北大数学物理方法(A)-数学物理方程教案13分离变量法总结

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Outline1›nù©lCþ{o(ÔnÆ2007cSC.S.Wu1›nù©lCþ{o(OutlineùLJ:1Sturm-Liouville.§Š¯KgŠŽÎŠ¯KSturm-Liouville.§Š¯KSturm-Liouville.§Š¯K¥{¿y–2lS-L.§Š¯Kw©lCþ{©lCþ{°Â©lCþ{uÐC.S.Wu1›nù©lCþ{o(OutlineùLJ:1Sturm-Liouville.§Š¯KgŠŽÎŠ¯KSturm-Liouville.§Š¯KSturm-Liouville.§Š¯K¥{¿y–2lS-L.§Š¯Kw©lCþ{©lCþ{°Â©lCþ{uÐC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...ReferencesÇÂÁ§5êÆÔn{6§§18.3—18.6ù&œ§5êÆÔn{6§§9.4nÎ!X1Á§5êÆÔn{6§§10.6C.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnùLJ:1Sturm-Liouville.§Š¯KgŠŽÎŠ¯KSturm-Liouville.§Š¯KSturm-Liouville.§Š¯K¥{¿y–2lS-L.§Š¯Kw©lCþ{©lCþ{°Â©lCþ{uÐC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnŠŽÎDefinitionLÚM½Â3˜½¼ê˜mS(‡©)ŽÎ§eéuT¼ê˜mS?¿ü‡¼êuÚv§ðk(v,Lu)=(Mv,u)=Zbav∗Ludx=Zba(Mv)∗udxK¡M´LŠŽÎC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.1L=ddxZbav∗dudxdx=v∗u ba−Zbadv∗dxudx¤±§uÚvÑ÷v.^‡y(a)=y(b)ž§ddxŠŽÎ´−ddxC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.1L=ddxZbav∗dudxdx=v∗u ba−Zbadv∗dxudx¤±§uÚvÑ÷v.^‡y(a)=y(b)ž§ddxŠŽÎ´−ddxC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.1L=ddxZbav∗dudxdx=v∗u ba−Zbadv∗dxudx¤±§uÚvÑ÷v.^‡y(a)=y(b)ž§ddxŠŽÎ´−ddxC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqn?ØeM´LŠŽÎ§Kéu?¿¼êuÚv§kZbav∗Mudx=Zba(Mu)∗vdx∗=Zbau∗Lvdx∗=Zba(Lv)∗udx¤±L´MŠŽÎC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqn?ØeM´LŠŽÎ§Kéu?¿¼êuÚv§kZbav∗Mudx=Zba(Mu)∗vdx∗=Zbau∗Lvdx∗=Zba(Lv)∗udx¤±L´MŠŽÎC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqn?ØeM´LŠŽÎ§Kéu?¿¼êuÚv§kZbav∗Mudx=Zba(Mu)∗vdx∗=Zbau∗Lvdx∗=Zba(Lv)∗udx¤±L´MŠŽÎC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.2L=d2dx2©ÜÈ©üg=Zbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx¤±§¼êuÚv÷vhv∗u0−(v∗)0uiba=0ž§d2dx2ŠŽÎÒ´§gC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.2L=d2dx2©ÜÈ©üg=Zbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx¤±§¼êuÚv÷vhv∗u0−(v∗)0uiba=0ž§d2dx2ŠŽÎÒ´§gC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.2L=d2dx2©ÜÈ©üg=Zbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx¤±§¼êuÚv÷vhv∗u0−(v∗)0uiba=0ž§d2dx2ŠŽÎÒ´§gC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~Ÿo^‡ehv∗u0−(v∗)0uiba=0º¼êuÚv÷v˜!!na.^‡α1y(a)+β1y0(a)=0α2y(b+β2y0(b)=0(Ù¥|α1|2+|β1|26=0,|α2|2+|β2|26=0)¼êuÚv÷v±Ï^‡y(a)=y(b)y0(a)=y0(b)C.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~Ÿo^‡ehv∗u0−(v∗)0uiba=0º¼êuÚv÷v˜!!na.^‡α1y(a)+β1y0(a)=0α2y(b+β2y0(b)=0(Ù¥|α1|2+|β1|26=0,|α2|2+|β2|26=0)¼êuÚv÷v±Ï^‡y(a)=y(b)y0(a)=y0(b)C.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~Ÿo^‡ehv∗u0−(v∗)0uiba=0º¼êuÚv÷v˜!!na.^‡α1y(a)+β1y0(a)=0α2y(b+β2y0(b)=0(Ù¥|α1|2+|β1|26=0,|α2|2+|β2|26=0)¼êuÚv÷v±Ï^‡y(a)=y(b)y0(a)=y0(b)C.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqngŠŽÎDefinitioneŽÎLŠŽÎÒ´§g§=éuT¼ê˜mS?¿ü‡¼êuÚv§ðk(v,Lu)=(Lv,u)=Zbav∗Ludx=Zba(Lv)∗udxK¡L´gŠŽÎC.S.Wu1›nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.3L=iddxZbav∗idudxdx=iv∗u ba−iZbadv∗dxudxu

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