TY - JOUR

T1 - A review of linear and nonlinear Cauchy singular integral and integro-differential equations arising in mechanics

AU - Cuminato, J.A.

AU - Fitt, A.D.

AU - McKee, S.

PY - 2007

Y1 - 2007

N2 - This study is primarily concerned with the presentation of a review of a collection (which could be regarded as a "test set") of linear and nonlinear singular integro-differential equations with Cauchy kernels, all of which arise from practical applications in Applied Mathematics and
Mathematical Physics. The main objective of this review is to provide numerical analysts and researchers interested in algorithm development with model problems of genuine scientific interest on which to test their algorithms.
Brief details of the methodology of derivation of the equations are provided and, where possible, existence, uniqueness and asymptotic results are discussed. References are also given to other studies that have dealt with similar problems. The importance of carrying out the necessary mathematical analysis is emphasized for one class of problems where it is shown that the solution abruptly ceases to exist as a parameter is varied. It is further shown that developing asymptotic estimates for the behavior of the solutions is very often a crucial component in the design of effective numerical methods. The importance of regularization is discussed for a class of problems, specific conclusions are drawn and recommendations are discussed. An appendix contains further related problems that may be used for further comparison purposes.

AB - This study is primarily concerned with the presentation of a review of a collection (which could be regarded as a "test set") of linear and nonlinear singular integro-differential equations with Cauchy kernels, all of which arise from practical applications in Applied Mathematics and
Mathematical Physics. The main objective of this review is to provide numerical analysts and researchers interested in algorithm development with model problems of genuine scientific interest on which to test their algorithms.
Brief details of the methodology of derivation of the equations are provided and, where possible, existence, uniqueness and asymptotic results are discussed. References are also given to other studies that have dealt with similar problems. The importance of carrying out the necessary mathematical analysis is emphasized for one class of problems where it is shown that the solution abruptly ceases to exist as a parameter is varied. It is further shown that developing asymptotic estimates for the behavior of the solutions is very often a crucial component in the design of effective numerical methods. The importance of regularization is discussed for a class of problems, specific conclusions are drawn and recommendations are discussed. An appendix contains further related problems that may be used for further comparison purposes.

KW - mechanics

KW - integral equations

KW - cauchy equations

KW - applied mathematics

UR - http://projecteuclid.org/euclid.jiea/1182525213

UR - http://rmmc.asu.edu/abstracts/jie/vol19-2/cumipag1.pdf

UR - http://dx.doi.org/10.1216/jiea/1182525213

U2 - 10.1216/jiea/1182525213

DO - 10.1216/jiea/1182525213

M3 - Article

VL - 19

SP - 163

EP - 207

JO - Journal of Integral Equations and Applications

JF - Journal of Integral Equations and Applications

SN - 0897-3962

IS - 2

ER -