北大数学物理方法(A)-数学物理方程教案04分离变量法3

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Outline1où©lCþ{(n)ÔnÆ2007cSC.S.Wu1où©lCþ{(n)OutlineùLJ:1šàg­½¯K«~{?˜ÚuÐ2šàg.^‡àgzÄg´AÏE|µ§9.^‡Óžàgz3­¡‹IXeLaplaceŽÎ΋IXeLaplaceŽÎ¥‹IXeLaplaceŽÎC.S.Wu1où©lCþ{(n)OutlineùLJ:1šàg­½¯K«~{?˜ÚuÐ2šàg.^‡àgzÄg´AÏE|µ§9.^‡Óžàgz3­¡‹IXeLaplaceŽÎ΋IXeLaplaceŽÎ¥‹IXeLaplaceŽÎC.S.Wu1où©lCþ{(n)OutlineùLJ:1šàg­½¯K«~{?˜ÚuÐ2šàg.^‡àgzÄg´AÏE|µ§9.^‡Óžàgz3­¡‹IXeLaplaceŽÎ΋IXeLaplaceŽÎ¥‹IXeLaplaceŽÎC.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesReferencesÇÂÁ§5êÆÔn{6§§14.6,15.1,15.2ù&œ§5êÆÔn{6§§8.3nÎ!X1Á§5êÆÔn{6§§10.4,12.1C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesReferencesÇÂÁ§5êÆÔn{6§§14.6,15.1,15.2ù&œ§5êÆÔn{6§§8.3nÎ!X1Á§5êÆÔn{6§§10.4,12.1C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesReferencesÇÂÁ§5êÆÔn{6§§14.6,15.1,15.2ù&œ§5êÆÔn{6§§8.3nÎ!X1Á§5êÆÔn{6§§10.4,12.1C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachùLJ:1šàg­½¯K«~{?˜ÚuÐ2šàg.^‡àgzÄg´AÏE|µ§9.^‡Óžàgz3­¡‹IXeLaplaceŽÎ΋IXeLaplaceŽÎ¥‹IXeLaplaceŽÎC.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachÝ/«S­½¯Kk½)¯K∂2u∂x2+∂2u∂y2=f(x,y)0xa,0ybu x=0=0u x=a=00≤y≤bu y=0=0u y=b=00≤x≤a^UƒAàg¯K¼êÐm{¦)C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachÝ/«S­½¯Kk½)¯K∂2u∂x2+∂2u∂y2=f(x,y)0xa,0ybu x=0=0u x=a=00≤y≤bu y=0=0u y=b=00≤x≤aŒu(x,y)=∞Xn=1Yn(y)sinnπaxf(x,y)=∞Xn=1gn(y)sinnπaxC.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachÝ/«S­½¯Kk½)¯K∂2u∂x2+∂2u∂y2=f(x,y)0xa,0ybu x=0=0u x=a=00≤y≤bu y=0=0u y=b=00≤x≤a“\§Ú.^‡§ŒY00n(y)−nπa2Yn(y)=gn(y)Yn(0)=0Yn(b)=0dd=Œ¦ÑYn(y)C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachÝ/«S­½¯Kk½)¯K∂2u∂x2+∂2u∂y2=f(x,y)0xa,0ybu x=0=0u x=a=00≤y≤bu y=0=0u y=b=00≤x≤aŒu(x,y)=∞Xm=1Xm(x)sinmπbyf(x,y)=∞Xm=1hm(x)sinmπbyC.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachÝ/«S­½¯Kk½)¯K∂2u∂x2+∂2u∂y2=f(x,y)0xa,0ybu x=0=0u x=a=00≤y≤bu y=0=0u y=b=00≤x≤a“\§Ú.^‡§ŒX00m(x)−mπb2Xm(x)=hm(x)Xm(0)=0Xm(a)=0dd½Œ¦ÑXm(y)C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachµãùü«‰{vkK O©Ì‡ØÓ´šàg‘gn(y)Úhm(x)¼ê/ªŒUØÓ§Ï3'uYn(y)ÚXm(x)šàgü‡~‡©§Y00n(y)−nπa2Yn(y)=gn(y)X00m(x)−mπb2Xm(x)=hm(x)¥ŒUk˜‡´u¦)C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproachùLJ:1šàg­½¯K«~{?˜ÚuÐ2šàg.^‡àgzÄg´AÏE|µ§9.^‡Óžàgz3­¡‹IXeLaplaceŽÎ΋IXeLaplaceŽÎ¥‹IXeLaplaceŽÎC.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproach­?êÐmk½)¯K∂2u∂x2+∂2u∂y2=f(x,y)0xa,0ybu x=0=0u x=a=00≤y≤bu y=0=0u y=b=00≤x≤a„Œ±č?˜Ú‰{§=òu(x,y)Úf(x,y)QU¼ê{Xn(x)}!qU¼ê{Ym(y)}Ðm(­?ê)u(x,y)=∞Xn=1∞Xm=1cnmsinnπaxsinmπbyf(x,y)=∞Xn=1∞Xm=1dnmsinnπaxsinmπbyÐmXêcnm–¦C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproach­?êÐm∂2u∂x2+∂2u∂y2=f(x,y)u(x,y)=∞Xn=1∞Xm=1cnmsinnπaxsinmπbyf(x,y)=∞Xn=1∞Xm=1dnmsinnπaxsinmπbyFϏf(x,y)®§cnm®C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproach­?êÐm∂2u∂x2+∂2u∂y2=f(x,y)u(x,y)=∞Xn=1∞Xm=1cnmsinnπaxsinmπbyf(x,y)=∞Xn=1∞Xm=1dnmsinnπaxsinmπbyFϏf(x,y)®§cnm®F3Š­?êÐmž§®²Ä.^‡C.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneousBVC’sLaplacianinOrthogonalCurvilinearCoordinatesIllustrativeExampleFurtherDevelopmentoftheSolutionApproach­?êÐm∂2u∂x2+∂2u∂y2=f(x,y)u(x,y)=∞Xn=1∞Xm=1cnmsinnπaxsinmπbyf(x,y)=∞Xn=1∞Xm=1dnmsinnπaxsinmπbyÏdIòþ¡Ðmª“\§−∞Xn=1∞Xm=1cnmnπa2+mπb2sinnπaxsinmπby=∞Xn=1∞Xm=1dnmsinnπaxsinmπbyC.S.Wu1où©lCþ{(n)ImhomogeneousSteadyStateProblemsHomogenizationofImhomogeneous

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