北大数学物理方法(A)-数学物理方程教案16Green函数2

Outline1›8ùGreen¼ê()ÔnÆêÆÔn{‘§|2007cSC.S.Wu1›8ùGreen¼ê()OutlineùLJ:1SPoisson§1˜Š¯KGreen¼ê©lCþ{”{2¹ž¯KGreen¼êJ{µ½)¯Ké¡5¹ž¯KGreen¼ê){Green¼ê¦{C.S.Wu1›8ùGreen¼ê()OutlineùLJ:1SPoisson§1˜Š¯KGreen¼ê©lCþ{”{2¹ž¯KGreen¼êJ{µ½)¯Ké¡5¹ž¯KGreen¼ê){Green¼ê¦{C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsReferencesÇÂÁ§5êÆÔn{6§§20.4—20.6ù&œ§5êÆÔn{6§§12.2,12.3nÎ!X1Á§5êÆÔn{6§§14.3,14.4,14.6C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesSPoisson§1˜Š¯KGreen¼ê∇22G(r;r0)=−1ε0δ(r−r0)|r|a,|r0|aG(r;r0)r=a=0Ù¥r2=x2+y2∇22=∂2∂x2+∂2∂y2C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesSPoisson§1˜Š¯KGreen¼ê∇22G(r;r0)=−1ε0δ(r−r0)|r|a,|r0|aG(r;r0)r=a=0Ù¥r2=x2+y2∇22=∂2∂x2+∂2∂y2C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesùLJ:1SPoisson§1˜Š¯KGreen¼ê©lCþ{”{2¹ž¯KGreen¼êJ{µ½)¯Ké¡5¹ž¯KGreen¼ê){Green¼ê¦{C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesIO‰{´µÄ§´˜‡šàg§§¤±òGreen¼êUƒAàg¯K¼êÐmæ^²¡4‹IX§‹I:˜3%G(r;r0)=R0(r)+∞Xm=1Rm1(r)cosmφ+Rm2(r)sinmφδ(r−r0)=1r0δ(r−r0)×12π+1π∞Xm=1cosmφcosmφ0+sinmφsinmφ0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesIO‰{´µÄ§´˜‡šàg§§¤±òGreen¼êUƒAàg¯K¼êÐmæ^²¡4‹IX§‹I:˜3%G(r;r0)=R0(r)+∞Xm=1Rm1(r)cosmφ+Rm2(r)sinmφδ(r−r0)=1r0δ(r−r0)×12π+1π∞Xm=1cosmφcosmφ0+sinmφsinmφ0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesIO‰{´µÄ§´˜‡šàg§§¤±òGreen¼êUƒAàg¯K¼êÐmæ^²¡4‹IX§‹I:˜3%G(r;r0)=R0(r)+∞Xm=1Rm1(r)cosmφ+Rm2(r)sinmφδ(r−r0)=1r0δ(r−r0)×12π+1π∞Xm=1cosmφcosmφ0+sinmφsinmφ0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesIO‰{´µÄ§´˜‡šàg§§¤±òGreen¼êUƒAàg¯K¼êÐmæ^²¡4‹IX§‹I:˜3%G(r;r0)=R0(r)+∞Xm=1Rm1(r)cosmφ+Rm2(r)sinmφδ(r−r0)=1r0δ(r−r0)×12π+1π∞Xm=1cosmφcosmφ0+sinmφsinmφ0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½R0(r)~‡©§½)¯K1rddrrdR0(r)dr=−12πε01r0δ(r−r0)R0(0)k.R0(a)=0R0(r)=(A0,rr0,B0lnra,rr0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½R0(r)~‡©§½)¯K1rddrrdR0(r)dr=−12πε01r0δ(r−r0)R0(0)k.R0(a)=0R0(r)=(A0,rr0,B0lnra,rr0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½R0(r)~‡©§½)¯K1rddrrdR0(r)dr=−12πε01r0δ(r−r0)R0(0)k.R0(a)=0R0(r)=(A0,rr0,B0lnra,rr0R0(r)r0+0r0−0=0dR0(r)drr0+0r0−0=−12πε01r0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½R0(r)~‡©§½)¯K1rddrrdR0(r)dr=−12πε01r0δ(r−r0)R0(0)k.R0(a)=0R0(r)=(A0,rr0,B0lnra,rr0A0=−12πε0lnr0aB0=−12πε0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½R0(r)~‡©§½)¯K1rddrrdR0(r)dr=−12πε01r0δ(r−r0)R0(0)k.R0(a)=0R0(r)=(A0,rr0,B0lnra,rr0R0(r)=(−12πε0lnr0arr0−12πε0lnrarr0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½Rm1(r)~‡©§½)¯Kh1rddrrddr−m2r2iRm1(r)=−δ(r−r0)πε0r0cosmφ0Rm1(0)k.Rm1(a)=0Rm1(r)=(Am1ramrr0Bm1hram−armirr0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½Rm1(r)~‡©§½)¯Kh1rddrrddr−m2r2iRm1(r)=−δ(r−r0)πε0r0cosmφ0Rm1(0)k.Rm1(a)=0Rm1(r)=(Am1ramrr0Bm1hram−armirr0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½Rm1(r)~‡©§½)¯Kh1rddrrddr−m2r2iRm1(r)=−δ(r−r0)πε0r0cosmφ0Rm1(0)k.Rm1(a)=0Rm1(r)=(Am1ramrr0Bm1hram−armirr0Rm1(r)r0+0r0−0=0dRm1(r)drr0+0r0−0=−1πε01r0cosmφ0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½Rm1(r)~‡©§½)¯Kh1rddrrddr−m2r2iRm1(r)=−δ(r−r0)πε0r0cosmφ0Rm1(0)k.Rm1(a)=0Rm1(r)=(Am1ramrr0Bm1hram−armirr0Am1=−12πε01mr0am−ar0mcosmφ0Bm1=−12πε01mr0amcosmφ0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½Rm1(r)~‡©§½)¯Kh1rddrrddr−m2r2iRm1(r)=−δ(r−r0)πε0r0cosmφ0Rm1(0)k.Rm1(a)=0Rm1(r)=(Am1ramrr0Bm1hram−armirr0Rm1(r)=(−12πε01mhrr0a2m−rr0micosmφ0rr0−12πε01mhrr0a2m−r0rmicosmφ0rr0C.S.Wu1›8ùGreen¼ê()GreenFunctionofHelmholtzEq...GreenFunctionsforTime-DependentProblemsSeparationofVariablesMethodofImagesû½Rm2(r)~‡©§½)¯Kh1rddrrddr−m2r2iRm2(r)=−δ(r−r0)πε0r0sinmφ0Rm2(0)k.Rm2(a)=0Rm2(