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HIKPLAPL.RNIKJ `L}RA[DJ7bKJMT{UDO jG`T `DjKR{IfJ~_K¤ÁSfLN`Db¯JMSKUZR{P`b?xA.JsR;opLqLWe9JMQ7YUZOMJhR{`¬R{IKJQ;UDLWJ /IKJMTNJ L©UQM`bKLWR{UDb R;CU|JMTW\áQM` " `b§UZbK_áSKLNJsjGSKOVQ;UZLNJDx HIKJd_KPÁ9SKLNP`b¦JMSKUZR{P`b D bK`´°R{UDnDJML¬RNIKJ^jG`T Nt (x, t) = D 4 N + f, /IKJMTNJ L"R{IKJ¦CUZeKOYUDQMYUDb`elJ7T{U=R{`TMxãóÄ` JMR{ JML`DRNIKJ7TQ7``TN_KPb4UZRNJLW\L}R{J L+UDTWJ eKTWJMjGJ7TNUD4 ]KOPJ«R{`dQ;UZTWR{JMLNUDb?xV£b+QM\OPbK_KTWQ;UZOUDbf_LNeKIKJMTNPQ;UDO?QM`Ä`DTN_KPb4UZR{JML7fjG`DTmJsrfU eKOJZÄ~J I4U;|J©R{IfJ©jG`DT SKOUDL ∂N 1 ∂2N 1 ∂ r + 4N = r ∂r ∂r r 2 ∂θ 2 ∂2N ∂ 1 ∂ 1 ∂N 1 2 ∂N 4N = 2 r + 2 sin φ + 2 2 r ∂r ∂r r sin φ ∂φ ∂φ r sin φ ∂θ 2 £bRNIKJ«Q7UDLNJi`Dj?LWeKIKJMTNQ7UDOlQ7``TN_KPb4UZRNJ7LV/PRNIjGSKbKQsR{P`bKL¾R{I4UZRXUDTWJiPbK_KJ7elJ7bf_KJ7b Rm`Zj.UDbK[OPJD R{IfJ¯.UDefOYUDQMYUDbLW eKO¤¥4J7LR{` ∂2N 2 ∂N 4N = + 2 ∂r r ∂r UDbK_bQs\Obf_KTNPQ;UDO?QM`Ä`DTN_KPb4UZR{JMLXR{` 4N = ∂2N 1 ∂N . + 2 ∂r r ∂r 01 ?1?EqßGڧدÜ4ÙMÚÄÙ%5M6CÙM8´Ü4698;Ü46CÚ<l@CÙ<9@.:=BÚÙ ñé{è J^L}RUDT}R//PR{IRNIKLeKSfTNJ© ∂N ∂2N = D 2 , −∞ < x < ∞, t > 0 ∂t ∂x N (x, 0) = f (x), −∞ < x < ∞. 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N (x, t) = f (x) ? √ 4πDt 4πDt −∞ ³=ë·ç4µ"!?³=ñÄé{æè± cX`DbKLNP_KJ7T%U+e9`Pb R^LN`SfTNQ7Jd/PRNI§b R{JMbKLN¤R£\ .R{I4UZR%PL7.UDb*bK¤R{UDO M _KPLWR{TW]KSÄR{`Db f (x) = M δ(x) /IKPQI [¤|JML¬LW`OSÄR{`Db N (x, t) = √ HCT{UDbKLWe9`DTWR¬HAJMT L £b R{TW`Ä_fSKQ7J¯UR{T{UZbKLNel`TWR/RNJ7T M x2 e− 4Dt . 4πDt b R{`RNIKJ©eKTN`]fOJ KLW`dRNI4UZR/R{IfJ SÄrR{JMT ö PLbK`´ Φ = −D 5 N + v0 , /IKJMTNJ L/UQ7`DbKLWR{UDb RiLWe9JMJ7_ UZRm/IKPQIR{IKJ UZR{JMTNUDO9PLm]lJ7bf[R{T{UZbKLNel`TWRNJ7_+LNeKUZR{UDOO¤\x 3£báIf[vIK0JMT¨_K JMbKLNP`bKLiRNIKL«LNelJ7J7_á~`SKOP_*]lJaU|DJ7O`Q7¤R£\ |DJ7QsR{`TMxv§HIKJaQ7`bfLNJ7T}|ZUZR{P`b LW\L}R{J bf`F ]9JMQ7` J7L ∂N ∂2N = D 2 − v0 N, −∞ < x < ∞, t > 0 ∂t ∂x N (x, 0) = f (x), −∞ < x < ∞. z°QIKUDbK[J¨`Zj|DUZTNYUZ]KOJML OPJ;UD_KLR{` RNIKJ¨JMSKUZR{P`b UDbK_IfJ7bKQMJ^]\ z = x − v t N = DN 0 t zz /I4UZR/~J©I4U;|J¯UDOTWJ;UD_f\_K`bKJZ¸U"LW`OSÄR{`Db Z 2 1 e−(x−v0 t−y) /(4Dt) f (y)dy. 4πDt −∞ # jqQM`SKTWLNJD4RNIKL[¤|JML¬SfLiUabKPQ7J^jG`T SKOYU¯bR{IKJ©Q7UDLNJ©R{I4U=R lbKU JMOP\D f (x) = M δ(x) N (x, t) = ∞ √ (x−v0 t)2 M e− 4Dt . 4πDt $mê íDë3êDò%©ë /ñ ±sëó=çúVë·µ·¶'&)(+*4³=çèsç4µyë3êDò-,mé{³=úVµ·¶ £b+R{IKPLQ;UDLWJ%RNIKJ.¾»¨ª ]9JMQ7` J7L 2 ∂ N 2 ∂N ∂N =D + + aN, ∂t ∂r 2 r ∂r /IKJMTNJ PL%U [TW`FRNIT{U=R{J3`T%U _fJ7Q;U;\T{UZRNJDx HA` [JMR UjGJMJ7O¾jG`T^R{IKJ[TN`´R{I*R{JMT UD[PbKaJRNI4UZRdR{IKJMTNJLdbK`*_KLWe9JMTNLW`b?]KSfRLN efOP\JÂrÄe9`bfJ7b R{UDO[TN`´RNI UZRUá¥frÄJ7_ el`b R;xhIKUZRXJ©R{IfJ7bI4U;|J%L/UDb # »¨ª x~HIKPL/LSKLWJMjGSKOPbel`eKSfOYUZRN`b dN/dt = aN Ä ` K _ M J O ¬ L D ` ¾ j D U f b D U O [ N T ´ ` N R I { R 4 I = U i R N R W T + \ { R ` Z U 7 Q 7 Q D ` K S b « R G j ` ¬ T N T M J eKTN`_KSKQsR{`DbáUDbK_¦_fLNelJ7TWLNP`bPb N (x, t) = √ ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32 4 U LNPbK[OPJ `Ä_KJMOÃxARQ;UDb+]lJ¨LNIf`F/bGLNJ7J"tvuwYR{IKUZR/U LW`OSfRN`bR{`R{IfLmeKTW`]KOPJ bfLWRUZbR{UDbKP`SKLel`b RLW`SKTNQMJ U=R UDbK_ L/[¤|J7b+] \ r=0 t = 0 M δ(r) r2 M exp at − N (x, t) = . 4πDt 4Dt /PRNI+UDb 0165.1'E78Aß3ÚáدÜ4Ù7ÚÙ9aد<96l8FEy6@C<l@CÙ:<l@C:=BÚÙ HIKJMLNJaLW`SKTNQMJ7L UDnZJ¯R{IfJ7TiUDeKelJ;UZT{UDbKQMJ7Lib]l`SKbf_4UDTW\QM`bK_K¤R{`DbKLi`DTiQM`b R{bSK`DSKL b RNJ7TNP`TmLW`SKTNQMJ7LMxJDopOO?L}R{PQnR{`aR{IfJ©jG`DT J7T7x $/èæsµ3ë·ò ë·çèNêD; é %©ë /ñ ±sëó=ç°úVë·µ·¶= ê <~³=çf±sµÃêDç4µ?>¨³=ñçlí êDésî@<~³=çlíDë·µ3ëó=ç ½iJ7TWJ*U®LW eKOJ Ä ` K _ M J ? 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N (r, t) = óÄ`XJ§Q7UDbRNTNJ7UZR+R{IKPLbKL}RUDb RUZbKJ7`SfL4πDt LW`SKTWQ7J*eKTN`]fOJ UDLU)eKTN`]fOJ /PR{IU SÄr ö U=R Q7`DbK_KPRN`bUZR¬R{IfJ¯OPJMj·R¬]9`DSKbK_4UDT}\¸]KSfR«T{UZRNIKJ7TRNI4UDbIKU´|DJUDb bKL}RUDb R{UDbKJ7`DSKLiLW`SKTWQ7J ¾~JQ7UDb®LNeKTWJ;UD_®`SfRaRNIKJ+eKSKOLWJ+`F|DJ7T"UáLNIK`DTWRdR{ JPb®U§L ``DRNIj3UDLNIKP`b] \ t_KJs= ¥4bK0bf[ 2 2 2M q(t) = √ e−t /(2σ ) σ 2π /IKJMTNJ PL/U"L UZOO9|ZUDTWYUDbfQ7JDx¾ä¬`DR{J¨RNI4UZR σ Z ∞ q(t)dt = M. 0 ó `¦~J/OPOTWJ7eKOUDQ7JR{IKJPbKLWR{UDb RUDbKJM`SKLLW`SKTWQ7J] \)U¦QM`b R{bSK`DSKLLN`DSKTNQMJR{I4UZRJ PR{L Ä UD]l`SfRR{IKJ«L{U J¨U `SKb R7 K`´|J7TmU¯TNJMOYUZRNP|DJ7OP\dLWIK`T}RXR{ J¬LNeKUDb] \SfLNbf[ R{IKJ«UD]l`F|DJ jG`T SKOYU jG`DT /PRNI UaLWSKM PR{UD]KOJ^QIf`QMJ%`Zj x q(t) σ QSR)Y3ÃÄLÅÇÆY;ZfxÈÂ`\CHAUDnZJ«U^Q7TWPRNQ;UZOfO``n UZRVR{IKJ/JÂrKU eKOPJ`DjkmUDbKnLit uKexVmÂZwyxVó`OP|DJ R{IfL©eKTN`D]KOJ bS JMTNPQ;UDOPOP\ SKLNPbK[RNIKJdjGSKbKQMRN`bá¥4OPJajGQ7bKTNUD_ SÄr© ¾ x xÉ ¼ `DS [I R©SKLWJ UDL/bR{IKJ%eKTWJM|`SfL¬JÂrfU efOJ©UDbK_ UDbb R{JMTW|ZUDOjGTW` ö RN`LW` JsR{IKPbK[dOPnDJ M = 1 1 150 UDOP`bK[/PRNI R{` QnUDb*bKL}RUDb RUZbKJ7`SfL%LW`SKTWQ7JDx"z«L%kmUZbKnL J7b R{P`bKLM?\`S Q7`DSKO_§SKLWJ σ = 1 ýF\ÄT©UZbK_ xa``n¦UZR^RNIKJd[T{UZeKIKL%UDbK_§LNJMJdPjR{IKJaTNUD_KSfL a = 2.65 D = 12.2 `DjR{IKJdjGTN`DbR%TW`SK[IKO¤\¦QM`TNTWJ7LNel`bK_fL%RN`RNIKJd_4UZRU+[¤|J7bx"I4U=R UZTNJaR{IKJ"UDefeKTN`;rÄ UZR{J el`eKSKOUZR{P`bKL/bRNIKJ \DJ;UDTWL UDbK_ /IKJMTNJ ~JQ7`DSKb R UDLi\J;UZ T 7JMTN] ` Ê®»«` 2, 4, 6, 8 10 1905 \`DS ]lJ7OPJM|DJ^R{IKPL `_KJ7ËO Ê QSR)Y3ÃÄLÅÇÆYZÄxv1 g \.óÄSfeKe9`DLNJmR{IKUZRq~J `_KJMO?GQ7TN Pb4UDOY_KTNSK[«L{UDOPJ7LVUDLqT{UD_KUZR{PbK[/jGTN` U^LNPbK[OJXLWR{J7UD_f\aLW`SKTWQ7J//IK`LNJ/OP`ÄQ7UZR{P`b ~J¬nbK`´ UDbK_"`]9Js\bK[©U^T{UZ_KYUDOÄ_KPÁ9SKLW`baOU´©x äi`´ TNJMeKTNJMLNJMbRNLaR{IKJ_KJMbKLN¤R£\`Dj¬_KTWSK[ábfQ7_fJ7b R{LMx½¬J7TNJ+UZTNJPLNLNSfJ7LR{`¦RNIKbfn NUD(r, UD]l`SfR¬ bf_t)~J^/OO9_KLWQ7SKLWL¬RNIKJ b+LNJ Pb4UDT7 ¡xq½i`´®Q;UDb~JLNJsRVSKeRNIKJXjGSKbKQsR{P`b¯¥4OPJ7LqRN`¨I4UDbf_KOJXU¨Q7`b R{PbSK`SKLhQM`bKL}RUDb RVLW`SKTNQMJ `DjlT{UZ_KYUDO_K¤ÁSKLW`bD[¤|J7b"U¨QM`bKLWR{UDb R¾_KPÁ9SKLW`bQM`ÄlJ P"Q7JMb RUDbK_R{IfJLN MJm`Dj¸R{IfJLN`SKTWQ7JZx I4UZRmUDTNJ/R{IfJ¨UDeKefTN`eKTWYUZRNJ/]9`SfbK_4UDT}\QM`bK_K¤R{P`bK¯L Êy»«`%R{IKPLjG`DTXU QM`b R{PbSf`SKL `_KJ7O UDL/bªqrfU eKO0 J Zfx¹ÄxLN`dR{I4U=Ri¤RLTWJ;UD_Ä\R{`"[`ajG`TmR{IfJ%TWJ7L}Ri`ZjARNIKLeKTW`]KOPJ xv ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32 ÌÅ x¾ä¬`´LNSKefe9`LWJ«R{IKUZR~~J¨UZTNJ¬[P|DJ7b_fTNSK[¯Q7TN J¬_4UZR{U%`DbU¯[DTN_ UDe`DjRNIKJ«QMPR£\ Ä TNJ7UDOO¤\fsxV½¬`F°Q;UDb+~J%RNT{UDbKLWOYUZRNJ«R{IfL/b R{`aPbfjG`T UZRN`bUD]9`DSfR Ê NLN(r, t)R{`J7L}R{ UZRNJdRNIKJ fxº©¤|JMbR{IfL©bfjG`DT UZR{P`báUD]l`SfR qIK`´ìQ7UDb*~JSf J"PR© N (r, t) _K¤ÁSKLW`bQM`ÄlJ P"Q7JMb Ri`ZjqR{IfJ `_KJ7ËO Ê Kxi3½«UDTW_KJ7+T Ív I4UZR¬¤jC~J%_K`b?o R/nbK`´ R{IKJ©QMJ7b R{JMT7¸]KSfR¬]lJ7OPJM|DJ¨R{IKJMTNJ©L`bfbJ Ê ÎÏÐÑuÏuÒsÓ-Ô1Õ-Ö×EØ Ì¢Ì
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