Boundary Integral Formulations

Abstract

An operator is a process which applied to a function or a set of functions produces another function, i.e.,

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaacI % cacaWG1bGaaiykaiabg2da9iaadkgaaaa!3B25!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ l(u) = b $$

(0.1)

where l (u) is the operator which applied to u produces b; u and b may be scalars or vectors: l () may be an ordinary differential operator such as

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaacI % cacaGGPaGaeyypa0JaamyyamaaBaaaleaacaaIWaaabeaakmaalaaa % baGaamizamaaCaaaleqabaGaaGOmaaaakiaacIcacaGGPaaabaGaam % izaiaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIaamyyamaa % BaaaleaacaaIXaaabeaakmaalaaabaGaamizaiaacIcacaGGPaaaba % GaamizaiaadIhaaaGaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaa % kiaacIcacaGGPaaaaa!4C3C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ l() = {a_0}\frac{{{d^2}()}}{{d{x^2}}} + {a_1}\frac{{d()}}{{dx}} + {a_2}() $$

(0.2)

a partial differential operator such as

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaacI % cacaGGPaGaeyypa0ZaaSaaaeaacqGHciITaeaacqGHciITcaWG4baa % amaabmaabaGaam4AamaalaaabaGaeyOaIyRaaiikaiaacMcaaeaacq % GHciITcaWG4baaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaeyOa % IylabaGaeyOaIyRaamyEaaaadaqadaqaaiaadUgadaWcaaqaaiabgk % Gi2kaacIcacaGGPaaabaGaeyOaIyRaamyEaaaaaiaawIcacaGLPaaa % aaa!5130!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ l() = \frac{\partial }{{\partial x}}\left( {k\frac{{\partial ()}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {k\frac{{\partial ()}}{{\partial y}}} \right) $$

(0.3)

or an integro operator

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaacI % cacaGGPaGaeyypa0Zaa8qCaeaacqaHfpqDcaGGOaGaamiDaiabgkHi % Tiabes8a0jaacMcacaGGOaGaaiykaiaadsgacqaHepaDaSqaaiaaic % daaeaacaWG0baaniabgUIiYdaaaa!4832!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ l() = \int\limits_0^t {\upsilon (t - \tau )()d\tau } $$

(0.4)

which can also be written in terms of convolution notation as

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacI % cacaGGPaGaeyypa0JaeqyXduNaaiikaiaadshacaGGPaGaaiOkaiaa % cIcacaGGPaaaaa!3F44!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ L() = \upsilon (t)*() $$

(0.5)