北大数学物理方法(A)-数学物理方程教案18变分法初步2

Outline1›lùC©{ÐÚ()ÔnÆêÆÔn{‘§|2007cSC.S.Wu1›lùC©{ÐÚOutlineùLJ:1¼^‡4Š(Y)Þ~2‡©§C©/ª‡©§½)¯K‡©§Š¯K3Rayleigh-Ritz{Äg´Þ~C.S.Wu1›lùC©{ÐÚOutlineùLJ:1¼^‡4Š(Y)Þ~2‡©§C©/ª‡©§½)¯K‡©§Š¯K3Rayleigh-Ritz{Äg´Þ~C.S.Wu1›lùC©{ÐÚOutlineùLJ:1¼^‡4Š(Y)Þ~2‡©§C©/ª‡©§½)¯K‡©§Š¯K3Rayleigh-Ritz{Äg´Þ~C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ReferencesÇÂÁ§5êÆÔn{6§121Ùù&œ§5êÆÔn{6§§15.1nÎ!X1Á§5êÆÔn{6§115ÙC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ReferencesÇÂÁ§5êÆÔn{6§121Ùù&œ§5êÆÔn{6§§15.1nÎ!X1Á§5êÆÔn{6§115ÙC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ReferencesÇÂÁ§5êÆÔn{6§121Ùù&œ§5êÆÔn{6§§15.1nÎ!X1Á§5êÆÔn{6§115ÙC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ExamplesùLJ:1¼^‡4Š(Y)Þ~2‡©§C©/ª‡©§½)¯K‡©§Š¯K3Rayleigh-Ritz{Äg´Þ~C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples~18.1¦¼I[y]=Z10xy02dx3.^‡y(0)k.,y(1)=0Úå^‡Z10xy2dx=1e4Š­‚=)æ^þ¡£ãLagrange¦f{§Œ±7‡^‡∂∂y−ddx∂∂y0xy02−λxy2
=0=ddxxdydx+λxy=0C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples~18.1¦¼I[y]=Z10xy02dx3.^‡y(0)k.,y(1)=0Úå^‡Z10xy2dx=1e4Š­‚=)æ^þ¡£ãLagrange¦f{§Œ±7‡^‡∂∂y−ddx∂∂y0xy02−λxy2
=0=ddxxdydx+λxy=0C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples‡©§ddxxdydx+λxy=0.^‡y(0)k.,y(1)=0¤Š¯KŠλi=μ2i,i=1,2,3,···(μi´l¼êJ0(x)1i‡:)ÐÒ´Lagrange¦fƒA¼êyi(x)=CJ0(μix)Ò´4Š¼ê~þCŒ±då^‡Z10xy2dx=1½ÑC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples‡©§ddxxdydx+λxy=0.^‡y(0)k.,y(1)=0¤Š¯KŠλi=μ2i,i=1,2,3,···(μi´l¼êJ0(x)1i‡:)ÐÒ´Lagrange¦fƒA¼êyi(x)=CJ0(μix)Ò´4Š¼ê~þCŒ±då^‡Z10xy2dx=1½ÑC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples‡©§ddxxdydx+λxy=0.^‡y(0)k.,y(1)=0¤Š¯KŠλi=μ2i,i=1,2,3,···(μi´l¼êJ0(x)1i‡:)ÐÒ´Lagrange¦fƒA¼êyi(x)=CJ0(μix)Ò´4Š¼ê~þCŒ±då^‡Z10xy2dx=1½ÑC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples‡©§ddxxdydx+λxy=0.^‡y(0)k.,y(1)=0¤Š¯KŠλi=μ2i,i=1,2,3,···(μi´l¼êJ0(x)1i‡:)ÐÒ´Lagrange¦fƒA¼êyi(x)=CJ0(μix)Ò´4Š¼ê~þCŒ±då^‡Z10xy2dx=1½ÑC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ExamplesϏC2Z10xJ20(μix)dx=C22J21(μi)=1¤±C=√2J1(μi)ù§Ò¦Ñ4Š¼êyi(x)=√2J1(μi)J0(μix)C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ExamplesϏC2Z10xJ20(μix)dx=C22J21(μi)=1¤±C=√2J1(μi)ù§Ò¦Ñ4Š¼êyi(x)=√2J1(μi)J0(μix)C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)ExamplesϏC2Z10xJ20(μix)dx=C22J21(μi)=1¤±C=√2J1(μi)ù§Ò¦Ñ4Š¼êyi(x)=√2J1(μi)J0(μix)C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples(¹k–½ëþ(Lagrange¦f)Euler-Lagrange§§Úàg.^‡|Ü3˜å§Ò¤Š¯KŠŠ¯K§§)§ŠÚ¼ê§káõ‡kü‡¯KI‡?ØC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples(¹k–½ëþ(Lagrange¦f)Euler-Lagrange§§Úàg.^‡|Ü3˜å§Ò¤Š¯KŠŠ¯K§§)§ŠÚ¼ê§káõ‡kü‡¯KI‡?ØC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples(¹k–½ëþ(Lagrange¦f)Euler-Lagrange§§Úàg.^‡|Ü3˜å§Ò¤Š¯KŠŠ¯K§§)§ŠÚ¼ê§káõ‡kü‡¯KI‡?ØC.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples˜!ùáõ‡¼êÑ´4Š¼ê=yδI[y]=Z10δy02xdx=2Z10y0(δy)0xdx?C©δ2I[y]=2Z10δy0δy0xdx=2Z10(δy0)2xdx0Ϗ¼I[y]?C©ðŠ§¤±ù4Š¼êþ¦¼4C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples˜!ùáõ‡¼êÑ´4Š¼ê=yδI[y]=Z10δy02xdx=2Z10y0(δy)0xdx?C©δ2I[y]=2Z10δy0δy0xdx=2Z10(δy0)2xdx0Ϗ¼I[y]?C©ðŠ§¤±ù4Š¼êþ¦¼4C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples˜!ùáõ‡¼êÑ´4Š¼ê=yδI[y]=Z10δy02xdx=2Z10y0(δy)0xdx?C©δ2I[y]=2Z10δy0δy0xdx=2Z10(δy0)2xdx0Ϗ¼I[y]?C©ðŠ§¤±ù4Š¼êþ¦¼4C.S.Wu1›lùC©{ÐÚIsoperimetricProblem(continued)VariationalFormulismofDifferentialEquation(ApproximationMethod(Rayleigh-Ritz)Examples˜!ùáõ‡¼êÑ´4Š¼ê=yδI[y]=Z10δy02xdx=2Z10y0(δy)0xdx?C©δ2I[y]=2Z10δy0δy0xdx=2Z10(δy0)2xdx0Ϗ¼I[y]?C©ð