Integral Boundary Conditions

Abstract

We first consider an expository linear example:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGa % amyDaiaac+cacaWGKbGaamiEamaaCaaaleqabaGaaGOmaaaakiabgU % caRiabeo7aNjaadwhacqGH9aqpcaaIWaaaaa!4AC4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${d^2}u/d{x^2} + \gamma u = 0$$

with conditions given as:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakqaabeqaaiaadwhadaqadaqaaiaadIhacqGH % 9aqpcqaH+oaEdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacq % GH9aqpcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSYaa8qmaeaa % cqaHYoGydaWgaaWcbaGaaGymaaqabaGccaWG1bWaaeWaaeaacaWG4b % aacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiabe67a4naaBaaameaa % caaIXaaabeaaaSqaaiabe67a4naaBaaameaacaaIYaaabeaaa0Gaey % 4kIipaaOqaaiaadwhadaqadaqaaiaadIhacqGH9aqpcqaH+oaEdaWg % aaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGIbWaaS % baaSqaaiaaikdaaeqaaOGaey4kaSYaa8qmaeaacqaHYoGydaWgaaWc % baGaaGOmaaqabaGccaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaa % GaamizaiaadIhaaSqaaiabe67a4naaBaaameaacaaIXaaabeaaaSqa % aiabe67a4naaBaaameaacaaIYaaabeaaa0Gaey4kIipaaaaa!742D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{gathered} u\left( {x = {\xi _1}} \right) = {b_1} + \int_{{\xi _1}}^{{\xi _2}} {{\beta _1}u\left( x \right)dx} \hfill \\ u\left( {x = {\xi _2}} \right) = {b_2} + \int_{{\xi _1}}^{{\xi _2}} {{\beta _2}u\left( x \right)dx} \hfill \\ \end{gathered} $$

γ, β ₁,, and β ₂ are assumed constants here although they can be functions of x with minor modifications to the procedure given. In decomposition format we have Lu + Ru = 0 or L⁻¹Lu = I:−L⁻¹ Ru or

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWG1bGaeyypa0Jaam4yamaaBaaaleaa % caaIWaaabeaakiabgUcaRiaadogadaWgaaWcbaGaaGymaaqabaGcca % WG4bGaeyOeI0Iaeq4SdCMaamysamaaDaaaleaacaWG4baabaGaaGOm % aaaakmaaqahabaGaamyDamaaBaaaleaacaWGUbaabeaaaeaacaWGUb % Gaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa!5475!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$u = {c_0} + {c_1}x - \gamma I_x^2\sum\limits_{n = 0}^\infty {{u_n}} $$

where EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGjbWaa0baaSqaaiaadIhaaeaacaaI % Yaaaaaaa!41D3!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$I_x^2$$ is a two-fold pure integration with respect to x.