变分法：bianfen03

ASBBB8BVC1 B2DADJ BIDBBKCUBVC1 1.1 APC2A3CR AHCNCDA3C8D1AGDAC2B1AGDBATCZBMA1BKCOCRBJ C2BMDEB1AGDBATA1CXA9CPBBC2CRCLBPAUA2BZA2AI AGBZB9DED3ABBFBQCRAIA2 CB 1.1.1(CFAV) 20 CABDBEA1 Heaviside CIC3C9 A0CWBAC3A1D6BFDDBMDBCGCWCWCSA1B6CZA5CWABBH CWA2CRD3CWCSBKBCCLBRAIAHCN h(x)= ? ? ? ? ? 1,x≥ 0, 0,x<0, BCBYA1ARA7BYCNBBA5 δ(x). BUA5CPCAC8D1AGC2D0 A4A1 h(x) ARAUCQBYA1BUBL δ(x) AUCQARCDAYDBBS BTAIC2AHCNA1D0BGDDANA5BMDEBBAIC7B1B0C8BHCW DIA1CICNBBBXCDABC1BSBTC2A2BUCDA1δ(x)CIC5BC DABICDC1BSBTC2A1C3BRAPBMDBD0APAUC2 “CQC3”BT A8A7BCA2 CB 1.1.2(Dirac BCBL) CIA2A8CAC4DAA1A7CQA8 B0BZC2?D9C2A7D1BZASAHCNCWAHCLA1AKBLBTC2AS AHCNCHC1B0C8 e iλx (x ∈ (?∞, +∞)), λ CDC5AZCNA1 ARCOA1BRAIB0C8C2B2D1 1 2π integraldisplay 2π ?2π e iλx dx. CRDBB2D1A5CauchyB2D1CWCCBTA1B7 1 2π integraldisplay 2π ?2π e iλx dx= lim n→∞ integraldisplay n ?n e iλx dx= lim n→∞ 1 π sin nλ λ . AKBKA1CRDEB4AMCIAYDBBSBTAIAUBOCIA2BKCOA1? D0BBBEBOA5CRDEB4AMCDB5AGCYD6BZC2 δ(λ), ARBO A5CD DiracD2AIA2D4AQA1CIDCABD5BBDAA1C7BMAVCR CLDDB5CNA7C4 δ(λ)C2CGCWCSCJA1ARABCVC5C6BZA2 CB 1.1.3(BIDBD1CQ) CICNBBAGC1C2CRCLDAA1BL C3B3BKBCBCAXCAC8D1AGDACLBMAUB1AGCGCWC6BZCU A3CYBFC2AMD8A220CABD30ASBRA1SobolevA5DDBJ 1 CCA2D1CWBAC3C2BOCIB2A0A4BMB2AAD7A1DBAFD1AW B2D1A0C8A1DFABDDAHCNCQA2B2C2DBATA1BPD4DDAB BTA2BWD0A4A1B0B7DDSobolevCSBJD0A4A2CRAOD4A9 ALBRA2D1CWBAD0A4C2BVC2A2 B1C4BXCLCBBUA1CUBBAHCNDBATA1A5ABBTAHCNBC COBIC5C2CNBBB1BHA1CLCNBBBED6BFDDAXC2D9CMA2 20 CABD 40 ASBRA1Schwartz A0B7DDCRBMBKCGC2BN AFA1BJD4DDABBTAHCNC2AHDDD0A4A1ARBUBLCOAZC1 1950ASCNBBAKDCBR —CZCPAABRA2 1.2 BPAQBKCUC6BU C0 ? CD R n C2C1C4CMB5A1BZ x =(x 1 ,x 2 ,···,x n ) APC9R n DAC2C7A2 CEAW L p CSBJCCBTA5 L p loc (?) = intersectiondisplay {L p (? prime ): ? prime ?? ?}, AZDA ? prime ?? ?BSA7A9 ? prime ? ?,B9 d(?? prime ,??) > 0. CL BNBM ? BXCCBTC2AHCNA1B6B5AL{x ∈ ?| f(x) negationslash=0} C2AJABA5 f C2A2BS, BBA5 suppf. COA1 ?BXCHC1CXB5AGC4 ?C2ACBBCQA2AHCNCS BJ C ∞ 0 (?) = {u ∈ C ∞ (?) : suppu ? ?}. AB?C1BQAIDCBKCCD0A2 B4C91.2.1 C ∞ 0 (?)CD L p (?)(1 <p<∞)C2BDAF ABB5A2 CIAJBVC7BBAIA3 D α = D α 1 1 ···D α n n ,D i = ? ?x i ,α=(α 1 ,···,α n ), APC9|α|BYC2A2D1CWABA1AZDA |α| = n summationtext i=1 α i . CID2A4SobolevCSBJDAC2AHCNC2ANAUB2D9C3A1 A1A1AJCLAAATAHCNCWAID9B2D9A1BKAPD2BZBDAFB2 AFCHBZB4AMA1CRCDB3BKBZAAATAHCNCIANDBBSBTAI AHC8DGCCC2AHCNA1A5BLAJBVC7ALAACWABA2 C0 ρ(x)A8AG (1) ρ ∈ C ∞ (R n ), 2 (2) suppρ ? B 1 (0), (3) integraldisplay R n ρ(x) dx=1, B6D0A5BHBNAC. AKBKCRBJC2AAATABCDBOCIC2A1D3 BR ρ(x)= ? ? ? ? ? c exp(|x| 2 ? 1) ?1 , BT|x| < 1, 0, BT |x| > 1, AZDAcA5B3CNA1A8AGDABO integraldisplay R n ρ(x) dx=1. BTBFh>0,h<d(x, ??), AOAA{h ?n ρ(xh ?1 )}CDA1B7 BMDEAAATABAHA2CL u ∈ L 1 loc (?),ANCJB2 (J h u)(x)=h ?n integraldisplay ? ρ( x ? y h )u(y) dy, B6D0A5 u C2CHBHCWAC, BLB6 u h (x)=(J h u)(x) A5 u C2C4A5BKCU (A0DIBOBKCU). CWAB J h C2ANBZCDA7 AHCNuALAAA2BTCI ?C2C5B5BXATBBCCBT u(x) ≡ 0, CJCQBQCWAI u h (x) ∈ C ∞ (R n ),B9BX suppuA5C1C4B5 C3A1u h ∈ C ∞ 0 (R n ). CIB5ALC ∞ 0 (?)BXCCBTCHD8B2BRAIA3 B4DB1.2.2 C0 {? m }?C ∞ 0 (?),? 0 ∈ C ∞ 0 (?),BR? (1) BOCIBMDEAOCLC4 ? C2C5ABB5 K ? ?, C6C1 supp? m ? K(m =1, 2,···); (2) CLC4BNBSD2AO α =(α 1 ,···,α n )AOC1 max x∈K |D α ? m (x) ?D α ? 0 (x)|→0(m →∞), B7 {D α ? m }CI K BXBMD6CHD8C4 D α ? 0 , CJB6B6DE {? m }CI C ∞ 0 (?) DACSCCC4 ? 0 . D5C6BX CLCHD8B2C2ANB2CSBJC ∞ 0 (?), B6A5BPAQBKCUC6BU D(?). 1.3 BIDBBKCUB0B4DBBMBPAQD9A6 B4DB1.3.1 D(?) BXC2BMB8D7B7ANB2CVAHCEB6 A5BIDBBKCU,B7ABBTAHCNCDCRBJC2CVAHf : D(?) → R 1 ,A8AG 3 (1) ANB2A3 〈f,λ 1 ? 1 + λ 2 ? 2 〉 = λ 1 〈f,? 1 〉 + λ 2 〈f,? 2 〉, ?? 1 ,? 2 ∈D(?),?λ 1 ,λ 2 ∈ R 1 ; (2) CLC4BNBSC2 {? m }∈D(?), BX ? m → ? 0 C3A1AO C1 〈f,? m 〉→〈f,? 0 〉(m →∞). BMB8ABBTAHCN f CYAJB7C2B5ALBBAND prime (?). f CIC7 ?BIC2D1 f(?)BBA5〈f,?〉,C4CD〈f,?〉 = f(?). BTCLC4BNBSC2C1C4CQB0B5 E ? ?, f(x)CI E BX A5 Lebesgue BSBTCDCQB2C2A1CJB6 f(x) CI ? BXCD C3ATC5BQB0,BBCRDBAHCNBHD8A5L 1 loc (?). f(x)CLBX A9BMDEABBTAHCN 〈f,?〉 = integraldisplay ? f(x)?(x) dx, ?? ∈D(?). A9A3ABBTAHCNCDCEAWCQB2AHCNC2DFABA2ACDE CEAWCQB2AHCNCLBXBMDEABBTAHCNA2ARAUCDCYC1C2 AHCNCECDABBTAHCNA2CBC5BXA1AYDBC2AUCQB0AHCN ARAUARCNB7CDABBTAHCNA2 CB 1.3.2 δ-AHCNA2C0 θ ∈ ?,CCBT 〈δ, ?〉 = ?(θ), ?? ∈D(?). δ AHCNCDBMDEABBTAHCNA2 B4DB 1.3.3 C0 {f m }?D prime (?),f 0 ∈D prime (?). BR? CLBMB8 ? ∈D(?),C1 lim m→∞ 〈f m ,?〉 = 〈f,?〉, CJB6{f m }CI D prime (?)DACSCCC4 f. CB 1.3.4 CI R 1 BX f m (x)= 1 π sin mx x ,m=1, 2,··· CDBMDE L 1 loc (R 1 ) AHCNA1BNCOCQBQCNANCDABBTAHCN DEA2AB?C1 f m → δ(m →∞). 1.4 BIDBAZCUBTCMD9A6 4 ABBTAHCNBCBYC2APCSCWCDC4A3C8D1AGDAC2D1 AWB2D1A2A5BLAB?AJAYA5BMAID1AWB2D1C2B1AGCS APA2C0f AJ?CECDCCBTCIRBXC2D7B7CQA2AHCNA1 ?CHC1C5CXB5A2C0D1AWB2D1A0C8A1C1 integraldisplay ∞ ?∞ f prime (x)?(x) dx= ? integraldisplay ∞ ?∞ f(x)? prime (x) dx. CRDEC3C8APAID2BZD1AWB2D1CQBQCLBMDEAHCNC2BC BYCGCWA5AUA5CLDIBMDEAHCNC2BCBYA2CRBMBLBTCO C3DCBKC2CBC5B2CRAB?A5BQAICWC8BVC7ABBTAHCN C2BYCNA2 B4DB1.4.1(CEAWCQB2AHCNC2ABBTBYCN) C0 u,v A5 ?BXCEAWCQB2AHCNA1BT integraldisplay ? uD α ?dx=(?1) α integraldisplay ? v?dx, ?? ∈ C ∞ 0 (?) B7D4A1CJB6 v A5 uC2 αBYBIDBAZCU. B4DB1.4.2(ABBTAHCNC2ABBTBYCN) C0f,g ∈D prime (?) CDABBTAHCNA1A8AG 〈f,D α ?〉 =(?1) α 〈g,?〉, ?? ∈D(?). CJ g B6A5f C2 αBYBIDBAZCU. B4C9 1.4.3 BNBMABBTAHCNC2CYC1BYABBTBYCN CEBOCICOB9CECDABBTAHCNA2 CB 1.4.4 C0 Heaviside AHCNhCYCCBTC2AHCN h(x)= ? ? ? ? ? 1,x≥ 0, 0,x<0 C2ABBTBYCN h prime = δ. A1CGA3CLC4BNBSC2 ? ∈D(R),C1 h prime (?)=?h(? prime )=? integraldisplay ∞ 0 ? prime dx= ?(0) = δ(?), BUBLh prime = δ. CB 1.4.5 |x| primeprime =2δ. A1CGA3CLC4BNBSC2 ? ∈D(R), BF a>0, C6C1 5 supp? ? (?a, a),CJC1 |x| primeprime (?)=|x|(? primeprime )= integraldisplay ∞ ?∞ |x|? primeprime (x) dx = ? integraldisplay 0 ?a x? primeprime (x) dx+ integraldisplay a 0 x? primeprime (x) dx = integraldisplay 0 ?a ? prime (x) dx? integraldisplay a 0 ? prime (x) dx = ?(0) + ?(0) = 2δ(?), BUBL|x| primeprime =2δ(?). B4C91.4.6 C0AHCNu ∈ L p (?),CJ uCHC1B2D1C2 CUD8D7B7B2A1B7CLC4BNBSC2 ε>0,BOCI δ>0,BX |h| <δC3 bardblu(x + h) ?u(x)bardbl p <ε. B4C91.4.7 C0 u ∈ L p (?),CJ bardblJ h ubardbl p ≤bardblubardbl p , lim h→0 bardblJ h u ? ubardbl p =0. A9A3BXD α u ∈ L p (?)C3A1C1 lim h→0 bardblD α u h ?D α ubardbl p = 0. B4C91.4.8(AND1CSB1AGBVD0) BR?u ∈ L p (?)A8 AG integraldisplay ? u(x)?(x) dx=0, ?? ∈ C ∞ 0 (?), CJ u =0CI ?BXB9AQBIBIB7D4 (BBA5u =0a.e.C4 ?,B0 u = 0 p. p. C4 ?). CEAWCQB2AHCNC2ABBTBYCNC3BHC4BQAICCBTA2 B4DB1.4.9 C0 u,v ∈ L p loc (?),BTCLC4BNBS ? prime ?? ?,BOCIAHCNDE u n ∈ C ∞ (? prime )A8AG bardblu n ? ubardbl p → 0, bardblD α u n ? vbardbl p → 0(n →∞), CJB6v CD uC2 αBYBIDBAZCU. A9A3ABBTBYCWABCDCDCAC8BYCWABCI L 1 loc C2AJ CUCNA1BUBLBTAHCNC1CAC8 αBYBYCNBOCIA1CJD0C7 ABBT αBYBYCNBMD6A2BUCDCQCQBLC1BDAQA3CAC8BY CNC0C4BYCCBTDCBYA1COABBT αBYBYCNCDD0BWDGBF CCBTA1BUBLDCBYABBTBYCNBOCIAUARDFBFC4BYBYCN BLBOCIA2 CEAWCQB2AHCNC2ABBTBYCNCHC1BRAIBEC3B2D9 6 (1) D α (au + bv)=aD α u + bD α v. (2) D α+β u = D α (D β u). (3) BT D α u =0CLBMB8 |α| = m B7D4C2BBD1AKBK DABOCD uB9AQBIBIC3C4BMDE (m ? 1)BMCNAQ C8A2 1.5 SobolevC6BU B4DB 1.5.1 C0 k A5D0D7CUCNA1 p ≥ 1, CCBT Sobolev C6BUBRAI W k p (?) = {u ∈ L p (?)| D α u ∈ L p (?),?|α|≤k}. W k p (?)DAC2CUCNCCBTA5 bardblubardbl k,p,? =( summationdisplay |α|≤k bardblD α ubardbl p p ) 1/p . SobolevCSBJBBAN W k p (?)B0 W k,p (?). ? W k p (?)CDA0?C2CSBJA2 ? BX k =0C3A1AKBK W 0 p (?) = L p (?). ? BX p =2C3A1W k 2 (?)CD HilbertCSBJA1BMA9BL BBA5 H k (?). CRCDC0C4CIH k (?) DACQBQCCBT AQB2 〈u,v〉 = summationdisplay |α|≤k 〈D α u,D α v〉 = summationdisplay |α|≤k integraldisplay ? D α uD α vdx. ? BX 1 <p<+∞C3A1CSBJW k p (?) CDACCTCQD1 C2BanachCSBJA2 ? C ∞ (?)CD W k p (?)C2BDAFB5A2 B4DB1.5.2 B6 C ∞ 0 (?)CI W k p (?)CUCNAIC2A0? AUCSBJA5 SobolevCSBJW k,p 0 (?). BX 1 <p<+∞C3A1C0C4C ∞ 0 (?) CD L p (?) C2BD AFABB5A1BUBL W k,p 0 (?) = L p (?) = W 0 p (?),BUCDA1BX k>0, ? negationslash= R n C3A1W k,p 0 (?)CDW k p (?)C2CSABCSBJA2 7 BDCC W k,p 0 (?) C2CWAMCZBICD W k,p 0 (?) DAC2BNAK AHCNCEuCQBQD7B7BEDHBZ W k p (R n ). A5BLA1D3BKBL ˉu(x)= ? ? ? ? ? u(x),x∈ ?, 0,x/∈ ?. u → ˉuA1B7W k,p 0 (?) → W k p (R n )C2D7B7BEDHA2 C0X 1 ,X 2 CDDBDED5CUANB2CSBJA1D0?C2CUCND1 AQA5bardbl·bardbl 1 ,bardbl·bardbl 2 ,BR?A8AGDABO (1) X 1 ? X 2 ; (2) BOCIB3CN c,C6C1bardblubardbl 2 ≤ cbardblubardbl 1 ,?u ∈ X 1 , CJB6CSBJX 1 B6BSBZCSBJX 2 ,BBANX 1 arrowhookleft→ X 2 . BR? B6BSAVA8AGAOC3CWABI : X 1 → X 2 CDC5C2A1CJB6CR DEB6BSCDC5C2A2 B4C91.5.3 B7D4 W 1,p 0 arrowhookleft→ ? ? ? ? ? ? ? ? ? ? ? ? ? C( ˉ ?),p>n, L q (?),p= n, 1 ≤ q<∞, L np/(n?p) (?),p<n, AZDABBAIarrowhookleft→BGAPC9ABAGA7AHCZDIA1AVAPC9B6BS CWABCDD7B7C2A2BLDIA1CLBNBS u ∈ W 1,p 0 (?)C1AUC3 C8 sup ? |u|≤C(n, p)|?| 1/n?1/p bardblDubardbl p ,p>n, bardblubardbl q ≤ C(n, q)|?| 1/q bardblDubardbl n ,p= n, 1 ≤ q<∞, bardblubardbl np/(n?p) ≤ CbardblDubardbl p ,p<n, AZDAbardblDubardbl p = summationtext |α|=1 bardblD α ubardbl p . D0CE1.5.4 A7C4W k,p 0 (?)(k>1)C2BAB0C1B6BS W k,p 0 arrowhookleft→ ? ? ? ? ? ? ? ? ? ? ? ? ? W l,s 0 (?),s= np n ? (k ?l)p , (k ? l)p<n, W l,q 0 (?), (k ?l)p = n, 1 ≤ q<∞, C l ( ˉ ?), 0 ≤ l ≤ k ? n p . D0CE 1.5.5 W k,p 0 (?) DACQBQCCBTBRAIC2C3BHCU CN bardblubardbl W k,p 0 (?) = summationdisplay |α|=k bardblD α ubardbl p . 8 B4DB1.5.6 BR?BOCIBMDEA6CCA8K ? ,C6ACDEC7 x ∈ ??BXC1BQxA5CBC7C7K ? BHC3C2A8K ? (x) ? ?, CJB6??A8AGCIATAACZBW. B4C91.5.7 BR???A8AGAQAWA8DABOA1CJ W k p (?) arrowhookleft→ ? ? ? ? ? L (np)/(n?kp) (?),kp<n, C l (?), 0 ≤ l<k? n p , W k p (?) arrowhookleft→ W l s (?),s= np n ? (k ?l)p , 0 < (k ? l)p<n. B6BSCCD0AVCQBQCI H¨older CSBJC1BZDIC9AFC2 AHCLA2 B4DB1.5.8 H¨olderCSBJC k,α ( ˉ ?)(C k,α (?))CDC k ( ˉ ?)(C k (?)) C2ABCSBJA1AZ k BYBYCNCHC1D2AO α(0 <α≤ 1)C2 H¨olderD7B7B2A2CYA9AHCN f CHC1D2CN αC2 H¨older D7B7B2A1BSD2 [f] α,? =sup x,y∈?,xnegationslash=y |f(x) ? f(y)| |x ? y| α < ∞, 0 <α≤ 1. CL C k,α (?)BXC2AHCN f,CQBQA6?CUCN bardblfbardbl C k,α (?) = bardblfbardbl k,∞,? +max 0≤|β|≤k [D β f] α,? COA1B7 BanachCSBJA2 B4C91.5.9 BR???A8AGAQAWA8DABOA1kp > n, CJC1H¨olderCSBJB6BS W k p (?) arrowhookleft→ ? ? ? ? ? C k?1,n/p (?), BX k ? n p CDD0CUCN C k?n/p?1,1 (?), BX k ? n p CDCUCN. B4C91.5.10 BR???A8AGAQAWA8DABOA1CJC1 W k p (?) arrowhookleft→ ? ? ? ? ? C l (?),kp>n,l<k? n p L s (?),s= np n ? kp ?ε,kp < n,ε < np n ? kp . A8AGBQBXDABOC2B6BSBYBZCDC5BYBZA2 A9A3CCD0B0CCD0CEBKBC??A8AGAQAWA8DABOA1 BUCDCLC4W k,p 0 (?),BLDABOCJAUAKBKA2CRCDCDW k,p 0 (?) AJ W k p (?)C2DCBKBDAQA2 B4C91.5.11(FriedrichsAUC3C8A1Poincar′eAUC3C8) BR?u ∈ W 1 2 (?),CJBOCIB3CN M>0,C6C1 ( integraldisplay ? |u| 2 dx) 1/2 ≤ M[ integraldisplay ? n summationdisplay i=1 ( ?u ?x i ) 2 + | integraldisplay ?? uds| 2 ] 1/2 . 9 D0CE1.5.12 CLAHCNB5AL B 1 0 = {u ∈ C 1 (?) intersectiondisplay C 0 ( ˉ ?)| u =0CI ??BX} C1AIAGC2AUC3C8 integraldisplay ? u 2 dx≤ C integraldisplay ? |Du| 2 dx, ?u ∈ B 1 0 , B7 bardblubardbl 2 ≤ CbardblDubardbl 2 , ?u ∈ B 1 0 , AZDAC CDC7uACA7COD3C7?C1A7C2B3CNA2 10 B2B7DJ ASBBC9CEDED6D9D1BBBAAUARA5D3CY 2.1 ASBBD3CY CLCVAHBCB4D1C2AAD7B6A5ASBBD3CY,C6CVAHBF B4D1C2AHCNB6A5ASBBD3CYB0BZ,BLB6A5BRA5BKCU B0BRA5B3. A4ADBDCCAND1AAD7C2BBCPB6A5ASBB B8. AKCVBSANAAD7CDA3C0 O C7 ACDDCCGAUDCB9AU CIDCBMB4BKANBXC2DBCCC7A1BR?ABC1AMAXAJCSB3 AID5A1BMD9C7CIDCD5ANBZAIBNOC7BGBMBEANBSA6 D5AC7A1AABEANB8AKDBB0A7C3A1D9C7BSA6C2C3BJ AKCJA4 C0CAAF O C7 A C2B4BKAWAGA5 XOY, OX A5CP AWDEA1 OY DEB4BKARAIA1 A C7C2AOAOA5 (a, b),B9 b>0. D9C7BNOCMC7CGCDA1D0C2CVCG v C7D0C2AF AOAOC1A7AHA3 v 2 =2gy, (2.1.1) AZDAg CDDCD5BFCVCGA2 C0D9C7BSA6BEANC2CWBAA5y = y(x),CJC0(2.1.1) C1 ds dt = v = radicalBig 2gy, C0BLC1 dt = ds √ 2gy = radicalBig 1+y prime 2 √ 2gy dx. CLBLC8B2D1A1C1BFD9C7BGBEAN y = y(x)C0 OBSD5 ACYB3C2C3BJA5 t = t[y(x)] = integraldisplay a 0 radicalBig 1+y prime 2 √ 2gy dx. (2.1.2) CRCDCRAIA1D9C7C0 OBSD5ACYB3C2C3BJtCDAHCN y(x)C2AHCNA1B6 tCDAHCNy(x)C2CVAHA1AKCVBSAN AAD7CDCDCIA8AGALC4DABO y(0) = 0,y(a)=b (2.1.3) C2CYC1D7B7AHCNy(x)DABCBFBMDEAHCNC6CVAH(2.1.2) BFAKATD1A2 11 A5DDBCBFAKCVBSANAAD7C2C3A1BQ (2.1.2)DAC2AF B2AHCNBBA5 F(y,y prime )= radicalBig 1+y prime 2 √ 2gy . (2.1.4) C0y(x)CDAKCVBSANAAD7C2C3A1B7y(x)A8AGALC4DA BO(2.1.3)B9C6 t[y(t)] = integraldisplay a 0 F(y,y prime ) dx=min. (2.1.5) CLC4A8AGALC4DABO ?(0) = ?(a)=0 (2.1.6) C2BNBSD7B7AHCN ?(x)B6BNBSC5CNε,AHCN y(x)+ε?(x) CLA8AGALC4DABO (2.1.3). BUBLA1CVAH t[y(x)+ε?(x)] BX ε =0C3BFAKATD1t[y(x)],BNCOC1 d dε t[y(x)+ε?(x)]| ε=0 =0. (2.1.7) C0 (2.1.5)B6D1AWB2D1C1BF d dε t[y(x)+ε?(x)] = d dε integraldisplay a 0 F(y + ε?, y prime + ε? prime ) dx = integraldisplay a 0 [F y (y + ε?, y prime + ε? prime )? +F y prime(y + ε?, y prime + ε? prime )? prime ] dx = integraldisplay a 0 [F y (y + ε?, y prime + ε? prime ) ? d dx F y prime(y + ε?, y prime + ε? prime )]?dx. BQBLC8BRBS (2.1.7)C1BF integraldisplay a 0 [F y (y,y prime ) ? d dx F y prime(y,y prime )]?dx=0. DHCFAND1CSB1AGBVD0A1C0BXC8C1BF F y (y,y prime ) ? d dx F y prime(y,y prime )=0. (2.1.8) 12 BNAND1AAD7BFCRBYBFC2A2D1CWBAB6A5D9AND1AAD7 C2CKC8BAAU. BUBLA1B3A2D1CWBA (2.1.8)CDAND1AA D7(2.1.5)C2AUCVCWBAA2C0 (2.1.8)CQC1 d dx [F(y,y prime ) ? y prime F y prime(y,y prime )] = F y (y,y prime )y prime + F y prime(y,y prime )y primeprime ?y primeprime F y prime(y,y prime ) ?y prime d dx F y prime(y,y prime ) =0. BUBLC1 F(y,y prime ) ? y prime F y prime(y,y prime )=c =B3CN. (2.1.9) CLC4AKCVBSANAAD7A1BQ (2.1.4)BRBSBXC8C1BF 1 radicalBig 2gy(1 + y prime 2 ) = c. C0BLC1 y(1 + y prime 2 )= 1 2gc 2 =2r. (2.1.10) BVC7ANCNBRAW x = x(θ),ARC0 y prime =cot θ 2 , (2.1.11) CJC0(2.1.10)C1 y =2r sin 2 θ 2 = r(1 ? cos θ). BLC8CL θ A2D1C1BF y prime dx dθ = r sin θ. BQ(2.1.11)BRBSBXC8C1 cot θ 2 dx dθ = r sin θ, C0BLC1BF dx dθ =2r sin 2 θ 2 = r(1? cos θ). B2D1BLC8A1AB?C1BFA2D1CWBA (2.1.10) C2DBC3C2 AZCNAPC9 x = r(θ ? sin θ)+x 0 , y = r(1 ? cos θ), 13 AZDA r C7 x 0 A5BNBSC5CNA2DHCFALC4DABO (2.1.3), B7C0BEANCAAFAOAOCBC7 O C1BFx 0 =0,CHC0BEAN CAAFA(a, b)CQBQBJCC rC2D1A2BUBLA1AKCVBSANCD B9A3ANC2BMCKA1D0CDBQ r A5AACBC2CCDDA3 x 2 +(y ?r) 2 = r 2 (2.1.12) BG XDEB9A5C3A1CCDDBXC2C7(0, 0)CGCDC2ADB3A2 BWB2D8C2AWANAAD7CDBMCYDCBKC2AND1AAD7A2 BWB2D8CJDID5ANBZCRC2ANB0A1ANB0DACRD3AQD5 (BWB2D8DFD9C7BJC2CECMAQD5) CYAMC2DJA1ANA5AR DCA1BOCIBWB2D8AQAWA1B6A5BWB2CCARB0ANB0ARA2 CIBGBGDID5COBWB2D8AXD6CBCWC2B0A7C3A1ANB0AR CDAYBFCLDIC4AMDJC2CWC8APALBFCWA2BWB2D5BBDA C2AKATCCARCBD0D2BFA3BWB2D8CIDID5ANBZAIA1CI CEALBPCYDABOC2BMB8A8BODAA1C6BWB2D8BIC4AWAN A7D1C2A8BOC6?CCAR E =ANB0AR -DID5CYAMC2DJ A5AKATA2 D3BRA1AB?BDCCAWAGBXALC4A6CCC2CLCFACAK (AUBAACDC)CJDID5ANBZAPC2AWANA8BOA2C5BICWAIA1 BWB2ACAKC2ANB0ARC7ACAKC2AGB2C2CKBFB7CVAIA1 CRDEAID3B3CNBVAMACAKC2CND5A2C0ACAKCYCIAWAG BDC8A5?, ALC4A5 ??. CIDID5ANBZAIA1ACAKCIC7 (x, y) ∈ ?BIC2BKD0A8BOBZu(x, y)APC9A1CJACAKC2 ANB0ARA5 T( integraldisplay ? integraldisplay radicalBig 1+u 2 x + u 2 y dxdy?|?|), (2.1.13) AZDAT A5ACAKC2CND5A1|?|A5BDC8 ?C2AGB2A2BU A5BWB2ANB0A5ATANB0A1BX u 2 x + u 2 y BBD1ATC3A1D2BZ C8CUA0C8 √ 1+ε ≈ 1+ ε 2 , ANB0AR (2.1.13)CQBQDAAWB7 T 2 integraldisplay ? integraldisplay (u 2 x + u 2 y ) dxdy, (2.1.14) 14 CHC0ACAKCIBTA8AGB2BXCYCJC2D5A5 f(x, y), CJBL DID5CYAMC2DJA5 integraldisplay ? integraldisplay f(x, y)u(x, y) dxdy. C4CDACAKC2?CCARA5 E(u)= T 2 integraldisplay ? integraldisplay (u 2 x + u 2 y ) dxdy? integraldisplay ? integraldisplay fudxdy, (2.1.15) D0CDAHCNu(x, y)C2CVAHA2C0C4ACAKC2ALC4CDA6CC C2A1CYBQC1ALC4DABO u =0, CI ??BX. (2.1.16) AKATCCARCBD0CRAIA1ACAKCJDID5 f(x, y)ANBZAPA1 CIA8AGALC4DABO (2.1.16) C2AHCNCYDAA1C6?CCAR (2.1.15) BFAKATD1C2A8BO u(x, y) CDCDACAKBPBZAW ANA8D7C3C2A8BOA2 CRBJBMCWA1BWB2ACAKAWANAAD7C2A8BOCDCDCIAL C4DABO (2.1.16)AIA1AND1AAD7 E(u)= T 2 integraldisplay ? integraldisplay (u 2 x + u 2 y ) dxdy? integraldisplay ? integraldisplay fudxdy =min (2.1.17) C2C3A2CYCPAKCVBSANAAD7C1BFCRDEAND1AAD7C2AU CVCWBACD PoissonCWBA ?triangleu = ? ? 2 u ?x 2 ? ? 2 u ?y 2 = f T , CI ?DA. (2.1.18) CRCDCRAIA1CIALC4DABO(2.1.16)AIA1AND1AAD7(2.1.17) C2C3CD PoissonCWBA(2.1.18)C2C3A2 CIBMCCBSBTAIA1ALD1AAD7 (2.1.18), (2.1.16)C3BH C4AND1AAD7 (2.1.17), (2.1.16). 2.2 ASBBBDCJ B4DB 2.2.1 C0 J[y(x)] CDCCBTCIAHCNB5AL Y = {y(x)}BXC2CVAHA1BX y(x),y 0 (x) ∈ Y C3A1B6 δy = δy(x)=y(x)? y 0 (x) A5ACANDC y(x)C2AND1A2 15 B4DB 2.2.2 BR? max|y(x) ? y 0 (x)|AMATA1CJB6 y(x)C7 y 0 (x)CHC1DHBYBWC8A2AHCNB5AL {y(x): |y(x) ?y 0 (x)| <δ} B6A5y 0 (x)C2DHBYδ-DGC8A2C0 k CDCVCUCNA1BR? max{|y(x)?y 0 (x)|,|y prime (x) ?y prime 0 (x)|,···,|y (k) (x) ? y (k) 0 |} AMATA1CJB6 y(x)C7 y 0 (x)C1 k BYBWC8A2AHCNB5AL {y(x): max{|y(x)?y 0 (x)|,|y prime (x)?y prime 0 (x)|,···,|y (k) (x)?y (k) 0 |} <δ} B6A5y 0 (x)C2 k BY δ-DGC8A2 B4DB2.2.3 BR?CLC4BNBSC2 ε>0, CL y 0 (x) C2 k BY δ-DGC8DAC2BNAK y(x),AOC1 |J[y(x)]? J[y 0 (x)]| <ε, CJB6 J[y(x)]CDCI y 0 (x) BICHC1 k BYBWC8CGC2D7B7 CVAHA2 B4DB2.2.4 C0F(x, y, y prime )CDA7C4BVDEANCAx, y, y prime C2CQBYD7B7CQA2C2AHCNA1C0 xBNBSA6CCA1η(x) CD BNBSCQA2AHCNA1ε CDATAZCNA1CJF(x, y, y prime )C2CKDC A5 ?F = F(x, y + εη,y prime + εη prime ) ?F(x, y, y prime ). C0 Taylor A0C8A1C1 ?F = ?F ?y εη + ?F ?y prime εη prime + R, AZDARCD ε → 0C2DCBYC2ACBBATA2B6 δF = ?F ?y εη + ?F ?y prime εη prime A5AHCN F(x, y, y prime )C2AND1A2 B4DB2.2.5 BR?CVAHJ[y(x)]C2CKDC?J = J[y + δy]? J[y]CQBQAPC9A5 ?J = J[y,δy]+β(y,δy)max|δy|, AZDA J[y,δy]CL δyCOBFCDANB2C2A1ARB9BX δy → 0 C3A1 β(y,δy) → 0, CJB6 J[y,δy] A5CVAH J[y] C2AN D1A1BBAN δJ,B7 δJ = J[y,δy]. 16 AIAGCHDGBFBMDBBUBUC2AND1C2CCBTA2COA1AGAZ CNαC2BMAHDEBQB5BEANy(x)+αδy,BQy(x),δyBNBS A6CCA1COA7CVAH J[y + αδy] CNB7AZCN α C2AHCNA1 BBAN Φ(α)=J[y + αδy]. B4DB2.2.6 BR?Φ prime (0) = ? ?α J[y +αδy]| α=0 BOCIA1 CJB6 Φ prime (0)A5CVAH J[y]C2AND1A1BLBBAN δJ,B7 δJ =Φ prime (0) = ? ?α J[y + αδy]| α=0 . B4DB2.2.7 C0 y 0 (x)CDCVAHJ[y]C2BQB5BEANB5 Y DAC2ANBMAHCNA1BTCLC4BNBSC2 y ∈ Y ,CEC1 J[y(x)] ≤ J[y 0 (x)](B0 J[y(x)] ≥ J[y 0 (x)]), CJB6CVAH J[y] CI y 0 (x) BIBPBZB4BQ (AT) D1A1ARB6 y 0 (x)A5 J[y]C2B4BQ (AT)D1BEANA2 BTCLC4y 0 (x)C2DHBYδ-DGC8AQC2CYC1AHCNy(x), CEC1 J[y(x)] ≤ J[y 0 (x)](B0J[y(x) ≥ J[y 0 (x)]), CJB6CVAH J[y]CI y 0 (x)BIBPBZB7B4BQ (AT)D1A2 BTCLC4y 0 (x)C2BMBYδ-DGC8AQC2CYC1AHCNy(x), CEC1 J[y(x)] ≤ J[y 0 (x)](B0J[y(x)] ≥ J[y 0 (x)]), CJB6CVAH J[y]CI y 0 (x)BIBPBZBUB4BQ (AT)D1A2 2.3 ASBBB8BPAQDCC9 BPAQDCC92.3.1 C0φ(x) ∈ C[a, b],BR?CLC4BNBS C2η(x) ∈ C 1 [a, b], η(a)=η(b)=0,AOC1 integraldisplay b a φ(x)η(x) dx=0, CJCIBDBJ [a, b]BXA1φ(x) ≡ 0. A1CGA3BZCTCWCSA2BGC0 φ(x)CI [a, b]DAC2ANC7 x 0 BIAUC3C4DHA1AUCXC0 φ(x 0 ) > 0. C0 φ(x)C2D7B7 B2CYA1AKCCBOCIBDBJ [x 1 ,x 2 ],C6x 0 ∈ [x 1 ,x 2 ] ? [a, b], COB9BX x ∈ [x 1 ,x 2 ]C3 φ(x) > 0. BF η(x)= ? ? ? ? ? ? ? ? ? ? ? ? ? 0,a≤ x ≤ x 1 (x ?x 1 ) 2 (x ? x 2 ) 2 ,x 1 <x<x 2 0,x 1 ≤ x ≤ x 1 17 CJ η(x 1 )=η(x 2 )=0,η prime (x)CI [x 1 ,x 2 ] BXD7B7A1BLCD CDCRη(x)CEALBVD0 1C2DABOA2BUCD integraldisplay b a φ(x)η(x) dx= integraldisplay x 2 x 1 φ(x)(x ?x 1 ) 2 (x ? x 2 ) 2 dx>0, CRC7BVD0C2BGC0A9CMA1A4 φ(x) ≡ 0. BPAQDCC9 2.3.2 C0 ? A5AWAGBDC8A1 ? C2ALC4 A5Γ, φ(x, y) ∈ C(?),BTCLBNBMCI?+ΓBXD7B7CQA2 B9CI ΓBXBFDHD1C2AHCN η(x, y),AOC1 integraldisplay ? integraldisplay φ(x, y)η(x, y) dxdy=0, CJCI?BX φ(x, y) ≡ 0. A1CGA3BZCTCWCSA2BGC0 φ(x, y)CI ?DAC2ANC7 (x 0 ,y 0 ) ∈ ? BIAUC3C4DHA1AUCXC0 φ(x 0 ,y 0 ) > 0. C0 φ(x, y)C2D7B7B2CYA1AKCCBOCI r>0, C6 φ(x, y)CI CC S r :(x ? x 0 ) 2 +(y ? y 0 ) 2 <r 2 AQBPA5CVA1COB9 S r ? ?. BF η(x, y)= ? ? ? ? ? 0, (x ? x 0 ) 2 +(y ? y 0 ) 2 ≥ r 2 , ((x ?x 0 ) 2 +(y ? y 0 ) 2 ? r 2 ) 2 , (x ?x 0 ) 2 +(y ? y 0 ) 2 <r 2 , CJ η(x, y) ∈ C 1 (?),B9 η(x, y)| Γ =0.BUCD integraldisplay ? integraldisplay φ(x, y)η(x, y) dxdy = integraldisplay ? integraldisplay φ(x, y)[(x? x 0 ) 2 +(y ? y 0 ) 2 ?r 2 ] 2 dxdy>0, CRC7BVD0C2BGC0A9CMA1A4 φ(x, y) ≡ 0. 18 B2CPDJ BGB4ARC0B0ASBBD3CY 3.1 ?BVAYCNC7D5B0 EulerBAAU AJD2A4AKB1AGC2CVAH J[y(x)] = integraldisplay x 1 x 0 F(x, y, y prime ) dx (3.1.1) C2B4D1AAD7A1AZDA F ∈ C 2 , y(x) ∈ C 2 ([x 0 ,x 1 ]),B9A8 AGALC4DABO y(x 0 )=y 0 ,y(x 1 )=y 1 . (3.1.2) C0 y(x)C6 J BPBZB4D1A1ALDFBY y(x)BXA8AGC2 DABOA2COA1BQC5CN α A5AZCNC2BMAHDEBQB5BEAN ˉy = y(x)+αδy,AZDAδyA5 y(x)C2AND1A1B7 δy| x=x 0 = δy| x=x 1 =0. AKBKˉy| x=x 0 = y 0 , ˉy| x=x 1 = y 1 , ˉy ∈ C 2 ([x 0 ,x 1 ]),BXα =0 C3A1 ˉy = y(x)CDC6CVAH (3.1.1)BPBZB4D1C2BEANA2 BQ ˉy BRBSC8(3.1.1),AMC1 ?(α)=J[y + αδy]= integraldisplay x 1 x 0 F(x, y + αδy,y prime + αδy prime ) dx. ?(α)CI α =0C3BFC1B4D1A1C0B4D1C2AKBKDABOA1 C1 ? prime (0) = ? ?α J[y + αδy]| α=0 = integraldisplay x 1 x 0 (F y δy + F y primeδy prime ) dx=0. (3.1.3) C0D1AWB2D1CSA1ARA3BSBZ δy| x=x 0 = δy| x=x 1 =0, C1 integraldisplay x 1 x 0 F y primeδy prime dx = integraldisplay x 1 x 0 F y primed(δy) = F y primeδy| x 1 x 0 ? integraldisplay x 1 x 0 δy d dx F y prime dx = ? integraldisplay x 1 x 0 δy d dx F y prime dx. BQBXC8BRBS (3.1.3)C8A1C1 ? prime (0) = δJ = integraldisplay x 1 x 0 [F y ? d dx F y prime]δy dx =0. 19 C0B1AGBVD0CYA1C6CVAH J[y]BPBZB4D1C2AHCN y(x) AKA8AGA2D1CWBA F y ? d dx F y prime =0, (3.1.4) B0 F y prime y primey primeprime + F yy primey prime + F xy prime ? F y =0. (3.1.5) CWBA(3.1.4)B0(3.1.5)B6A5CVAHJ[y]C2EulerCWBAA2 ADBXCYCLANC1A3 B4C93.1.1 CVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ) dx CI y(x) BPBZB4D1C2AKBKDABOCD y(x) A8AGCWBAC8 (3.1.4)B0 (3.1.5)C8A2 CB 3.1.2 CWAIEulerCWBA(3.1.5)CHC1B0C8 d dx (F ? y prime F y prime) ? F x =0. (3.1.6) A1CGA3 dF dx = F x + F y y prime + F y primey primeprime , d dx (y prime F y prime)=y primeprime F y prime + y prime (F xy prime + F yy primey prime + F y prime y primey primeprime ), DBC8AOBMA1C1 d x (F ? y prime F y prime)=F x ? y prime (F y prime y primey primeprime + F yy primey prime + F xy prime ? F y ). CHC0(3.1.5)AMC1BZCWAIA2 CB 3.1.3 BCCVAH J[y(x)] = integraldisplay 1 0 (y prime2 +12xy) dx C2B4D1BEANA2 BZ F(x, y, y prime )=y prime2 +12xy,AZ EulerCWBAA5 12x ? 2y primeprime =0. C3BLCWBAA1C1 y = x 3 + c 1 x + c 2 . C0 y(0) = 0,y(1) = 1 C1A3 c 1 = c 2 =0,BUBL J[y(x)] C2B4D1BEANA5 y = x 3 . 20 CB 3.1.4 BCCVAH J[y(x)] = integraldisplay 1 0 (y prime2 ? y 2 ? 2xy) dx A8AGALC4DABO y(0) = y(1) = 0C2B4D1BEANA2 BZ F(x, y, y prime )=y prime2 ? y 2 ? 2xy, AZ EulerCWBAA5 ?2y ? 2x ? 2y primeprime =0. C3BLCWBAA1C1 y = c 1 cos x + c 2 sin x ?x. C0ALC4DABO y(0) = y(1) = 0,C1 c 1 =0,c 2 = 1 sin 1 , C4CDJ[y(x)]C2B4D1BEANA5 y = sin x sin 1 ? x. CB3.1.5 CID7BWDBC7A(x 0 ,y 0 ),B(x 1 ,y 1 )C2CYC1 AWAGBEANDAA1BCB4CGAKCJC2BEANA2 BZ AAD7CQBQA5AUA5CIALC4DABOy(x 0 )=y 0 ,y(x 1 )= y 1 AIBCCVAH J[y]= integraldisplay x 1 x 0 radicalBig 1+y prime2 dx C2B4ATD1A2 B4D1BEANA5D0ANy = c 1 x + c 2 ,BRBSALC4DABOBJ CCc 1 ,c 2 ,C1 y = y 0 + y 1 ?y 0 x 1 ?x 0 (x ?x 0 ). CB 3.1.6 BCCVAH J[y]= integraldisplay x 1 x 0 √ 1+y prime2 x dx,y(x 0 )=y 0 ,y(x 1 )=y 1 C2B4D1BEANA2 BZ F(x, y, y prime )= √ 1+y prime2 x 21 AUAKAG y,A4 EulerCWBAA5 d dx y prime x √ 1+y prime2 =0, BEB2D1A5 y prime x √ 1+y prime2 = c B0 x = y prime c √ 1+y prime2 . BVC7AZCNA1DJ y prime =tant,CJ x = tant c √ 1+tan 2 t = 1 c sin t = c 1 sin t, c 1 = 1 c , dy = y prime dx =tantc 1 cos tdt = c 1 sin tdt, y = ?c 1 cos t + c 2 . A4C1 ? ? ? ? ? x = c 1 sin t, y = ?c 1 cos t + c 2 . ASBGAZCN t,C1 x 2 +(y ? c 2 ) 2 = c 2 1 . CB3.1.7 BCC1ANAAD7C2C3A1B7BC J[y]= integraldisplay a 0 √ 1+y prime2 √ 2gy dx,y(0) = 0,y(a)=b C2B4D1BEANA2 BZ F(x, y, y prime )= √ 1+y prime2 √ 2gy AUAKAG x,EulerCWBAA5 d dx (y prime F y prime ? F)=0. C4CDA1 EulerCWBACHC1BEB2D1 y prime F y prime ?F = c, B7 y prime y prime √ 2gy √ 1+y prime2 ? √ 1+y prime2 √ 2gy = c. 22 AUBLAPA1C1 y(1 + y prime2 )=c 1 . DJ y prime =cott/2,CJC1 y = c 1 1+y prime2 = c 1 sin 2 t 2 = c 1 2 (1 ? cos t), dx = dy y prime = c 1 2 sin t cot t 2 dt = c 1 2 (1 ? cos t)dt. A4 ? ? ? ? ? x = c 1 2 (t ? sin t)+c 2 , y = c 1 2 (1 ? cos t). C0 t =0C3 x =0,C1 c 2 =0,CO c 1 C0 y(a)=b BJ CCA1C4CDC1ANAAD7C2C3A5 ? ? ? ? ? x = c 1 2 (t ? sin t), y = c 1 2 (1 ? cos t). CRCDBMDAAFDBC7C2A8ANA2 CB3.1.8 BCAKATB9A5BEAGAGB2C2AAD7A1B7BCCV AH J[y]=2π integraldisplay x 1 x 0 y radicalBig 1+y prime2 dx,y(x 0 )=y 0 ,y(x 1 )=y 1 C2B4D1BEANA2 BZ EulerCWBACHC1BEB2D1 y prime F y prime ?F = c, B7 2πy y prime2 √ 1+y prime2 ? 2πy radicalBig 1+y prime2 = c. AUBLAPCQC1 y = c radicalBig 1+y prime2 . DJ y prime =sinht,BRBSBXC8C1 y = c 1 cosh t, dx = dy y prime = c 1 sinh t sinh t dt = c 1 dt. C4CDA1CYBCBEAGCDC0AWAGBEAN ? ? ? ? ? x = c 1 t + c 2 , y = c 1 cosh t, 23 BM xDEB9A5COB7C2A1CIBXC8DAASBGAZCN t,C1 y = c 1 cosh x ?c 2 c 1 , AZDAB3CNc 1 ,c 2 C0 y(x 0 )=y 0 ,y(x 1 )=y 1 BJCCA2C0BL CQBNA1AKATBEAGCDB8D9AGA2 3.2 BJDDB6BFBKCUB0B9BKB0ASBBD3CY C0CVAHA5 J[y(x),z(x)] = integraldisplay x 1 x 0 F(x, y, z, y prime ,z prime ) dx, (3.2.1) ALC4DABOA5 ? ? ? ? ? y(x 0 )=y 0 , z(x 0 )=z 0 , ? ? ? ? ? y(x 1 )=y 1 , z(x 1 )=z 1 , (3.2.2) BCCVAH(3.2.1)C8CIALC4DABO (3.2.2)C8AIC2B4D1A2 AJCOBF J BFC1B4D1C2AKBKDABOA2C0 F A7C4CY AGANDCCHC1CQBYD7B7AVBYCNA1ARC0 J CIBEAN y = y(x),z= z(x)BXBFC1B4D1A2C0 ? ? ? ? ? ˉy = y(x)+αδy, ˉz = z(x)+βδz, A5 ? ? ? ? ? y = y(x), z = z(x), C2DGC8BEANA1AZDA y(x),z(x) ∈ C 2 ([x 0 ,x 1 ]),δy,δzCD y,zC2AND1A1B7 δy| x 0 = δz| x 0 = δy| x 1 = δz| x 1 =0. BQ ˉy, ˉz BRBS (3.2.1)C8A1C1 J[ˉy, ˉz]= integraldisplay x 1 x 0 F(x, y+αδy,z+βδz,y prime +αδy prime ,z prime +βδz prime ) dx. BXC8CI α = β =0C3BFC1B4D1A1CYBQ ?J ?α vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle α=β=0 =0, ?J ?β vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle α=β=0 =0, 24 B7 integraldisplay x 1 x 0 (F y δy + F y primeδy prime ) dx= integraldisplay x 1 x 0 (F y ? d dx F y prime)δydx =0, integraldisplay x 1 x 0 (F z δz + F z primeδz prime ) dx= integraldisplay x 1 x 0 (F z ? d dx F z prime)δz dx =0, C0C4δy,δzCDBNBSAHCNA1DHCFB1AGBVD0CY ? ? ? ? ? F y ? d dx F y prime =0, F z ? d dx F z prime =0. (3.2.3) CWBA(3.2.3)C8B6A5CVAH (3.2.1)C8C2 EulerCWBAA2 B4C93.2.1 CVAH (3.2.1)CI y(x),z(x) BFC1B4D1 C2AKBKDABOCD y(x),z(x)A8AG EulerCWBA(3.2.3). CB 3.2.2 BCCVAH J[y,z]= integraldisplay π 2 0 (y prime2 + z prime2 +2yz) dx A8AGALC4DABO y(0) = z(0) = 0,y( π 2 )=1,z( π 2 )=?1 C2B4D1BEANA2 BZ BUA5 F = y prime2 + z prime2 +2yz, A4 EulerCWBAA5 ? ? ? ? ? 2z ? d dx (2y prime )=0, 2y ? d dx (2z prime )=0, B7 ? ? ? ? ? z ? y primeprime =0, y ?z primeprime =0. C0BLCWBAASBG z,C1 y (4) ? y =0, AZDBC3A5 ? ? ? ? ? y = c 1 e x + c 2 e ?x + c 3 cos x + c 4 sin x, z = y primeprime = c 1 e x + c 2 e ?x ?c 3 cos x ? c 4 sin x. 25 C0ALC4DABOCQBQBJCC c 1 = c ? 2=c 3 =0,c 4 =1,A4 CYBCC2B4D1BEANA5 ? ? ? ? ? y =sinx, z = ?sin x. CB 3.2.3 C0 F y prime y primeF z prime z prime ? F 2 y prime z prime negationslash=0,BCCVAH J[y,z]= integraldisplay x 1 x 0 F(y prime ,z prime ) dx C2B4D1BEANA2 BZ BUA5F y = F z =0,A4 EulerCWBAA5 ? ? ? ? ? d dx F y prime =0, d dx F z prime =0, B7 ? ? ? ? ? F y prime y primey primeprime + F y prime z primez primeprime =0, F y prime z primey primeprime + F z prime z primez primeprime =0. DHCFBGC0DABOA1BLCWBAC6C1DHC3 y primeprime =0,z primeprime =0. B7 ? ? ? ? ? y = c 1 x + c 2 z = c 3 x + c 4 CDCYBCC2B4D1BEANA2 B4C93.2.4 CVAH J = integraldisplay x 1 x 0 F(x, y 1 ,···,y n ,y prime 1 ,···,y prime n ) dx CIA8AGALC4DABO y i (x 0 )=y i0 ,y i (x 1 )=y i1 ,i=1, 2,···,n. AIBFC1B4D1C2AKBKDABOCD y 1 (x),y 2 (x),···,y n (x) A8 AGEulerCWBA F y i ? d dx F y prime i =0,i=1, 2,···,n. A9A3BMA9CWCRA1BXCLEulerCWBAC2DBC3AGC12n DEBNBSB3CNA1D0?CQC0CYDGC2ALC4DABOBJCCA2 26 3.3 BJDDD2A3BKCUB0BEBYAZCUB0ASBBD3CY AJD2A4CVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ,y primeprime ) dx (3.3.1) C2B4D1AAD7A1AZALC4DABOA5 ? ? ? ? ? y(x 0 )=y 0 , y(x 1 )=y 1 , ? ? ? ? ? y prime (x 0 )=y prime 0 , y(x 1 )=y prime 1 . (3.3.2) C0 F ∈ C 3 ,y∈ C 4 ,ARC0y(x)CDC6J BFC1B4D1C2AH CNA1DJ ˉy = y(x)+αδy,AZDAδyCD y C2AND1A1B7 δy| x 0 = δy| x 1 = δy prime | x 0 = δy prime | x 1 =0. BQ ˉy BRBS(3.3.1)C8A1CL αBCBYAPCHDJ α =0,C1 δJ[y]= integraldisplay x 1 x 0 (F y δy + F y primeδy prime + F y primeprimeδy primeprime ) dx. D2BZD1AWB2D1CSA1C1 integraldisplay x 1 x 0 F y primeδy prime dx= F y primeδy vextendsingle vextendsingle vextendsingle vextendsingle x 1 x 0 ? integraldisplay x 1 x 0 δy d dx F y prime dx= ? integraldisplay x 1 x 0 d dx F y prime·δy dx, integraldisplay x 1 x 0 F y primeprimeδy primeprime dx= ? integraldisplay x 1 x 0 d dx F y primeprime·δy prime dx= integraldisplay x 1 x 0 d 2 dx 2 F y primeprime·δydx, C4CD δJ[y]= integraldisplay x 1 x 0 (F y ? d dx F y prime + d 2 dx 2 F y primeprime)δy dx. C0 δyC2BNBSB2B6B1AGBVD0CY F y ? d dx F y prime + d 2 dx 2 F y primeprime =0. (3.3.3) CWBA(3.3.3)B6A5 Euler-PoissonCWBAA2C4CDC1 B4C93.3.1 CVAH (3.3.1)CI y(x)BFC1B4D1C2AK BKDABOCD y(x)A8AGEuler-PoissonCWBA(3.3.3). A9A3 BMA9CWCRA1 E-P CWBACDCTBYB3A2D1CW BAA1AZDBC3DAAGC1CTDEBNBSB3CNA1D0?C0ALC4DA BO(3.3.2)BJCCA2 CB 3.3.2 BCCVAH J[y]= integraldisplay 1 0 (1 + y primeprime2 ) dx 27 A8AGALC4DABO y(0) = 0,y prime (0) = 0,y(1) = 1,y prime (1) = 1 C2B4D1BEANA2 BZ BUA5 F =1+y primeprime2 , A4AZ E-PCWBAA5 d 2 dx 2 (2y primeprime )=0, B7 y (4) =0, AZDBC3A5 y = c 1 x 3 + c 2 x 2 + c 3 x + c 4 . C0ALC4DABOCQC1 c 1 = c 2 = c 4 =0,c 3 =1,BUBLA1CY BCB4D1BEANA5y = x. CB 3.3.3 BCCVAH J[y]= integraldisplay π 2 0 (y primeprime2 ? 2y prime2 + y 2 ) dx A8AGALC4DABO y(0) = y prime (0) = 0,y( π 2 )=1,y prime ( π 2 )= π 2 C2B4D1BEANA2 BZ BUA5 F = y primeprime2 ? 2y prime2 + y 2 , A4AZ E-PCWBAA5 2y ? d dx (?4y prime )+ d 2 dx 2 (2y primeprime )=0, B7 y (4) +2y primeprime + y =0, AZDBC3A5 y =(c 1 + c 2 x)cosx +(c 3 + c 4 x)sinx. 28 C0ALC4DABOCQC1 c 1 =0,c 2 = ?1,c 3 =1,c 4 =0,BUBL CYBCB4D1BEANA5 y = ?x cos x +sinx. CB 3.3.4 BCCVAH J[y]= integraldisplay l ?l ( 1 2 μy primeprime2 + ρy) dx A8AGALC4DABO y(?l)=y prime (?l)=y(l)=y prime (l)=0 C2B4D1BEANA2 BZ AGD7CIBCDBCIB6A2C2BWB2DJB0DAC2DJBEC3 C9BZA2BR?DACDCLCFC2A1CJ μρCDB3CNA1A4AZ E-P CWBAA5 ρ + d 2 dx 2 (μy primeprime )=0, B7 y (4) = ? ρ μ , AZDBC3A5 y = ? ρ 24μ x 4 + c 1 x 3 + c 2 x 2 + c 3 x + c 4 . C0ALC4DABOCQC1 c 1 = c 3 =0,c 2 = ρl 2 12μ ,c 4 = ? ρl 4 24μ , BUBLCYBCB4D1BEANA5 y = ? ρ 24μ x 4 + ρl 2 12μ x 2 ? ρl 4 24μ = ? ρ 24μ (x 2 ?l 2 ) 2 . CLC4AGC1A6CYAHCNC2DIDCBYBYCNA1B0AGCNDEA6 CYAHCNC2A6CCALC4C2AND1AAD7A1CIF AGA2AAATC2 DABOAIA1CYCUC5CQBQC1BZAIDEC2?A2 B4C93.3.5 BR?CVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ,···,y (n) ) dx, CJy(x)C6J[y]BFC1B4D1C2AKBKDABOCDEuler-Poisson CWBA F y ? d dx F y prime + d 2 dx 2 F y primeprime ?···+(?1) n d n dx n F y (n) =0 29 B7D4A2 B4C93.3.6 BTCVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ,···,y (n) ) dx, CJy(x),z(x)C6J[y,z]BFC1B4D1C2AKBKDABOCDEuler- PoissonCWBA ? ? ? ? ? F y ? d dx F y prime + d 2 dx 2 F y primeprime ?···+(?1) n d n dx n F y (n) =0, F z ? d dx F z prime + d 2 dx 2 F z primeprime ?···+(?1) n d n dx n F z (n) =0, B7D4 B4C93.3.7 BR?CVAH J[y 1 ,y 2 ,···,y m ]= integraldisplay x 1 x 0 F(x, y 1 ,···,y (n 1 ) 1 ; y 2 ,···,y (n 2 ) 2 ;···; y m ,···,y (n m ) m ) dx, CJ y 1 (x),y 2 (x),···,y m (x) C6 J[y 1 ,y 2 ,···,y m ] BFC1B4 D1C2AKBKDABOCD Euler-PoissonCWBA F y i ? d dx F y prime i + d 2 dx 2 F y primeprime i ?···+(?1) n d n dx n F y (n) i =0,i=1,···,m. B7D4A2 3.4 B6DFBKCUB0B9BKBRA5D3CY COA1CQCAAHCNC2CVAH J[u(x, y)] = integraldisplay D integraldisplay F[x, y, u(x, y),u x (x, y),u y (x, y)] dσ (3.4.1) C2B4D1AAD7A1AZDAF CDCHC1CQBYD7B7AVBYCNC2AD CAAHCNA1COAHCN u(x, y)BKBCCIAWAGBXC2AJBDC8 D AQC1CQBYD7B7AVBYCNA1B9CIDC2ALC4ΓBXBFBPCY D1A2DJ Y = {u(x, y): u ∈ C 2 (D),u| Γ =BPCYD1}. ALCIC2AAD7CDA1CIY DACOBMDEAHCNu(x, y),C6CVAH J[u(x, y)]BFB4D1A2BT u ∈ Y C6C1 J[u] BPBZB4D1A1 30 AOAAu(x, y) CHC1C4AAD4CTAPA4CLC4C5CN α, COA1 AIBUAHCNB5AL ˉu = u + αη, AZDAη = δu = u 2 ?u 1 ,u 1 ,u 2 ∈ Y,η| Γ = u 2 | Γ ?u 1 | Γ =0. BUBLA1 ˉu ∈ Y . CLC4BNBSA6CCC2η(x, y),COA1AHCN ?(α)=J[ˉu]= integraldisplay D integraldisplay f(x, y, u + αη,u x + αη x ,u y + αη y ) dσ. CICYBGCCC2DABOAIA1 ?(α) CHC1D7B7BYCNA1B9BX α =0C3 ?(α)BFC1B4D1A1BUCO ? prime (α)| α=0 =0.CABA CWC1 ? prime (α)= integraldisplay D integraldisplay (F u η + F u x η x + F u y η y ) dσ. A3BSA1BXC8C2AFB2AHCNDA F u ,F u x ,F u y CI x, y, u + αη,u x + αη x ,u y + αη y BIBFB4D1A1BQ α =0BRBSBX C8A1C1 ? prime (0) = integraldisplay D integraldisplay (F u η + F u x η x + F u y η y ) dσ=0. (3.4.2) (3.4.2) C8C2AFB2AHCNDA F u ,F u x ,F u y CI x, y, u, u x ,u y BIBFD1A1u(x, y)A5C6J[u]BPBZB4D1C2AHCNA2A5DD BXBZB1AGBVD0A1AB?AKB4D1CZBF η(x, y)CWA1BUBL BKC3DG η CL x, y C2AVBYCNA1AB?BXBZ GreenA0C8 contintegraldisplay Γ Pdx+ Qdy= integraldisplay D integraldisplay ( ?Q ?x ? ?P ?y ) dσ. BUF u x η x C7F u y η y ARAUCDANDEAHCNC2AVBYCNA1COCD F u x η,F u y η C2AVBYCNC2BMAWD1A2CBC5BX ? ?x (F u x η)=F u x η x + η ? ?x F u x , ? ?y (F u y η)=F u y η y + η ? ?y F u y , 31 CYBQ integraldisplay D integraldisplay (F u x η x + F u y η y ) dσ = integraldisplay D integraldisplay [ ? ?x (F u x η)+ ? ?y (F u y η)] dσ ? integraldisplay D integraldisplay η( ? ?x F u x + ? ?y F u y ) dσ = contintegraldisplay Γ (?F u y η) dx+(F u x η) dy ? integraldisplay D integraldisplay η( ? ?x (F u x + ? ?y (F u y ) dσ = ? integraldisplay D integraldisplay η( ? ?x F u x + ? ?y F u y ) dσ BQBXC8BRBS (3.4.2)C8C1 ? prime (0) = integraldisplay D integraldisplay η[F u ? ( ? ?x F u x + ? ?y F u y )] dσ=0. BUA5η(x, y)CDBNBSC2A1C0B1AGBVD0CY F u ? ? ?x F u x ? ? ?y F u y =0. C4CDC1AIAGC2CCD0A2 B4C93.4.1 BTAHCNu(x, y) ∈ Y C6CVAH(3.4.1)BF C1B4D1A1CJ u(x, y)AKA8AGAVA2D1CWBA F u ? ? ?x F u x ? ? ?y F u y =0. (3.4.3) CWBA(3.4.3)B6A5CVAHJ[u]C2A6CFCWBAA1CRCDBM DECQBYAVA2D1CWBAA2 A9A3BX u(x, y)C6CVAH J[u]= integraldisplay D integraldisplay F(x, y, u, u x ,u y ) dσ BPBZB4D1C3A1 u(x, y)AKCDAVA2D1CWBACCC3AAD7 ? ? ? ? ? F u ? ? ?x F u x ? ? ?y F u y =0, u(x, y)| Γ =BPCYD1, C2C3A2CRCRAIA1BCAVA2D1CWBAC2CCC3AAD7CQBQA5 AUA5BCANDECVAHC2B4D1AAD7A1CRCDCDBCC3AVA2D1 CWBAC2AND1CSC2B1BHA2 CB 3.4.2 BCBFCVAH J[u]= integraldisplay D integraldisplay [u 2 x + u 2 y +2uf] dσ 32 C2A6CFCWBAA2 BZ BUA5 F = u 2 x + u 2 y +2uf,F u =2f,F u x =2u x ,F u y =2u y , A4A6CFCWBAA5 F u ? ? ?x F u x ? ? ?y F u y =2f ? 2u xx ? 2u yy =0, B0 triangleu = ? 2 u ?x 2 + ? 2 u ?y 2 = f(x, y). B7BLCVAH J[u]C2A6CFCWBAA5 PoissonCWBAA2 A7C4CNCAAHCNC2CVAHA1C1AIDEBVDEC2A4A2 B4C93.4.3 CLC4DBDECQCAAHCNC2CVAH J[u(x, y),v(x, y)] = integraldisplay D integraldisplay F(x, y, u, v, u x ,v x ,u y ,v y ) dσ, AZA6CFCWBAA5 F u ? ? ?x F u x ? ? ?y F u y =0,F v ? ? ?x F v x ? ? ?y F v y =0. B4C9 3.4.4 CLC4CHC1BMDECQCAAHCN u(x, y) B9 F DAAGC1D0C2DCBYAVBYCNC2CVAHA1BR J[u]= integraldisplay D integraldisplay F(x, y, u, u x ,u y ,u xx ,u xy ,u yy ) dσ, AZA6CFCWBAA5 F u ?( ? ?x F u x + ? ?y F u y )+( ? 2 ?x 2 F u xx + ? 2 ?x?y F u xy + ? 2 ?y 2 F u yy )=0. B4C93.4.5 BVCAAHCNC2CVAH J[u(x, y, z)] = integraldisplayintegraldisplay ? integraldisplay F(x, y, z, u, u x ,u y ,u z ) dv, C2A6CFCWBAA5 F u ? ? ?x F u x ? ? ?y F u y ? ? ?z F u z =0. CB 3.4.6 BCCVAH J[u(x, y, z)] = integraldisplayintegraldisplay ? integraldisplay (u 2 x + u 2 y + u 2 z +2uf) dv 33 C2A6CFCWBAA2 BZ BUA5 F = u 2 x +u 2 y +u 2 z +2uf,F u =2f,F u x =2u x ,F u y =2u y ,F u z =2u z , A4 J[u]C2A6CFCWBAA5 2f ? 2 ? 2 u ?x 2 ? 2 ? 2 u ?y 2 ? 2 ? 2 u ?z 2 =0, B0 triangleu = ? 2 u ?x 2 + ? 2 u ?y 2 + ? 2 u ?z 2 = f(x, y, z). CB 3.4.7 BCBFCVAH J[u(x, y)] = integraldisplay D integraldisplay (u 2 xx +2u 2 xy + u 2 yy ) dσ C2A6CFCWBAA2 BZ BUA5 F = u 2 xx +2u 2 xy + u 2 yy , F u =0,F u x =0,F u y =0,F u xx =2u xx ,F u xy =4u xy ,F u yy =2u yy , A4 J[u]C2A6CFCWBAA5 F u ? ( ? ?x F u x + ? ?y F u y )+( ? 2 ?x 2 F u xx + ? 2 ?x?y F u xy + ? 2 ?y 2 F u yy ) =0? ( ?0 ?x + ?0 ?y )+[ ? 2 ?x 2 (2u xx )+ ? 2 ?x?y (4u xy )+ ? 2 ?y 2 (2u yy )] =0, B0 triangletriangleu = ? 4 u ?x 4 +2 ? 4 u ?x 2 ?y 2 + ? 4 u ?y 4 =0 CRDECWBAB6A5DCCAAJCWBAA1BLBTBBA5 triangletriangleu =0. 34 B2CVDJ C5B5ARC0B0ASBBD3CY 4.1 ASBBD3CYB0CXAW D2A4CVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ) dx (4.1.1) C2B4D1A2ALCIBGCCDBDEALC4C7A(x 0 ,y 0 ),B(x 1 ,y 1 )DA C2BMDEB0DBDEARA2BOCDA1CRC3CQBFBEANC2CUA3CU BQA2BUBLA1C6C1A6CCALC4AAD7BPBZB4D1C2B1AGAK BKDABOBXBXC1BZA8AGA1B7AHCNy(x)BXBXA8AGEuler CWBA F y ? d dx F y prime =0. (4.1.2) CRCDBMDECQBYB3A2D1CWBAA1D0C2DBC3y = y(x, c 1 ,c 2 ) DAABAGDDDBDEBNBSB3CNA1BKAPBJCCD0?B3BKDBDE DABOA2CIA6CCALC4C2AND1AAD7DAA1CRDBDEDABOCD y(x 0 )=y 0 AJ y(x 1 )=y 1 , COCIALC4CQANCDC2BAB0 AIA1 x 0 AJ x 1 BLCDBSCCC2A1A5DDBJCCD0?A1B3BK BCCOAZCZDABOA2AB?AJD2A4ALCIC7A(x 0 ,y 0 )A6CC COC2CIC7B(x 1 ,y 1 )CQBQANCDC2BAB0A2 C0CVAH (4.1.1) C8DAC2 y(x) C1DCBMDEA6CCC2AL CIC7A(x 0 ,y 0 ), B7 y(x 0 )=y 0 , COC0C2CIC7 B(x 1 ,y 1 ) CQBQCIANBEANw(x, y)=0BXBOCDA1CJC1DABOw(x 1 ,y 1 ) =0.BLC3x 1 CDBSCCC2A2 C0CVAH (4.1.1) C2B4D1BEANA5y = y(x), COA1BX ALC4C7 B(x 1 ,y 1 ) C2A8D7BOCDBZC7B 1 (x 1 + δx 1 ,y 1 + δy 1 )=y(x) C2A8D7C3CVAH (4.1.1) C2AND1A2A5BLA1 AB?CICVAHC2CKDC ?J DABFBFCLδx 1 AJ δy 1 COBF A5ANB2DIAWC2AOBMAWD1A2CVAH J C2CKDC ?J = integraldisplay x 1 +δx 1 x 0 F(x, y + δy,y prime + δy prime ) dx? integraldisplay x 1 x 0 F(x, y, y prime ) dx = integraldisplay x 1 +δx 1 x 1 F(x, y + δy,y prime + δy prime ) dx + integraldisplay x 1 x 0 [F(x, y + δy,y prime + δy prime ) ? F(x, y, y prime )] dx. (4.1.3) CLC4(4.1.3)C8C2CIC2C6BMAQA1C0B2D1DAD1CCD0B6 35 F C2D7B7B2A1C1 integraldisplay x 1 +δx 1 x 1 F(x, y + δy,y prime + δy prime ) dx = F| x=x 1 +θδx 1 · δx 1 = F(x, y, y prime )| x=x 1 · δx 1 + ε 1 δx 1 , (4.1.4) AZDA 0 <θ<1,COBXδx 1 → 0C3 ε 1 → 0. CLC4(4.1.3)C8C2CIC2C6CQAQA1C0 Taylor A0C8CL CMA1C1 integraldisplay x 1 x 0 [F(x, y + δy,y prime + δy prime ) ?F(x, y, y prime )] dx = integraldisplay x 1 x 0 [F y (x, y, y prime )δy + F y prime(x, y, y prime )δy prime ] dx+ R 1 , (4.1.5) AZDA R 1 CDBU δy AJ δy prime DCBYC2ACBBATDCA2D2BZD1 AWB2D1CSB6 Euler CWBA(4.1.2),ARA3BSBZ δy| x 0 =0, C1 integraldisplay x 1 x 0 [F y δy + F y primeδy prime ] dx =[F y primeδy]| x 1 x 0 + integraldisplay x 1 x 0 [F y ? d dx F y prime]δy dx =[F y primeδy]| x=x 1 . (4.1.6) A3BSBMA9CWCRA1δy| x=x 1 negationslash= δy 1 ,CRCDBUA5δy 1 CDBXAL C4C7BOCDBZ(x 1 + δx 1 ,y 1 + δy 1 )A8D7C3y 1 C2CKDCA1 CO δy| x=x 1 CDBXDBAF (x 0 ,y 0 ) AJ (x 1 ,y 1 ) DBC7C2B4D1 BEANBOBZDBAF (x 0 ,y 0 )AJ (x 1 + δx 1 ,y 1 + δy 1 )DBC7C2 B4D1BEANC3A1CIC7 x 1 BIAFAOAOC2CKDCA2 AKBKA1 δy| x=x 1 ≈ δy 1 ? y prime (x 1 )δx 1 . BLC3C8CUC3C8C7C9BJC3C8AOB2BMDEBU δx 1 A5DCBY C2ACBBATA2C4CDC0 (4.1.4)C8A1ARBQ (4.1.6)C8BRBS (4.1.5)C8A1C1 integraldisplay x 1 +δx 1 x 0 Fdx≈ F| x=x 1 δx 1 , integraldisplay x 1 x 0 [F(x, y + δy,y prime + δy prime ) ? F(x, y, y prime )] dx ≈ F y prime| x=x 1 (δy 1 ?y prime (x 1 )δx 1 ). AZDAC8CUC3C8C7C9BJC3C8AOB2BU δx 1 AJ δy 1 A5DC BYC2ACBBATDCA2BQBQBXDBC8BRBS (4.1.3)C8A1B7CQ 36 C1BZ δJ = F| x=x 1 δx 1 + F y prime| x=x 1 (δy 1 ? y prime (x 1 )δx 1 ) =(F ? y prime F y prime)| x=x 1 δx 1 + F y prime| x=x 1 δy 1 . COB4D1C2B1AGAKBKDABO δJ =0CDB7A5 (F ? y prime F y prime)| x=x 1 δx 1 + F y prime| x=x 1 δy 1 =0. (4.1.7) BT δx 1 AJ δy 1 CDAOASACA7C2A1CJC1 (F ?y prime F y prime)| x=x 1 =0,F y prime| x=x 1 =0. BUCDB3B3C1AKBKCOA1AND1 δx 1 C7 δy 1 AOA7C2BA B0A2C0C2CIC7CQBQBGANBMBEAN C 1 : w(x, y)=0BO CDA1BLC3 w x 1 δx 1 + w y 1 δy 1 =0. (4.1.8) BGCC w y 1 negationslash=0,C0 (4.1.8)C3BF δy 1 APBRBS (4.1.7),AR C0δx 1 C2BNBSB2A1C1 [(F ? y prime F y prime)w y ? F y primew x ]| x=x 1 =0. (4.1.9) BTBGCC w x 1 negationslash=0,DCBJCQBQC1BZ(4.1.9). CWBA(4.1.9) B6A5B4D1BEAN y = y(x) C7BEAN C 1 C2AVBZDABOA2B7BT y = y(x)(x 0 ≤ x ≤ x 1 ) A5ANCD C2CIC7C2CVAHC2B4D1BEANA1CJC2CIC7 x 1 AKA8AG (4.1.7)C8A2AIAGCOA1B9DBDCBKC2D4CKBAB0A2 (1) C0C2CIC7B CQBQBGBEAN C 1 : y = ?(x) BO CDA2CRC3 w = y ? ?(x),w x = ?? prime (x),w y =1, C4CD (4.1.9)C8B7A5 [(? prime ? y prime )F y prime + F]| x=x 1 =0. (4.1.10) CRDEDABOBPD4DDCIALC4C7BI? prime AJ y prime CQAVA2CZBJ C2BNBNA7AHA2 (2) C0C2CIC7B CQBQCID0ANx = x 1 BXBOCDA2 CRC3 B B6A5ACC0CIC7A1CO δx 1 =0,C4CDC0 (4.1.7) C8B0 (4.1.9)C8CQC1 F y prime| x=x 1 =0. (4.1.11) 37 CRCRAIA1BT y = y(x) CD J[y] C2B4D1BEANA1CJ F y prime BGA9y = y(x)CI x = x 1 BIC2D1A5DHA2CRBJC2DABO (4.1.11)B6A5J[y]C2ACBKALC4DABOA2 (3) C0C2CIC7B CQBQCID0ANy = y 1 BXBOCDA1 C4CDδy 1 =0.C0 (4.1.7)C8B0 (4.1.9)C8CQC1 [F ? y prime F y prime]| x=x 1 =0. (4.1.12) AKAPA1BTCQBFBEANC2ALCIC7A(x 0 ,y 0 )CDCQANCD C2A1B0CQDBDECIC7A, BCECDCQANCDC2A1CJBQBXCL DCBJC2D2A4BXBZC4D9C7BXA1CQCY x 0 ,x 1 A5DFDBAU DCBAB0D1AQA8AGDABO (4.1.9)-(4.1.12)C8A2 CB 4.1.1 BCD7BWAWAGBXDBDABEAN x + y +1=0 C7 xy =1C2AKCJBEANA2 BZ C0CYBCAWAGBEANA5 y = f(x), ALCIC2AAD7 CDBCCVAH J[y]= integraldisplay x 1 x 0 radicalBig 1+y prime2 dx C2B4D1BEANA2 C 2 : y = ψ(x)=?(1 + x),A(x 0 ,ψ(x 0 )) ∈ C 2 ; C 1 : y = ?(x)= 1 x ,B(x 1 ,?(x 1 )) ∈ C 1 . J[y]C2 EulerCWBAC2C3A5 y = c 1 x + c 2 . CRC3C1 y prime = c 1 , F = radicalBig 1+y prime2 = radicalBig 1+c 2 1 , F y prime = y prime √ 1+y prime2 = c 1 radicalBig 1+c 2 1 . CICRDBDEALC4C7CEANCDC2BACTAIA1C0AVBZDABO(4.1.10) C7CYDGALC4DABOA1BXC1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [(? prime ? y prime )F y prime + F]| x=x 1 =0, [(ψ prime ? y prime )F y prime + F]| x=x 0 =0, y(x 0 )=y 0 , y(x 1 )=y 1 , 38 B7 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (? 1 x 2 1 ?c 1 ) c 1 radicalBig 1+c 2 1 + radicalBig 1+c 2 1 =0, (?1 ?c 1 ) c 1 radicalBig 1+c 2 1 + radicalBig 1+c 2 1 =0, c 1 x 0 + c 2 = ?(1 + x 0 ), c 1 x 1 + c 2 = 1 x 1 . C3D6D4CWBAC1A3 x 0 = ? 1 2 ,x 1 = ±1,c 1 =1,c 2 =0. BTCOA1C2CDCOBEANBXAOBXC4x>0C2AOBMD1CXA1 AOAAD6D4CWBAC1A4BMC2BMAJC3 x 0 = ? 1 2 ,x 1 =1,c 1 =1,c 2 =0. C4CDCYBCBEANA5 y = x, ? 1 2 ≤ x ≤ 1. CB 4.1.2 BCCVAH J[y]= integraldisplay a 0 √ 1+y prime2 √ 2gy dx C2B4D1BEANA2 BZ BUA5 F = √ 1+y prime2 √ 2gy , BUBLA1C1 F y prime = y prime √ 2gy √ 1+y prime2 . BNCOA1A8AGALC4DABOy(0) = 0C2EulerCWBAC2C3A5 ? ? ? ? ? x = c 1 2 (t ? sin t), y = c 1 2 (1 ? cos t). D7DACLy(a)ABC1ANBNAKBKBCA1CRBSA7A9CYDGBFC2 CDACBKALC4DABOA1C4CDC1 y prime (x) radicalBig 2gy(x) radicalBig 1+[y prime (x)] 2 | x=a =0, B7 y prime (a)=0. 39 CRCRAIA1CICYBCBEANBXC2C7 (a, y(a)) BIA1B8ANBX C7X DEAWB1A2CYBQC7 (a, y(a))BXCDA8ANC2CBC7A1 COA8ANC2CBC7CLBXC4t = π,BUBL a = c 1 2 (π ? sin π),c 1 = 2a π . C4CDCYBCB4D1BEANA5 ? ? ? ? ? x = a π (t ? sin t), y = a π (t ? cos t), 0 ≤ t ≤ π. 4.2 integraldisplay x 1 x 0 F(x, y, z, y prime ,z prime ) dxD8B9BKB0B0C5B5ARC0 D3CY AGC0D2A4CVAH J[y,z]= integraldisplay x 1 x 0 F(x, y, z, y prime ,z prime ) dx (4.2.1) C2CQCDALC4C2B4D1AAD7A2A5DDCWAMB1BNA1AB?BG CCJ[y,z]C2BMDEALC4C7 A(x 0 ,y 0 ,z 0 )CDA6CCC2A1CO DIBMDEALC4C7 B(x 1 ,y 1 ,z 1 )CQBQBOCDA2C0 (4.2.1)C8 C2B4D1BEANA5 ? ? ? ? ? y = y(x), z = z(x), x 0 ≤ x ≤ x 1 (4.2.2) CYCUC4A6CCALC4AAD7C2D2A4A1y(x),z(x) AKB4A8AG EulerCWBA ? ? ? ? ? F y ? d dx F y prime =0, F z ? d dx F z prime =0. (4.2.3) CRDECWBAC2DBC3DAAGC1CTDEBNBSB3CNA1C0C4B C7ANCDA1C3CNDDBMDEA6CYDCx 1 . CRBJA1A5DDARA2BJ CCA4BMC3A1CDBKBJCCADDEB3CNA2C0C4AC7A6CCA1 BMA9AVCQBQBMBYDBDEBNBSB3CNA1A5DDBJCCDIDIBV DEB3CNA1AKB4AVC1BVDECWBAA2AIAGDHCF BC2AUDC BACTA1D1AQDGBFCRAUCWBAA2 CLC4ALCIC7A(x 0 ,y 0 ,z 0 )A6CCCOC2CIC7B(x 1 ,y 1 ,z 1 ) CQBQANCDC2BAB0A1CRC3 J[y,z]BFB4D1C2AKBKDABO 40 δJ =0CAAUBLAPA5 [F?y prime F y prime?z prime F z prime]| x=x 1 δx 1 +F y prime| x=x 1 δy 1 +F z prime| x=x 1 δz 1 =0. (4.2.4) BXC8DAδx 1 ,δy 1 ,δz 1 CDBNBSC2A1B7C7 BCQBQA5BNBS CWC8BOCDA1AIAGAB?D1BVDBBAB0D2A4A2 (1) BT δx 1 ,δy 1 ,δz 1 AOASACA7A1C0 (4.2.4)C8CQC1 ? ? ? ? ? ? ? ? ? ? ? [F ?y prime F y prime ? z prime F z prime]| x=x 1 =0, F y prime| x=x 1 =0, F z prime| x=x 1 =0. (4.2.5) BCBF Euler CWBA(4.2.3) C2DBC3A1C0 (4.2.5) C8B6 A A6CCCWBJCCCRDBBAB0AIC2B4D1BEANB6 x 1 C2D1A2 (2) BTC2CIC7B BGBEAN y = ?(x),z = ψ(x) BO CDA1CRC3 δx 1 ,δy 1 ,δz 1 A8AGA7AHC8 δy 1 = ? prime (x 1 )δx 1 ,δz 1 = ψ prime (x 1 )δx 1 . BRBS (4.2.4)C8A1C1 [F +(? prime ?y prime )F y prime +(ψ prime ? z prime )F z prime]| x=x 1 · δx 1 =0. C0 δx 1 C2BNBSB2A1C1 [F +(? prime ? y prime )F y prime +(ψ prime ?z prime )F z prime]| x=x 1 =0. (4.2.6) BXC8B6A5CVAH J[y,z]C2B4D1BEANC7CIC7BEANC2AV BZDABOA2BCBFEulerCWBA(4.2.3)C2DBC3A1CHC0(4.2.6) C8A1y 1 = ?(x 1 ),z 1 = ψ(x 1 )B6 AA6CCCDCQBQBJCCCR DBBAB0AIC2B4D1BEANAJ x 1 C2D1A2 (3) BTCIC7BCIANBEAG?(x, y, z)=0BXBOCDA1 BLC3δx 1 ,δy 1 ,δz 1 A8AGA7AHC8 ? x 1 δx 1 + ? y 1 δy 1 + ? z 1 δz 1 =0. BR?? x 1 negationslash=0,CJC1 δ x 1 = ? x 1 ? z 1 δx 1 ? ? y 1 ? z 1 δy 1 . BRBS (4.2.4)C8A1ARA3BSBZ δx 1 ,δy 1 C2BNBSB2C1 ? ? ? ? ? [F ? y prime F y prime ? z prime F z prime ? F z prime ? x ? z ] x=x 1 =0, [F y prime ? F z prime ?F z prime ? y ? z ] x=x 1 =0, (4.2.7) 41 (4.2.7) C8B6A5CVAH J[y,z] C2B4D1BEANC7CIC7BEAG C2AVBZDABOA2BCBFEulerCWBA(4.2.3)C2DBC3A1CHC0 (4.2.7)C8A1?(x 1 ,y 1 ,z 1 )=0AJ A(x 0 ,y 0 ,z 0 )CDA6CCC2 CQBQCCBFB4D1BEANAJ x 1 C2D1A2 AKAPA1BTALC4C7 A(x 0 ,y 0 ,z 0 )CDCQBQANCDC2A1BQ BXCLDCBJC2CWCSBXBZC4D9C7BXA1CDC1BZA0BHCYCU C2DABOA2BXBKBLCQBQCYCUC5D2A4DBDEALC4C7DCC3 ANCDC2BAB0A2 CB 4.2.1 BCCVAH J[y,z]= integraldisplay x 1 0 [y prime2 + z prime2 +2yz] dx C2B4D1BEANA1BPCYy(0) = 0,z(0) = 0,B9C7B(x 1 ,y 1 ,z 1 ) CIAWAGx = x 1 BXBOCDA2 BZ AGD3CD A(0, 0, 0)A6CCA1B(x 1 ,y 1 ,z 1 )CIAWAG x = x 1 BXANCDC2BAB0A2 AJBC EulerCWBAC2DBC3A2C0C4 F = y prime2 + z prime2 +2yz, F y =2z,F z =2y,F y prime =2y prime ,F z prime =2z prime , EulerCWBAA5 ? ? ? ? ? F y ? d dx F y prime =2z ? 2y primeprime =0, F z ? d dx F z prime =2y ? 2z primeprime =0. D0C2DBC3A5 ? ? ? ? ? y = c 1 cosh x + c 2 sinh x + c 3 cos x + c 4 sin x, z = c 1 cosh x + c 2 sinh x ?c 3 cos x ? c 4 sin x. C0ALC4DABO y(0) = 0,z(0) = 0 C1 c 1 = c 3 =0,C4 CD ? ? ? ? ? y = c 2 sinh x + c 4 sin x, z = c 2 sinh x ? c 4 sin x. (4.2.8) C0C4B CIAWAGx = x 1 BXBOCDA1A4 δx 1 =0,CO δy 1 ,δz 1 BNBSA2C0BMA9DABO (4.2.4),C1 ? ? ? ? ? F y prime| x=x 1 =0, F z prime| x=x 1 =0, 42 B7 ? ? ? ? ? y prime (x 1 )=0, z prime (x 1 )=0. C0 (4.2.8)C8A1C1 ? ? ? ? ? c 2 cosh x 1 + c 4 cos x 1 =0, c 2 cosh x 1 ?c 4 cos x 1 =0. BT cos x 1 negationslash=0,CJC0BXC8CQBQC3BFA3 c 2 = c 4 =0, C4CDA1CYBCB4D1BEANA5 ? ? ? ? ? y =0, z =1. BT cos x 1 =0,CJ x 1 = nπ + π 2 , nA5CUCNA1 c 2 =0,B4D1BEANA5 ? ? ? ? ? y = c 4 sin x 1 , z = ?c 4 sin x. AZDA c 4 A5BNBSB3CNA1D0CDAWAGz = ?y BXC2BNBS BMDABEANA2BQBRBICWA1CIBXCLDBDBBAB0AICEC1 J[y,z]=0. CB 4.2.2 BCD7BWC7 B(1, 0, 0) C7BEAG summationtext : z = x 2 + y 2 + 1 4 C2AKCJBEANA2 BZ C0CYBCBEANA5 ? ? ? ? ? y = y(x), z = z(x), x 0 ≤ x ≤ 1, D0AKC6CVAH J = integraldisplay 1 x 0 radicalBig 1+y prime2 + z prime2 dx BFAKATD1A2CRDECVAHC2BMDEALC4C7 B(1, 0, 0) A6 CCA1CODIBMDEALC4C7 A(x 0 ,y 0 ,z 0 ) CICSBJBEAG summationtext : z = x 2 + y 2 + 1 4 BXBOCDA2 F = radicalBig 1+y prime2 + z prime2 , 43 F y =0,F y prime = y prime √ 1+y prime2 + z prime2 , F z =0,F z prime = z prime √ 1+y prime2 + z prime2 . J C2 EulerCWBAA5 ? ? ? ? ? ? ? ? ? ? ? F y ? d dx F y prime = ? d dx y prime radicalBig 1+y prime2 + z prime2 =0, F z ? d dx F z prime = ? d dx z prime radicalBig 1+y prime2 + z prime2 =0, B7 ? ? ? ? ? ? ? ? ? ? ? y prime radicalBig 1+y prime2 + z prime2 =0, z prime radicalBig 1+y prime2 + z prime2 =0, CYBQz prime = ky prime ,BRBSBXC8A1C1 y prime = g 1 ,z prime = g 2 ,C4CDC1 ? ? ? ? ? y = g 1 x + h 1 , z = g 2 x + h 2 , AZDA k,g 1 ,g 2 ,h 1 ,h 2 CECDB3CNA1D0C2B9AKDEB0A5CS BJD0ANA2C0ALC4DABO y(1) = 0,z(1) = 0,C1 h 1 = ?g 1 ,h 2 = ?g 2 , CYBQ ? ? ? ? ? y = g 1 (x ? 1), z = g 2 (x ? 1). BUA5C7ACIBEAG summationtext BXBOCDA1CHC0AVBZDABO(4.2.7), C1 ? ? ? ? ? ? ? ? ? ? ? ? ? [F ?y prime F y prime ?z prime F z prime ?F z prime ? x ? z ]| x=x 0 =0, [F y prime ?F z prime ? y ? z ]| x=x 0 =0, z 0 ?x 2 0 ? y 2 0 ? 1 4 =0. B7 ? ? ? ? ? ? ? ? ? ? ? ? ? [1 + 2xz prime ]| x=x 0 =0, [y prime +2yz prime ]| x=x 0 =0, z 0 ? x 2 0 ? y 2 0 ? 1 4 =0. C0BLC3C1 g 1 =0,g 2 = ?1,x 0 = 1 2 , 44 C4CDCYBCD0ANA5 ? ? ? ? ? y =0, z = ?(x ? 1), 1 2 ≤ x ≤ 1. C7 AC2AOAOA5( 1 2 , 0, 1 2 ). BQCYBCD0ANBRBSCVAHA1B7 CQC1BZAKCJCICZ |AB| = radicalBigg ( 1 2 ? 1) 2 +( 1 2 ) 2 . 4.3 integraldisplay x 1 x 0 F(x, y, y prime ,y primeprime ) dxD8B9BKB0C5B5ARC0D3 CY BDCCBXCVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ,y primeprime ) dx (4.3.1) C2ALC4CQBQANCDC3C2B4D1AAD7A2C0C7 A(x 0 ,y 0 ,y prime 0 ) A5A6CCC7A1COC7B(x 1 ,y 1 ,y prime 1 )CDCQBQANCDC2A2C0CV AH (4.3.1) C2B4D1BEANA5 y = y(x), CJD0AKCCA8AG Euler-PoissonCWBA F y ? d dx F y prime + d 2 dx 2 F y primeprime =0. (4.3.2) CIBMA9BAB0AIA1CRCDBMDECTBYB3A2D1CWBAA1DBC3 DAAGC1CTDEBNBSB3CNA2C0C4BC7ANCDA1CYBQx 1 BL BSCCA2BUA5AC7A6CCA1C0 y(x 0 )=y 0 ,y prime (x 0 )=y prime 0 CQ BQBJCCBFEuler-Poisson CWBADBC3DAC2DBDEBNBSB3 CNA2BUBLA1A5DDBJCCA4BMC3A1B7A5DDBJCCDIDIBVDE BSCCB3CNA1CDAKB4CHCOBFBVDECWBAA1CRAUCWBACQ BQC0CVAHBFB4D1C2B1AGAKBKDABO δJ =0C1BZA2AB ?AVCDAJCDBMA9BAB0C7B1D2A4A1BACWCVAH (4.3.1) C2CKDC ?J,BKAPD1CZBFANB2DIAW δJ. C0CVAH(4.3.1)C8CIBEANy = y(x)BXBFC1B4D1A1 BNBFBMDACQBFBEAN y = y(x)+δy,BX x = x 0 C3CLBX C4C7A,BXx = x 1 +δx 1 C3CLBXC4C7 B prime (x 1 +δx 1 ,y 1 + δy 1 ,y prime 1 + δy prime 1 ). C7B5AGC2DFBYCYCUA1C4CDC1 ? ? ? ? ? δy| x=x 1 = δy 1 ?y prime (x 1 )δx 1 , δy prime | x=x 1 = δy prime 1 ? y primeprime (x 1 )δx 1 . (4.3.3) 45 ?J = integraldisplay x 1 +δx 1 x 0 F(x, y + δy,y prime + δy prime ,y primeprime + δy primeprime ) dx ? integraldisplay x 1 x 0 F(x, y, y prime ,y primeprime ) dx = integraldisplay x 1 +δx 1 x 0 F(x, y + δy,y prime + δy prime ,y primeprime + δy primeprime ) dx + integraldisplay x 1 x 0 [F(x, y + δy,y prime + δy prime ,y primeprime + δy primeprime ) ?F(x, y, y prime ,y primeprime )] dx. BXBZDAD1CCD0A1ARC0 F BQB6y(x),y prime (x),y primeprime (x)C2D7 B7B2A1CDC1BZ ?J = F(x, y, y prime ,y primeprime )| x=x 1 · δx 1 + integraldisplay x 1 x 0 [F y δy + F y primeδy prime + F y primeprimeδy primeprime ] dx+ R, AZDA R CDBU |δx 1 |,|δy 1 |,|δy|,|δy prime |,|δy primeprime | DGCNDACZAK BQBYCNA5DIDCBYC2ACBBATDCA2BUBL δJ = F| x=x 1 · δx 1 + integraldisplay x 1 x 0 [F y δy + F y primeδy prime + F y primeprimeδy primeprime ] dx. BQBXC8C2CIB2D1AIAIC2C6CQAQD1AWB2D1BMBMA1C6 BVAQD1AWB2D1DBBMA1ARA3BSBZC2CIC7A6CCC3δy| x=x 0 = 0,δy prime | x=x 0 =0,CHC0(4.3.2)C8A1AB?C1BZ δJ =[Fδx 1 + F y primeδyF y primeprimeδy prime ? d dx (F y primeprime)δy]| x=x 1 . BQ (4.3.3)C8BRBSBXC8A1CRC3CVAH (4.3.1)BFB4D1C2 AKBKDABO δJ =0AMB7A5 [F ? y prime F y prime ? y primeprime F ? y primeprime + y prime d dx F y primeprime]| x=x 1 · δx 1 + [F y prime ? d dx f y primeprime]| x=x 1 · δy 1 + F y primeprime| x=x 1 · δy prime 1 =0. (4.3.4) CRCRAIA1ALCIC7A6CCA3 y(x 0 = y 0 ,y prime (x 0 )=y prime 0 ,C2CI C7AUA9BRAKANCDA1B7AUA9 δx 1 ,δy 1 ,δy prime 1 A5AKD1A1BG C6 J[y] BFB4D1C2BEANCI x = x 1 BI (4.3.4) C8?B7 D4A2CRC3BLCQD1B7BVDBBACTCWD2A4A3 (1) BT δx 1 ,δy 1 ,δy prime 1 DAA5y prime 1 AOASACA7A1CI (4.3.4) C8DAA1C0 δx 1 ,δy 1 ,δy prime 1 C2BNBSB2CYA1D0?C2AHCNCI C7x = x 1 BIBXD9A5DHA1B7 ? ? ? ? ? ? ? ? ? ? ? [F ? y prime F y prime ? y primeprime F y primeprime + y primeprime F y primeprime d dx F y primeprime| x=x 1 =0, [F y prime ? d dx F y primeprime]| x=x 1 =0, F y primeprime| x=x 1 =0. (4.3.5) 46 (2) BT y 1 ,y prime 1 CEBNCXC4 x 1 , CIC7B(x 1 ,y 1 ,y prime 1 ) CI BEANy = ?(x)BXBOCDA1B9 y prime = ψ(x),CRC3 δy 1 = ? prime (x 1 )δx 1 ,δy prime 1 = ψ prime (x 1 )δx 1 . BRBS (4.3.4)C8A1ARA3BSBZ δx 1 C2BNBSB2A1C1 [F +(? prime ?y prime )(F y prime ? d dx F y primeprime)+(ψ prime ? y primeprime )F y primeprime] x=x 1 =0. (4.3.6) A4CI x = x 1 BIBLC1BVDEDABOA3 y 1 = ?(x 1 ),y prime 1 = ψ(x 1 )B6 (4.3.6)C8A2 (3) BT x 1 ,y 1 ,y prime 1 BOCIA9A7AHC8 Φ(x 1 ,y 1 ,y prime 1 )=0, CJC1 Φ x 1 δx 1 +Φ y 1 δy 1 +Φ y prime 1 δy prime 1 =0. BGCC Φ y prime 1 negationslash=0,CJ δy prime 1 = ? Φ x 1 Φ y prime 1 δx 1 ? Φ y 1 Φ y prime 1 δy 1 . BRBS (4.3.4)C8A1CHC0δx 1 ,δy 1 C2BNBSB2A1C1 ? ? ? ? ? ? ? [F ? y prime (F y prime ? d dx F y primeprime) ? (y primeprime + Φ x Φ y prime F y primeprime]| x=x 1 =0, [F y prime ? d dx F y primeprime ? Φ y Φ y prime F y primeprime]| x=x 1 =0. (4.3.7) A4CI x = x 1 BIBPC1BVDEDABOA1B7 Φ(x 1 ,y 1 ,y prime 1 )=0 B6 (4.3.7)C8A2 CB 4.3.1 C0CVAH J[y]= integraldisplay 1 0 (1 + y primeprime2 ) dx,y(0) = 0,y prime (0) = 1,y(1) = 1, CO y prime (1)BNBSA1CGBCJ[y]C2B4D1BEANA2 BZ C0C4F =1+y primeprime2 ,A4 Euler-PoissonCWBAA5 F y ? d dx F y prime + d 2 dx 2 F y primeprime =2y (4) =0, DBC3A5 y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 . C0 y(0) = 0 C1 c 1 =0;C0 y prime (0) = 1 C1 c 2 =1;C0 y(1) = 1 C1 c 3 + c 4 =0.BUA5x 1 =1,y 1 =1CDA6CC C2A1CYBQ δx 1 =0,δy 1 =0,C4CDBMA9DABO (4.3.4)AU A5 F y primeprime| x=1 δy prime 1 =0 47 B0 y primeprime | x=1 δy prime 1 =0. C0C4δy prime 1 CDBNBSC2A1A4y primeprime | x=1 =0,COy primeprime =2c 3 +6c 4 x, BX x =1C3C12c 3 +6c 4 =0.BLC8C7c 3 + c 4 =0D6D4 CQC3C1 c 3 = c 4 =0.BUBLA1B4D1D3ARCID0ANy = x BXBPBZA2 48 B2D4DJ CZBWBRA5B0ASBBD3CY 5.1 DHDGCTCZBW ? =0D5B0ASBBD3CY DABOB4D1C2AND1AAD7CDCDCICVAHCYBNCXC2AH CNBXD8BFBMAUCECMDABOCWBCCVAHC2B4D1AAD7A2 DIBKBDCCCVAH J[y,z]= integraldisplay x 1 x 0 F(x, y, z, y prime ,z prime ) dx, (5.1.1) A8AGALC4DABO ? ? ? ? ? y(x 0 )=y 0 , z(x 0 )=z 0 , ? ? ? ? ? y(x 1 )=y 1 , z(x 1 )=z 1 , (5.1.2) AJCECMDABO ?(x, y, z)=0 (5.1.3) C2B4D1AAD7A1DFBYBFCVAH J C2B4D1BEANCYBXA8AG C2DABOA2BMA9BAB0CQBQCYDFA2 CRCYAAD7C2B9AKBSBTCDA3CIBEAG ?(x, y, z)=0 BXBCBMDABEAN Γ: ? ? ? ? ? y = y(x), z = z(x), x 0 ≤ x ≤ x 1 , C6CVAH(5.1.1)CI ΓBXBFC1B4D1A2CRDBDABOB4D1C2 BCC3AAD7CYCUC4CNCAAHCNBCB4D1C2LagrangeB9CN CSA1CQBQA5AUA5ACDABOB4D1CWBID0A2 B4C95.1.1(Lagrange) C0y(x),z(x)CDCVAH(5.1.1) CIALC4DABO(5.1.2)AJCECMDABO (5.1.3)AIC2B4D1AH CNA2BR?CIBEAN Γ: ? ? ? ? ? y = y(x), z = z(x), BX? y ,? z D5BYC1BMDEAUA5DHA1CJAKBOCIAHCNλ(x)), C6 y(x),z(x)A8AGCVAH J ? [y,z]= integraldisplay x 1 x 0 (F + λ?) dx:= integraldisplay x 1 x 0 F ? dx (5.1.4) 49 C2 EulerCWBA F ? y ? d dx F ? y prime =0,F ? z ? d dx F ? z prime =0, (5.1.5) AZDAF ? = F + λ?. A1 AUCXC0CIBEAN Γ BX ? z negationslash=0,C0BWAHCNBOCI CCD0A1CQC0 (3)C8BJCCBMDEAHCN z = ψ(x, y). BQBLAHCNBRBS (5.1.1)C8A1C1 J = integraldisplay x 1 x 0 F[x, y, ψ(x, y),y prime ,ψ prime x + ψ y y prime ] dx := integraldisplay x 1 x 0 Φ(x, y, y prime ) dx. (5.1.6) AZDAΦ(x, y, y prime )=F[x, y, ψ(x, y),y prime ,ψ prime x +ψ y y prime ]. CRBJCD A7CVAH (5.1.1)C2DABOB4D1AAD7A5AUA5CVAH (5.1.6) C2ACDABOB4D1AAD7A2 CVAH (5.1.6)C2 EulerCWBAA5 Φ y ? d dx Φ y prime =0, (5.1.7) BACW Φ y , Φ y prime, d dx Φ y prime,C1 Φ y = F y + F z ψ y + F z prime[ψ xy + ψ yy y prime ], Φ y prime = F y prime + F z primeψ y , d dx Φ y prime = d dx F y prime + ψ y d dx F z prime + F z prime[ψ yz + ψ yy y prime ], BRBS (5.1.7)C8A1C1 Φ y ? d dx Φ y prime = F y + ψ y [F z ? d dx F z prime] ? d dx F y prime =0. A3BSBZ ? z negationslash=0,ψ y = ?z ?y = ? ? y ? z , C4CDC1 F y ? d dx F y prime ? ? y ? z [F z ? d dx F z prime]=0, B0 F y ? d dx F y prime ? y = F z ? d dx F z prime ? z := ?λ(x). 50 BUBLA1BOCI λ(x), C6C1B4D1AHCN y(x),z(x) A8AGCW BA F y + λ(x)? y ? d dx F y prime =0,F z + λ(x)? z ? d dx F z prime =0. DJ F ? = F + λ?,CJ F ? y = F y + λ? y ,F ? y prime = F y prime,F ? z = F z + λ? z ,F ? z prime = F z prime, C4CDBXC8ANA5 F ? y ? d dx F ? y prime =0,F ? z ? d dx F ? z prime =0. COBLCWBACDCDBQF ? A5AFB2AHCNC2CVAHC2EulerCW BAA2 CB5.1.1 CICSBJBEAG summationtext : z ? 1 2 x 2 =0BXBCD7BW DBC7O(0, 0, 0)AJ B(1, 1 2 , 1 2 )C2AKCJBEANA2 BZ AAD7ACC2A5CIBEAG summationtext : z ? 1 2 x 2 =0BXBCBM BEAN Γ: ? ? ? ? ? y = y(x), z = z(x), 0 ≤ x ≤ 1, C6AZA8AGALC4DABO ? ? ? ? ? y(0) = 0, z(0) = 0, ? ? ? ? ? y(1) = 1 2 , z(1) = 1 2 , B9C6CVAH J[y,z]= integraldisplay 1 0 radicalBig 1+y prime2 + z prime2 dx BFB4ATD1A2C0 LagrangeCCD0A1AND4A0CVAH J ? = integraldisplay 1 0 [ radicalBig 1+y prime2 + z prime2 +λ(x)(z? 1 2 x 2 )] dx= integraldisplay 1 0 F ? dx, AZDA F ? = radicalBig 1+y prime2 + z prime2 + λ(x)(z ? 1 2 x 2 ), C0 EulerCWBAC8 (5.1.5)B6CECMDABOA1AB?C1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? F ? y ? d dx F ? y prime = ? d dx ( y prime radicalBig 1+y prime2 + z prime2 )=0, (I) F ? z ? d dx F ? z prime = λ(x) ? d dx ( z prime radicalBig 1+y prime2 + z prime2 )=0, (II) z ? 1 2 x 2 =0. (III) 51 C0 (I)C8C1 y prime radicalBig 1+y prime2 + z prime2 = c 1 , BQ (III)C8BRBSBLC8C1 y prime radicalBig 1+y prime2 + x 2 = c 1 , B0 y prime = radicaltp radicalvertex radicalvertex radicalbt c 2 1 1 ?c 2 1 √ 1+x 2 := c √ 1+x 2 . B2D1APCQC1 y = c 2 [x √ 1+x 2 +ln(x + √ 1+x 2 )] + c 2 . BUA5CYBCBEANAFC7 O(0, 0, 0)AJ B(1, 1 2 , 1 2 ), C0BLCQ BQBJCC c 2 =0,c= 1 √ 2+ln(1+ √ 2) . (IV) C4CDCYBCBEANA5 ? ? ? ? ? ? ? y = x √ 1+x 2 +ln(x + √ 1+x 2 ) 2[ √ 2+ln(1+ √ 2)] , z = 1 2 x 2 , 0 ≤ x ≤ 1. D7BWC7 O C7 B C2ARCKC2B4CGA5 J = integraldisplay 1 0 √ 1+y prime2 + z prime2 dx = integraldisplay 1 0 radicalBig 1+c 2 (1 + x 2 )+x 2 dx = √ 1+c 2 integraldisplay 1 0 √ 1+x 2 dx = √ 1+c 2 1 2 [x √ 1+x 2 +ln(x + √ 1+x 2 )]| 1 0 = √ 1+c 2 2c , AZDAcA5 (IV)C8A2 CB 5.1.2 CGBCCCDJAG x 2 + y 2 = R 2 BXD7BWC7 P 1 (x 1 ,y 1 ,z 1 )C7C7P 2 (x 2 ,y 2 ,z 2 )C2AKCJBEANA2 BZ C0CCDJAGC2AZCNCWBAA5 ? ? ? ? ? x = R cos t, y = R sin t, 0 ≤ t ≤ 2π. BUA5C7 P 1 ,P 2 CICCDJAGBXA1A4CYBCBEANC2 x, y AO AOC7CCDJAGC2AODCA1D3BKBCBF z AOAO z = z(t) B7 52 CQA2C0 P 1 ,P 2 C7CLBXC2AZCNA5 t 1 <t 2 , CJ P 1 P 2 C2 ARB4A5 l = integraldisplay t 2 t 1 radicalBig x prime2 (t)+y prime2 (t)+z prime2 (t) dt. C4CDAAD7AUA5A3BCAF P 1 ,P 2 C7B9A8C4CCDJAGBXC2 BEANA1C6CVAH lBFB4ATD1A2 AND4A0CVAH l ? = integraldisplay t 2 t 1 [ radicalBig x prime2 (t)+y prime2 (t)+z prime2 (t)+λ(t)(x 2 +y 2 ?R 2 )] dt. BQ x = R cos t, y = R sin tBRBSBXC8A1C1 l ? = integraldisplay t 2 t 1 radicalBig R 2 + z prime2 (t) dt, AZDA F ? = √ R 2 + z prime2 ,F ? z =0,F ? z prime = z prime √ R 2 + z prime2 =0,l ? C2 EulerCWBAA5 F ? z ? d dt F ? z prime = ? d dt z prime √ R 2 + z prime2 =0. B2D1A1C1 z prime √ R 2 + z prime2 = c, CJ z prime = cR √ 1 ? c 2 := c 1 . CHB2D1A1C1 z = c 1 t + c 2 . A4CYBCAKCJBEANCDCCDJA5AN ? ? ? ? ? ? ? ? ? ? ? ? ? x = R cos t, y = R sin t, z = c 1 t + c 2 , AZDAc 1 ,c 2 C0 P 1 (x 1 ,y 1 ,z 1 ),P 2 (x 2 ,y 2 ,z 2 )BJCCA2 A9A3 BTBQ LagrangeCCD0DAC2CECMDABOAWA5A3 ?(x, y, z, y prime ,z prime )=0,CO ? y prime B0 ? z prime DAD5BYC1BMDEAUA5 DHA1CJCCD0C2C2A4BPB7D4A2 LagrangeCCD0C2BMA9B0C8A5A3 C0AHCN y 1 (x),y 2 (x),···,y n (x)CDCVAH J[y 1 ,y 2 ,···,y n ]= integraldisplay x 1 x 0 F(x, y 1 ,y 2 ,···,y n ,y prime 1 ,y prime 2 ,···,y prime n ) dx 53 CIALC4DABO y i (x 0 )=y i0 ,y i (x 1 )=y i1 ,i=1, 2,···,n, AJCECMDABO ? j (x, y 1 ,y 2 ,···,y n )=0,j=1, 2,···,m; m<n, AIC2B4D1AHCNA2BGCC? j (j =1, 2,···,m)AOASCFD4A1 B7D5BYC1BMDE mBYAHCNB1DEC8AUA5DHA1AIBR D(? 1 ,? 2 ,···,? m ) D(y 1 ,y 2 ,···,y m ) = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?? 1 ?y 1 ?? 1 ?y 2 ··· ?? 1 ?y m ?? 2 ?y 2 ?? 1 ?y 2 ··· ?? 2 ?y m ··· ··· ··· ··· ?? m ?y 1 ?? m ?y 2 ··· ?? m ?y m vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle negationslash=0, CJBOCI m DEAHCN λ 1 (x),λ 2 (x),···,λ m (x), C6C1B4D1 AHCNy 1 (x),y 2 (x),···,y m (x)A8AGCVAH J ? = integraldisplay x 1 x 0 [F + m summationdisplay j=1 λ j ? j ] dx= integraldisplay x 1 x 0 F ? dx C2 EulerCWBA F ? y i ? d dx F ? y prime i =0,i=1, 2,···,n. BXCLC2A4APAIA1C6CVAH J BPBZB4D1C2AHCNBQ DCC3C6CVAH J ? BPBZACDABOB4D1A2BR?BQBXCLC2 CECMDABODAA5 ψ j (x, y 1 ,···,y n ,y prime 1 ,···,y prime n )=0,j=1,···,m; m<n, COBGCC D(ψ 1 ,ψ 2 ,···,ψ m ) D(y prime 1 ,y prime 2 ,···,y prime m ) negationslash=0, CJBXCLC2A4DCBJB7D4A2 5.2 B1A8D3CY BMA9C2C3DDAAD7CDD2CICVAHCECM (C3DDDABO) K[y]= integraldisplay x 1 x 0 G(x, y, y prime ) dx= l (5.2.1) 54 B6ALC4DABO y(x 0 )=y 0 ,y(x 1 )=y 1 (5.2.2) AIC2CYC1CQBYD7B7BYCNC2AHCNy(x)DAA1B7CIB5AL Y = {y(x): integraldisplay x 1 x 0 G(x, y, y prime ) dx= l,y(x i )=y i ,i=0, 1,y∈ C 2 } DABCCVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ) dx (5.2.3) C2B4D1AHCNA1AZDA G, l, y i C3CLCDDGCCC2AHCNB0B3 CNA2 A7C4CRDBB0C8C2C3DDAAD7B4D1BOCIC2AKBKDA BOA1AB?C1BRAIC2 Euler CCD0A1D0A7DABOB4D1C2 AND1AAD7A5AUA5ACDABOB4D1C2AND1AAD7A2 B4C95.2.1(Euler) BTBEAN Γ:y = y(x) CIC3DD DABO(5.2.1)B6ALC4DABO (5.2.2)AIC6CVAH (5.2.3)BP BZB4D1A1CJBOCIB3CN λ,C6 y(x)A8AGCVAH J ? = integraldisplay x 1 x 0 (F + λG) dx:= integraldisplay x 1 x 0 F ? dx (5.2.4) C2 EulerCWBA F ? y ? d dx F ? y prime =0, (5.2.5) AZDAF ? = F + λG. A1 BUA5B5AGC0A2D1CWBACECMDABOC2B4D1AAD7 BPCAC1BZC3CKA2ALCIACBKBQCVAHCECMANB7A2D1CW BACECMC2BACTCWBID0A2A5BLA1DJ u(x)= integraldisplay x 1 x 0 G(x, y, y prime ) dx, D0A8AG u(x 0 )=0,u(x 1 )=l, u prime (x)=G[x, y(x),y prime (x)]. BR?BQJ[y]CNB7DBDEAHCN y(x),u(x)C2CVAH J[y,u]= integraldisplay x 1 x 0 F(x, y, y prime ) dx, (5.2.6) ARDJ Φ(x, y, y prime ,u,u prime )=G(x, y, y prime ) ?u prime , 55 CJC3DDAAD7C2B4D1BEAN y = y(x) AKCDCVAH (5.2.6) CICECMDABO Φ(x, y, y prime ,u,u prime )=0 AJALC4DABO y(x 0 )=y 0 ,y(x 1 )=y 1 ,u(x 0 )=0,u(x 1 )=l AIC2B4D1AHCNCZBMA2 C0B5BMC0CYA1BT y(x),u(x) A5BLDABOAIC2B4D1 AHCNA1CJAKBOCI λ(x),C6 y(x),u(x)A8AGCVAH J ?? = integraldisplay x 1 x 0 [F(x, y, y prime )+λ(x)Φ(x, y, y prime ,u,u prime )] dx = integraldisplay x 1 x 0 [F(x, y, y prime )+λ(x)(G(x, y, y prime ) ?u prime )] dx := integraldisplay x 1 x 0 F ?? (x, y, y prime ,u,u prime ) dx C2 EulerCWBA ? ? ? ? ? F ?? u ? d dx F ?? u prime =0, (5.2.7) F ?? y ? d dx F ?? y prime =0, (5.2.8) AZDA F ?? = F9x, y, y prime )+λ(x)[G(x, y, y prime ) ? u prime ]. BUA5 F ?? u =0,F ?? u prime = ?λ(x),CRC3 (5.2.7)C8B7A5 d dx λ(x)=0, CYBQA1 λA5B3CNA2C0C4 F ?? y = F y + λG y ,F ?? y prime = F y prime + λG y prime, C0 (5.2.8)C8C1 F y ++λG y ? d dx (F y prime + λG y prime)=0. COBLCWBACDCD F ? y ? d dx F ? y prime =0. C0 EulerCCD0A1AB?C1BZBCCVAH (5.2.3)CIC3DD DABO(5.2.1)B6ALC4DABO(5.2.2)AIC2B4D1AHCNC2C3 D7CWCSA2 AND4A0CVAH J ? [y]= integraldisplay x 1 x 0 [F + λG] dx:= integraldisplay x 1 x 0 F ? dx, 56 AZDAλA5BSCCB3CNA2BKAPC0 J ? C2 EulerCWBA F ? y ? d dx F ? y prime =0 C7C3DDDABOB6ALC4DABOD6D4BCC3A2 CB5.2.2 CIXOYAWAGBXBCDBAFDBCCC7A(x 0 ,y 0 ), B(x 1 ,y 1 ) B9B4CGA5CCD1 l C2BEAN y = y(x), C6D0CY A3B7C2BEALD5B0C2AGB2AKATA2 BZ AAD7CDCIC3DDDABOB6ALC4DABO integraldisplay x 1 x 0 radicalBig 1+y prime2 dx= l, y(x 0 )=y 0 ,y(x 1 )=y 1 AIBCCVAH J = integraldisplay x 1 x 0 ydx C2B4ATD1A2 AND4A0CVAH J ? = integraldisplay x 1 x 0 [y + λ radicalBig 1+y prime2 ] dx, CJC1 F ? = y + λ radicalBig 1+y prime2 ,F ? y =1, F ? y prime = λy prime √ 1+y prime2 , A4 J ? C2 EulerCWBAA5 1 ? d dx λy prime √ 1+y prime2 =0, B7 x = λy prime √ 1+y prime2 + c 1 . DJ y prime =tant,CJ x = λ tant sec t + c 1 B0 x = λ sint + c 1 . C0 dy= y prime dx,C1 dy=tant ·λ cos tdt= λ sin tdt. C0BLC1 y = ?λ cos t + c 2 . 57 C4CDC1 ? ? ? ? ? x = λ sin t + c 1 , y = ?λ cos t + c 2 , AUA5D0BTAOAOCWBA (x ? c 1 ) 2 +(y ? c 2 ) 2 = λ 2 . CRCRAIA1B4D1BEANCDBQ (c 1 ,c 2 )A5CCAYA1λA5AA CBC2CCDAC2BMCKCCARA2C0ALC4DABOC7CECMDABOCQ BQBJCCc 1 ,c 2 AJ λ. A3BSA1BTBJCCC2CCARAUA4BMA1 CJBMDEC6J BFB4BQD1A1CODIBMDEC6 J BFB4ATD1A2 AKAPA1AB?DGBFEulerCCD0C2BMA9BAB0A2 C0AHCN y 1 (x), y 2 (x), ···, y n (x)CIC3DDDABO integraldisplay x 1 x 0 G i (x, y 1 ,···,y n ,y prime 1 ,···,y prime n ) dx= l, i =1,···,m AJALC4DABO y j (x 0 )=y j0 ,y j (x 1 )=y j1 ,j=1, 2,···,n AIC6CVAH J[y 1 ,y 2 ,···,y n ]= integraldisplay x 1 x 0 F(x, y 1 ,···,y n ,y prime 1 ,···,y prime n ) dx BPBZB4D1A1CJAKBOCIB3CN λ 1 , λ 2 , ···, λ m , C6 y 1 (x), y 2 (x), ···, y n (x)A8AGCVAH J ? = integraldisplay x 1 x 0 [F + m summationdisplay i=1 λ i G i ] dx:= integraldisplay x 1 x 0 F ? dx C2 EulerCWBA F ? y j ? d dx F ? y prime j =0,j=1, 2,···,n. 58 B2CDDJ B9BKBRD7D3CYDED6D9D1BBBAAU 6.1 PoissonBAAUARA5D3CYDEASBBD3CY C0 m A5D0D7CUCNA1AB?B3BZBZAIAGBMAUC0D7 B7AHCNAJB7C2AHCNCSBJA2 C m (?) = {u| D α uCI?AQD7B7,?|α|≤m}. C ∞ (?) = ∞ intersectiondisplay m=0 C m (?) = {u| D α uCI ?AQD7B7,?α}. C m 0 (?) = {u ∈ C m (?)| uCI ?DAC1C5CXB5}. C ∞ 0 (?) = {u ∈ C ∞ (?)| uCI ?DAC1C5CXB5}. C m ( ˉ ?) = {u| D α uCI ?BXD7B7, ?|α|≤m}. AGC0CWAIBMA9C2PoissonCWBADirichletAAD7 ?triangleu = f(x),CI?DA, (6.1.1) u = g,CI ?BX, (6.1.2) C3BHC4BMDEAND1AAD7A2A5BLAB?AJCWAIAIAGC2BV D0A2 ASBBB8BPAQDCC96.1.1 BR?AHCN u ∈ C 0 (?) A8 AG integraldisplay ? u(x)?(x) dx=0, ?? ∈ C ∞ 0 (?), (6.1.3) CJCI?DAu ≡ 0. AYBZBBAI B 2 = {v ∈ C 2 (?) intersectiondisplay C 1 ( ˉ ?)| v = gCI ??BX, B 2 0 = {v ∈ C 2 (?) intersectiondisplay C 1 ( ˉ ?)| v =0CI ??BX, I(v)= integraldisplay ? ( 1 2 |Dv| 2 ? fv) dx. (6.1.4) AIAGC2CCD0CRAIA1BCALD1AAD7 (6.1.1), (6.1.3) CI B 2 0 DAC2C3C3BHC4CI B 2 0 DABCCVAH (6.1.4) C2B4D1 AHCNA2 B4C96.1.2C0?CHC1C 1 ALC4??,u∈ B 2 . AOAAA1 uCD PoissonCWBA(6.1.1)C2C3C2BBBKDABOCDA3uCD AND1AAD7 I(u)=min v∈B 2 I(v), (6.1.5) 59 C2C3A2 A1CGA3AJCWAIDABOCDAKBKC2A2C0u ∈ B 2 CDPois- sonCWBA(6.1.1)C2C3A2CIAUCVA0C8 (EulerCCD0) integraldisplay ? (D 1 w 1 +···+D n w n dx= integraldisplay ?? [w 1 cos(ν,x 1 )+···+w n cos(ν,x n )] ds, (6.1.6) (AZDAν A5DICSARBTA8ARDC) DABF (w 1 ,···,w n )=(vD 1 u,···,vD n u)=vDu, v ∈ B 2 . C1BFDDDF (Green)A0C8 integraldisplay ? vtriangleudx+ integraldisplay ? Dv · Dudx = integraldisplay ?? v ?u ?ν ds. (6.1.7) C0C4??u = f,C0BXC8C1BF integraldisplay ? (Du· Dv ?fv) dx=0, ?v ∈ B 2 0 . (6.1.8) CLBNBS v ∈ B 2 , DJ v ? u = ?, CJ v = u + ?, ? ∈ B 2 0 . C0(6.1.4)B6(6.1.8)C1 I(v)=I(u + ?) = 1 2 integraldisplay ? |Du + D?| 2 dx? integraldisplay ? f(u + ?) dx = I(u)+ integraldisplay ? (Du· D?? f?) dx+ 1 2 integraldisplay ? |D?| 2 dx = I(u)+ 1 2 integraldisplay ? |D?| 2 dx≥ I(u). BLC8APAI uCDAND1AAD7 (6.1.5)C2C3A2 CHCWAIDABOCDBBD1C2A2C0 u ∈ B 2 CDAND1AAD7 (6.1.5)C2C3A2BNBFAHCN ? ∈ C ∞ 0 (?),C4CDCLBNBSAZ CNt ∈ RC1 u + t? ∈ B 2 . C0 (6.1.5)CYC0 I(u)=min t∈R I(u + t?). BUBLC1 d dt I(u + t?)| t=0 =0. (6.1.10) C0 I C2CCBT (6.1.4)C1 I(u + t?)= integraldisplay ? [ 1 2 |Du + tD?| 2 dx? f(u + t?)] dx = I(u)+t integraldisplay ? (Du· D??f?) dx+ t 2 2 integraldisplay ? |D?| 2 dx. BQBLC8BRBS (6.1.10)C1BF integraldisplay ? (D?· Du? f?) dx=0. (6.1.11) 60 CIDDDFA0C8 (6.1.7)DAAJv = ? ∈ C ∞ 0 (?)C1BF integraldisplay ? ? ·triangleudx+ integraldisplay ? D?· Dudx =0. (6.1.12) C0(6.1.12)BMBG (6.1.11)C1 integraldisplay ? ?(triangleu + f) dx=0. C0C4? ∈ C ∞ 0 (?)CDBNBSC2A1CLBXC8D2BZAND1CSB1 AGBVD0CYC0 triangleu + f ≡ 0, B7 u A8AG Poisson CWBA (6.1.1). CWA0A2 6.2 A0CWACDEB7AXB9BKBRD7D3CY AIAGCLBGC0 H CDC5HilbertCSBJA1D A CD H C2 BMDEBDAFANB2ABCSBJA1 A CD D A BZ H DAC2ANB2 CWAB (AUAKCDC1C4C2). B4DB6.2.1 BR?CWAB AA8AG 〈Au, v〉 = 〈u,Av〉, ?u,v ∈ D A , CJB6AA5 D A BXC2CLB6CWABA2 B4DB6.2.2 C0 ACD D A BXC2CLB6CWABA2BR?CW ABAA8AG 〈Au, u〉 > 0,?0 negationslash= u ∈ D A , CJB6AA5 D A BXC2CVCWABA2BR?BOCICVB3CN cC6 C1 〈Au, u〉≥cbardblubardbl 2 , ?u ∈ D A , CJB6AA5 D A BXC2CVCCCWABA2 CB6.2.3 C0C1C4BDC8?C1C 1 ALC4??,H = L 2 (?). C0C4 D A = {u ∈ C 2 ( ˉ ?)| u =0CI ??BX} CI C ∞ 0 (?)DABDAFA1CO C ∞ 0 (?) ? D A ? L 2 (?), BUBLD A CI H DABDAFA2 A = ??CD D A BXC2CVCC CWABA2 61 A1CGA3A = ?? AKBKCDD A BXC2ANB2CWABA2BX u,v ∈ D A C3A1C0 GreenA0C8C1 integraldisplay ? v?udx+ integraldisplay ? Du·Dv dx = integraldisplay ?? v ?u ?n ds, (6.2.1) integraldisplay ? u?vdx+ integraldisplay ? Du· Dv dx = integraldisplay ?? u ?v ?n ds, AZDAnA5DICSARDCC2BTA8ARDCA2BQBXDEDBC8AOBM C1C6CQGreenA0C8 integraldisplay ? (v?u ? u?v) dx= integraldisplay ?? (v ?u ?n ? u ?v ?n ) ds. C0C4CI??BXC1u = v =0,BXC8CQBQAWB7 integraldisplay ? v?udx= integraldisplay ? u?vdx B0 〈??u,v〉 = 〈u,??v〉. CRCRAIA = ??CDD A BXC2CLB6CWABA2CHC0(6.2.1), FriedrichsAUC3C8C1BFA3CLBNBS u ∈ D A C1 〈Au, u〉 = 〈??u,u〉 = ? integraldisplay ? u?udx = integraldisplay ? |Du| 2 dx≥ c 2 integraldisplay ? u 2 dx= c 2 bardblubardbl 2 2 , AZDA c>0. BXC8CRAIA = ?? CD D A BXC2CVCCCW ABA2 B4C96.2.4 BR?ACD D A BXC2CVCWABA1 f ∈ H, CJCWBA Au = f (6.2.2) CI D A DAAKCND3ARC1BMDEC3A2 A1CGA3C0CWBA(6.2.2)C1DBDEC3 u,v ∈ D A ,C0BL CY A(u ? v)=Au ? Av =0, BNCOC1 〈A(u ? v),u?v〉 =0. C0C4ACDCVCWABA1BUBL u ? v =0,B7 u = v. B4C96.2.5 C0ACDD A BXC2CVCWABA1u ∈ D A ,f∈ H. uCDCWBA Au = f 62 C2C3BXB9C6BXA3 uCDAND1AAD7 I(u)=min v∈D A I(v)(6.2.3) C2C3A1AZDA I(v)= 1 2 〈Av, v〉?〈f,v〉 CDBMDECQBMCVAHA2 A1CGA3AJCWAIDABOCDAKBKC2A2C0uCDCWBAAu = f C2C3A1CJCLC4BNBSC2 v ∈ D A ,DJ ?v ? uC1BF I(v)=I(u + ?)= 1 2 〈A(u + ?),u+ ?〉?〈f,u + ?〉 = I(u)+〈Au ? f,?〉+ 1 2 〈A?, ?〉≥I(u). CHCWDABOCDBBD1C2A2C0uCDAND1AAD7(6.2.3)C2 C3A1CJCLC4BNBSC2 ? ∈ D A C1 I(u)=min t∈R I(u + t?). C0BLCY d dt I(u + t?) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle t=0 =0. (6.2.4) CHC0I(v)C2CCBTC1BZ I(u + t?)= 1 2 〈A(u + t?),u+ t?〉?〈f,u + t?〉 = I(u)+t〈Au ? f,?〉 + t 2 2 〈A?, ?〉. BQBLC8BRBS (6.2.4)C1BF 〈Au ?f,?〉 =0. BLC8CLC4BNBSC2 ? ∈ D A B7D4A1BUBL Au = f,BLB7 CRAIuCDCWBAAu = f C2C3A2 6.3 ADCOARC0CZBW CLC4BMA9C2C6BVALD1AAD7 (RobinAAD7) ??u = f CI ?DA (6.3.1) ?u ?n + α(x)u = ψ(x), CI ??BX (6.3.2) 63 AB?C1AIAGC2CCD0A2 B4C96.3.1 C0 ?CHC1C 1 ALC4 ??, f ∈ C( ˉ ?),α∈ C(??), ψ ∈ C(??),u∈ D ? = C 2 (?) ∩ C 1 ( ˉ ?),BB I(v)= integraldisplay ? ( 1 2 |Dv| 2 ? fv) dx+ integraldisplay ?? ( 1 2 α(x)v 2 ?ψv) ds. BR?uCDAND1AAD7 I(u)= min v∈D ? I(v)(6.3.3) C2C3A1CJ u CD Poisson CWBAC6BVALD1AAD7 (6.3.1), (6.3.2) C2C3A2CTCZA1BR?CI ?? BXC1 α(x) ≥ 0, B9 u CD Poisson CWBAC6BVALD1AAD7 (6.3.1), (6.3.2) C2 C3A1CJ uCDAND1AAD7 (6.3.3)C2C3A2 64 B2CLDJ ASBBD3CYA7B0A4BXB8 7.1 CAAB (Ritz)B8 D1AACSC2B1AGCSAPCDBZBACCC2AHCNB6DEC2AN B2AJALAHC8AND1AAD7C2B4D1BEANA2 COB1BRAICYAZC2AND1AAD7A2C0CVAH J[y]= integraldisplay x 1 x 0 F(x, y, y prime ) dx (7.1.1) C2ALC4DABOA5 y(x 0 )=0,y(x 1 )=0. (7.1.2) CRDBALC4DABOB6A5B0BMALC4DABOA2BR?CYD2A4C2 CVAHC2ALC4AUCDB0BMC2A1BR y(x 0 )=y 0 ,y(x 1 )=y 1 , CJCQBQDJ y(x)=z(x)+ x ?x 1 x 0 ? x 1 y 0 + x ? x 0 x 1 ?x 0 y 1 , (7.1.3) CRC3C1z(x 0 )=z(x 1 )=0.BZ(7.1.3)C8CLCVAH(7.1.1) ANANAWA1B7CQBQAAD7 (7.1.1), (7.1.2)C8AUA5B0BMAL C4C2AND1AAD7 J[y]:=J 1 [z]= integraldisplay x 1 x 0 F 1 (x, z, z prime ) dx, z(x 0 )=z(x 1 )=0. C0CVAH J[y] C2B4D1BEANCIAHCNB5 Y AQA1ARB9 Y A1B7ANB2CSBJA2BZD1AACSBCCVAH J[y]C2C8CUC3 C2AVDFBRAIA3 (1) BABFY C2B1AHCN ? 1 (x),? 2 (x), ···,? n (x),··· (7.1.4) CLBNBMy ∈ Y , yCECQBQAPC9A5{? i }C2C1AMB0ACAM ANB2AJALA2D4AQA1CLB4D1AHCN f(x),BLC1 f(x)=c 1 ? 1 (x)+c 2 ? 2 (x)+···+ c n ? n (x)+···. (2) CLACDEn,COA1C0? 1 (x), ? 2 (x), ···, ? n (x)C2 B7C2ANB2ABCSBJY n ,C0 y n = n summationdisplay i=1 a i ? i ∈ Y n , 65 C0CVAH J[y]CDBJCCDD nCAAHCN J[a 1 ,···,a n ]:=J[y n ]= integraldisplay x 1 x 0 F[x, n summationdisplay i=1 a i ? i (x), n summationdisplay i=1 a i ? prime i (x)] dx. (3) CLACDE n, BABF a (n) 1 , a (n) 2 , ···, a (n) n , C6 J[y n ] BFB4D1A1BLCDCDC0CWBA ? ?a i J[y n ]=0,i=1, 2,···,n CWBJCC a (n) 1 ,a (n) 2 ,···,a (n) n ,BKAPBZC1BZC2AHCN f n = n summationdisplay i=1 a (n) i ? i ANA5AND1AAD7C2C8CUC3A2 BXCLC2 f n CDB6DE (7.1.4)C8B5nDEAHCNCYC1CQ ARC2ANB2AJALDAC6CVAH J[y n ] BPBZB4D1C2AHCNA2 CRBJC1BZC2B6DEf 1 , f 2 , ···, f n , ···B6A5J[y]C2B4AT AUB6DEA2BUA5 Y 1 ? Y 2 ?···?Y n ?···, CYBQ J[f 1 ] ≥ J[f 2 ] ≥···≥J[f n ] ≥···, ARB9 J[f n ] ≥ J[f],n=1, 2,···, CIBMA9BACTAIA1C1 lim n→∞ J[f n ]=J[f]. B7C6CRBJA1BLAUARADCW lim n→∞ J[f n ]BOCIA1B7C6 lim n→∞ J[f n ] BOCIA1D0BLAUBMCCCHD8C4 J[y] C2B4D1AHCN f. BU CLB3BZAHCNCWCRA1 f n AUC6DHC7CHD8COB9BMD6CH D8C4f. BUBLA1BR?AUBGA0B7B4AMCGCWA1COD3AMC4 B5AGC2 nAQ f n ,CJ f n CDCDAND1AAD7C2C8CUC3A2 CLC4AAD7(7.1.1), (7.1.2)C8A1B1AHCNDBB3BABFAI DEC2BVDEAHCNAHA3 ? n (x)=(x ? x 0 ) n (x 1 ?x),n=1, 2,···; ? n (x)=(x 1 ?x) n (x ? x 0 ),n=1, 2,···; 66 ? n (x)=sin nπ(x ?x 0 ) x 1 ?x 0 ,n=1, 2,···. CB 7.1.1 BZ RitzCSBCAND1AAD7 J[y]= integraldisplay 1 0 (y prime2 ? y 2 ? 2xy) dx, y(0) = y(1) = 0 C2C8CUC3A2 BZ BF ? n (x)=x n (1? x),n=1, 2,···, AKBK ? n (0) = ? n (1) = 0. (1) BCBMBYC8CUC3A2 AN y 1 = c 1 ? 1 (x)=c 1 x(1 ? x), CJ y prime = c 1 (1 ? 2x). BRBSCVAHA1C1 J[y 1 ]=J[c 1 ] = integraldisplay 1 0 [c 2 1 (1 ? 2x) 2 ?c 2 1 x 2 (1? x) 2 ? 2xc 1 x(1 ? x)] dx = 3 10 c 2 1 ? 1 6 c 1 . BABFc 1 ,C6 J[c 1 ]BFB4D1A2A5BLA1DJ dJ dc 1 = 6 10 c 1 ? 1 6 =0, C1 c 1 = 5 18 ,A4BMB8C8CUC3A5 f 1 = 5 18 x(1 ? x). (2) BCCQB8C8CUC3 AN y 2 = c 1 ? 1 (x)+c 2 ? 2 (x) = c 1 x(1 ?x)+c 2 x 2 (1 ? x) = x(1 ? x)(c 1 + c 2 x). CJ y prime 2 =(1? 2x)(c 1 + c 2 x)+c 2 x(1 ?x). BRBSCVAHA1C1 J[y 2 ]=J[c 1 ,c 2 ] = integraldisplay 1 0 {[(1 ? 2x)(c 1 + c 2 x)+c 2 x(1 ?x)] 2 ?x 2 (1 ?x) 2 (c 1 + c 2 x) 2 ? 2x 2 (1 ? x)(c 1 + c 2 x)}dx. 67 BQ J[y 2 ]D1AQCLc 1 AJ c 2 BCAVBYCNA1BKAPCHDJ ?J ?c 1 =0, ?J ?c 2 =0, C1CWBA ? ? ? ? ? 3 10 c 1 + 3 20 c 2 = 1 12 , 3 20 c 1 + 13 105 c 2 = 1 20 . C3BLCWBAC1 c 1 = 71 369 ,c 2 = 7 41 , A4CQB8C8CUC3A5 f 2 = 71 369 x(1 ? x)+ 7 41 x 2 (1 ?x) = 1 41 x(1 ?x)( 71 9 +7x). (3) C8CUC3C7C9BJC3C2AIBUA2 C3 EulerCWBAA1CQC1C9BJC3A5 y = sin x sin 1 ? x. ALBQC9BJC3A1BMB8C8CUC3A1CQB8C8CUC3AIBUBR AIA3 x C9BJC3 BMB8C8CUC3 CQB8C8CUC3 y = sin x sin 1 ?x y 1 = 5 18 x(1 ? x) y 2 = 1 41 x(1 ? x)( 71 9 +7x) 0 0 0 0 0.2 0.0361 0.0444 0.0362 0.4 0.0627 0.0667 0.0626 0.6 0.0710 0.0667 0.0709 0.8 0.0525 0.0444 0.0526 1 0 0 0 68