北大数学物理方法(A)-数学物理方程教案19二阶线性偏微分方程的分类

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Outline1›Êù‚5 ‡©§©aÔnÆêÆÔn{‘§|2007cSC.S.Wu1›ÊùC©{ÐÚOutlineùLJ:1‚5 ‡©§IO/ªgCþC†e ‡©§2‚5 ‡©§©a˜‡½nV­.§ý .§Ô.§C.S.Wu1›ÊùC©{ÐÚOutlineùLJ:1‚5 ‡©§IO/ªgCþC†e ‡©§2‚5 ‡©§©a˜‡½nV­.§ý .§Ô.§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)ReferencesÇÂÁ§5êÆÔn{6§122Ùù&œ§5êÆÔn{6§§7.3nÎ!X1Á§5êÆÔn{6§§9.2C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)‘§o?Øn«a. ‡©§½)¯K)µÅЧ9D§­½¯K§XLaplace§§Poisson§§Helmholtz§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)‘§o?Øn«a. ‡©§½)¯K)µÅЧ9D§­½¯K§XLaplace§§Poisson§§Helmholtz§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)‘§o?Øn«a. ‡©§½)¯K)µÅЧ9D§­½¯K§XLaplace§§Poisson§§Helmholtz§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)‘§o?Øn«a. ‡©§½)¯K)µÅЧ9D§­½¯K§XLaplace§§Poisson§§Helmholtz§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)ùna§§£ØÓÔnL§§§‚)ÑLyшgØÓA:3êÆþ§ùna§©áV­.!Ô.Úý .na‚5 ‡©§§´ÄҐkùn«a.ºC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)ùna§§£ØÓÔnL§§§‚)ÑLyшgØÓA:3êÆþ§ùna§©áV­.!Ô.Úý .na‚5 ‡©§§´ÄҐkùn«a.ºC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)ùna§§£ØÓÔnL§§§‚)ÑLyшgØÓA:3êÆþ§ùna§©áV­.!Ô.Úý .na‚5 ‡©§§´ÄҐkùn«a.ºC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsùLJ:1‚5 ‡©§IO/ªgCþC†e ‡©§2‚5 ‡©§©a˜‡½nV­.§ý .§Ô.§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms±ü‡gCþ‚5 ‡©§~éuõ‡gCþœ/§¯K‡E,˜§?ØĐ{ƒÓC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms±ü‡gCþ‚5 ‡©§~éuõ‡gCþœ/§¯K‡E,˜§?ØĐ{ƒÓC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsIO/ªa∂2u∂x2+2b∂2u∂x∂y+c∂2u∂y2+d∂u∂x+e∂u∂y+fu+g=0a,b,c,d,e,fÚg´x,y®¼êÏ~b§‚ëYŒ‡¼êa,b,c¥§–k˜‡Øð0§ÄKÒؤُ ‡©§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsIO/ªa∂2u∂x2+2b∂2u∂x∂y+c∂2u∂y2+d∂u∂x+e∂u∂y+fu+g=0a,b,c,d,e,fÚg´x,y®¼êÏ~b§‚ëYŒ‡¼êa,b,c¥§–k˜‡Øð0§ÄKÒؤُ ‡©§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsIO/ªa∂2u∂x2+2b∂2u∂x∂y+c∂2u∂y2+d∂u∂x+e∂u∂y+fu+g=0a,b,c,d,e,fÚg´x,y®¼êÏ~b§‚ëYŒ‡¼êa,b,c¥§–k˜‡Øð0§ÄKÒؤُ ‡©§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsIO/ªa∂2u∂x2+2b∂2u∂x∂y+c∂2u∂y2+d∂u∂x+e∂u∂y+fu+g=0a,b,c,d,e,fÚg´x,y®¼êÏ~b§‚ëYŒ‡¼êa,b,c¥§–k˜‡Øð0§ÄKÒؤُ ‡©§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsùLJ:1‚5 ‡©§IO/ªgCþC†e ‡©§2‚5 ‡©§©a˜‡½nV­.§ý .§Ô.§C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsÄaÚ(½)cØð0œ/ؔa6≡0ŠC†ξ=φ(x,y),η=ψ(x,y)yξÚηE´ÕáCþ§ù˜|C†7L÷v∂(ξ,η)∂(x,y)6=0ea=c=0§K®áue¡‡Ñœ/ƒ˜C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsÄaÚ(½)cØð0œ/ؔa6≡0ŠC†ξ=φ(x,y),η=ψ(x,y)yξÚηE´ÕáCþ§ù˜|C†7L÷v∂(ξ,η)∂(x,y)6=0ea=c=0§K®áue¡‡Ñœ/ƒ˜C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsÄaÚ(½)cØð0œ/ؔa6≡0ŠC†ξ=φ(x,y),η=ψ(x,y)yξÚηE´ÕáCþ§ù˜|C†7L÷v∂(ξ,η)∂(x,y)6=0ea=c=0§K®áue¡‡Ñœ/ƒ˜C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsÄaÚ(½)cØð0œ/ؔa6≡0ŠC†ξ=φ(x,y),η=ψ(x,y)yξÚηE´ÕáCþ§ù˜|C†7L÷v∂(ξ,η)∂(x,y)6=0ea=c=0§K®áue¡‡Ñœ/ƒ˜C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransformsÄaÚ(½)cØð0œ/ؔa6≡0ŠC†ξ=φ(x,y),η=ψ(x,y)yξÚηE´ÕáCþ§ù˜|C†7L÷v∂(ξ,η)∂(x,y)6=0ea=c=0§K®áue¡‡Ñœ/ƒ˜C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms3ù˜|C†e§k∂u∂x=∂ξ∂x∂u∂ξ+∂η∂x∂u∂η=∂φ∂x∂u∂ξ+∂ψ∂x∂u∂η∂u∂y=∂φ∂y∂u∂ξ+∂ψ∂y∂u∂ηC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms∂2u∂x2=∂φ∂x2∂2u∂ξ2+2∂φ∂x∂ψ∂x∂2u∂ξ∂η+∂ψ∂x2∂2u∂η2+∂2φ∂x2∂u∂ξ+∂2ψ∂x2∂u∂η∂2u∂x∂y=∂φ∂x∂φ∂y∂2u∂ξ2+∂φ∂x∂ψ∂y+∂φ∂y∂ψ∂x∂2u∂ξ∂η+∂ψ∂x∂ψ∂y∂2u∂η2+∂2φ∂x∂y∂u∂ξ+∂2ψ∂x∂y∂u∂η∂2u∂y2=∂φ∂y2∂2u∂ξ2+2∂φ∂y∂ψ∂y∂2u∂ξ∂η+∂ψ∂y2∂2u∂η2+∂2φ∂y2∂u∂ξ+∂2ψ∂y2∂u∂ηC.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms§CA∂2u∂ξ2+2B∂2u∂ξ∂η+C∂2u∂η2+D∂u∂ξ+E∂u∂η+Fu+G=0A=a∂φ∂x2+2b∂φ∂x∂φ∂y+c∂φ∂y2B=a∂φ∂x∂ψ∂x+b∂φ∂x∂ψ∂y+∂φ∂y∂ψ∂x+c∂φ∂y∂ψ∂yC=a∂ψ∂x2+2b∂ψ∂x∂ψ∂y+c∂ψ∂y2······C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms§CA∂2u∂ξ2+2B∂2u∂ξ∂η+C∂2u∂η2+D∂u∂ξ+E∂u∂η+Fu+G=0A=a∂φ∂x2+2b∂φ∂x∂φ∂y+c∂φ∂y2B=a∂φ∂x∂ψ∂x+b∂φ∂x∂ψ∂y+∂φ∂y∂ψ∂x+c∂φ∂y∂ψ∂yC=a∂ψ∂x2+2b∂ψ∂x∂ψ∂y+c∂ψ∂y2······C.S.Wu1›ÊùC©{ÐÚLinearPDE(2order)ClassificationofPDE’s(2order)CanonicalFormsPDEunderVariableTransforms§CA∂2u∂ξ2+2B∂2u∂ξ∂η+C∂2u∂η2+D∂u∂ξ+E∂u∂η+Fu+G=0A=a∂φ∂x2+2b∂φ∂x∂φ∂y+c∂φ∂y2B=a∂φ∂x∂ψ∂x+b∂φ∂x∂ψ∂y+∂φ∂y∂ψ∂x+c∂φ∂y∂ψ∂yC=a∂ψ∂x2+2b

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