decades, from roughly 1964 to 2001. I, too, have been very privileged to become involved with this field in the 1970’s and later to work side by side with Frank, especially in that part of the development of the methodology for what is now referred to as “hypersingular� integral equations. I’m sure that all the present authors can recall numerous occasions and conversations with Frank on a technical point or issue regarding the application of “his� boundary integral method to their own problem of interest. Throughout his productive career, his easy going, collegial, engaging, yet rigorous style earned him respect and admiration that surely befits the “father� of modern boundary integral methods. This commemorative sequence of two issues represents only a small token of tribute and recognition that Frank Rizzo so much deserves for his “singular� contributions to the field that he virtually invented, developed, promoted and nurtured to maturity. Thomas J. Rudolphi Iowa State University <br /><br /><br /> </td> </tr> </table>

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