?=?s` ?Dt ?Z? a0 a1 star a2a3a4a5a6a7a8a9a10 6 ? E=c` ?Dt ?Z???3Hq?1: ? E=c` ?Dt ?Z???3Hq ?Dt ?Z?Yè·Vt ?D#  eò ê1 ? SDa/ ? SD? ?á 3¥ ê±sZ? μ H9 ?DNμ1¥sZ?a±ssZ??è±sZ? è ? ?áè ]?? ? ] ?@¥LaplaceZ?PoissonZ? ?o¥.l ? ?@¥o?Z? ? £.?ù5? ? ?ù5?¥ £.?Z? ? ??oé ?D?¥Navier–StockesZ?F?EulerZ?F ? íèH??M?¥MaxwellZ?F ?T1±4té?'? p¥Schr¨odingerZ??DiracZ? ??? ?D?¥Saint-VenantZ?F ???tZ?(F) ^=¨L? ê±sZ?(F) ?[' [?|"?) ?+???¥=¨ L? ê±sZ? 'cVBtt ?ù5?Bt??¥=¨L? ê±sZ?[a) ??tZ?¥B??é #3E x12.1?¥???Z??2: x12.1?¥???Z? μB?? ? ? è¥ ( ? £ ü°L*?7a[ ?ZE? P?]B? ü ?  Tl?? ?¥???Z? m12.1`?¥??? tan 1 = @u @x ? x ; tan 2 = @u @x ? x+dx |?¥ ü?ê?1xà O 7 ?US1x = 0Dx = l !u(x;t) ^US1x¥? B?t H Y¥(?_)êM? ? ?é1dx¥Bl (? í)?í¥?é@l[à? V[ü ? A? ^é? s?í s ? ? ? ?x#x+dx) s?f ?¥T¨ y1? ^? ? ? è¥#o s? M_? ?f ?T¥T¨7 àμE_? ?] H { ? × ?¥T¨ yNμ (T sin )x+dx ?(T sin )x = dm @ 2u @t2 ; (T cos )x+dx ?(T cos )x = 0: l??í ?x+dxDx ?W ?B H Y?_êM-μu(x+dx;t)?u(x;t)DdxM 1 ^B?l ' j@u=@xj? 1: l??í ?/ sin … tan = @u@x { ? @u@x¥ ?)[ ? ; cos … 1 { ? @u@x¥=)[ ? : ?"üμ (T)x+dx ?(T)x = 0'(T)x+dx = (T)x; a üT? ?xM???ò?¥f ?M?? ^ ‰dx@ 2u @t2 = T " @u @x ? x+dx ? @u @x ? x # = T @ 2u @x2dx; x12.1?¥???Z??3: ' ‰@ 2u @t2 ?T @2u @x2 = 0; ?‰ ^?¥L á(?êé¥é )?l a = rT ‰; 5Z? V[? @2u @t2 ?a 2@2u @x2 = 0: aü ^?¥??.l ? ? V[£ ül??í ?/f ?TDtí1? ^y1?í¥ %é ds?dx = p du2 +dx2 ?dx = 2 4 s 1+ @u @x ?2 ?1 3 5dx = O @u @x ?2? ; ?[ ??@u=@x¥B)[('l??í ?)¥Hq/?í¥é? ? HWM ?yN?vHooke? pT9? ? HWM? - ??Xü£ üVT9? ?xM? ?[T ^B?è ? ?T??_('êMu¥?_) ? s?? ?¥T¨ !?êé ? s¥? ?1f5_v - ?¥w?μ ‰dx@ 2u @t2 = T @2u @x2dx+fdx: yN @2u @t2 ?a 2@2u @x2 = f ‰; ?¥d Q[f=‰ ^?êé  ? s¥? ? x12.2?¥:??Z??4: x12.2?¥:??Z? I nB ( %??éZ_Tl??L !<°?éZ_¥ ?B? ? ò?¥?? f ?('êM)? ?M] m12.2`?¥:??`? ?D?M F ?m12.2 |?éZ_1xàZ_<°??éZ_¥ò? ? (¨ ?¥ ü?ê?xS: F ?B H YtN? ?M? ü?ê?¥êM1u(x;t) F??? ?Bl (x; x+dx)s s ? ?YV? ?x s??? ?P(x;t)S¥T¨ ?YV? ?x+dx s??? ?P(x+dx;t)S¥T¨ P(x;t)1?ê ? ? s¥?? ?(? ?)xZ_1? yN? Newton?=? pü¤? dm@ 2u @t2 = [P(x+dx;t)?P(x;t)]S: ??¥ á1‰5dm = ‰dx¢S ‰@ 2u @t2 = @P @x: ?T { ?<°?éZ_¥?M? Hooke? p? ?PD?M@u=@x??1 P = E@u@x; 1 è" ?E?1?¥Young   ? ^B?téè ??"ü¤? ?¥:??Z? @2u @t2 ?a 2@2u @x2 = 0; ? a = rE ‰ : ?¥:??D?¥??? ?i?? ?M]? ? ì ?@¥ ê±sZ?¥? T ?? ?B "?B ?Z?d?1o?Z? x12.2?¥:??Z??5: ÷B?1 ?? bW?¥o?Z? ^ @2u @t2 ?a 2r2u = 0; ? r2 · @ 2 @x2 + @2 @y2 + @2 @z2 ?1Laplace ?? r2 = r¢r'r2u = r¢(ru): x12.3 £.?Z??6: x12.3 £.?Z? w? £.?Z? ?¨¥ ?DZE?  ?¥? ?M]?]-)? 8¥t ?? p ?]? ú¨?¥ ^ £DZ ?¥ ?'? p' ?  o?? p? £.?¥Fourier? p £.?¥Fourier? p` !μB v ??oé |?B?US"i¨u(x;y;z;t)V Uoé = bWUS1(x;y;z)¥B?t H Y¥? ?xZ_μB?¥?μxZ_9üB?μ £ ¥.?V?4  A?ê HW =YV<°xZ_¥?ê ?¥ £ qD?¥ bWM? q? ?1' q = ?k@u@x; q?1 £ @ ák?1? £ q kDoé¥é μ17 O?ì a ?D?u9μ1"? ?T?¥M?S? ?v5 V[í ?1|k A?Duí1  ? T?¥μ|V U £ @¥Z_??M?¥Z_?zMQ' £ ?ú? @_?? ù? ??ò_]?oé?¥ £.?oé? ??Z_ ?i?μ5μ qx = ?k@u@x; qy = ?k@u@y; qz = ?k@u@z;  q= ?kru; ' £ @ á O qD?0ru??1 ? Fourier? p? ?  o?? pw? ( ò_]?oé?¥ £.?Z? oé =?? ?B? ü? B ?8(nm12.3) B? ???US ?×? m12.3` £.?Z? ê?(x;y;z)?¥l B ?8 F ?t HW =xZ_ @ ? B ?8¥ £  £(q x)x ?(qx)x+?x ??y?z?t =h k@u @x ? x+?x ? k@u@x ? x i ?y?z?t =k@ 2u @x2?x?y?z?t: x12.3 £.?Z??7: F ?t HW =yZ_ @ ? B ?8¥ £  h (qy)y ?(qy)y+?y i ?x?z?t = k@ 2u @y2?x?y?z?t; F?t HW =zZ_ @ ? B ?8¥ £  £(q z)z ?(qz)z+?z ??x?y?t = k@2u @z2?x?y?z?t: ?T B ?8 = àμ  e £  ?÷h{5? ?  o?? pê @ ?¥ £ ????oé N HW =? 6ú ?31¥ £  k @2u @x2 + @2u @y2 + @2u @z2 ? ?x?y?z?t = ‰?x?y?z ¢c¢?u: ?[ @u @t ? k ‰cr 2u = 0; ?‰ ^oé¥ ác ^1 £ ? 7? = k=‰c5μ @u @t ??r 2u = 0; ???1 ? ? q?.? q ?Toé =μ £ á 3( è ?μ?DQ?? 3Yμè @¢¢¢¢¢¢)?ê HW =? ê8oé?á 3¥ £ 1F(x;y;z;t)5μ kr2u?x?y?z?t+F(x;y;z;t)?x?y?z?t = ‰?x?y?z ¢c¢?u; @u @t ??r 2u = 1 ‰cF(x;y;z;t) = f(x;y;z;t): ?Toé? ( 5? £ qkDUS(x;y;z)μ1? H £.?Z?üM1 ‰c@u@t ?r¢(kru) = F(x;y;z;t): £.?Z?¥ 6B?? T` 7j = ‰cu?1 £ @( <)5 @j @t +r¢q= F(x;y;z;t): ??Z?è?1 ???Z? ?T ^ò_s?oé5Fourier? p??? q= ?K¢ru: x12.3 £.?Z??8: ? ú¥K ^B?3£3 ? ??ru? ?eE¥?5Meq?ru? ^  O . M?1 £.?Z?M1 ‰c@u@t ?r¢(K¢ru) = F(x;y;z;t): Vs0?¥? A?¥ú? ^s0 £? ?¥Q?s0 £?¥? ü? YV ·?D ? ?4 üVC1 £ ¥.? V[ !X ?Toé =iY?? (  ? è ?té i¥? ( YVs0¥?9?? 3té¥?D??4 üVC1s0 ¥ ? ???±4 ? ¥M ??ü %?  ? ?Z?? £.?Z?μM]¥? T @u @t ?Dr 2u = f(x;y;z;t); ?¥u(x;y;z;t)}Vs0 iD ^ ? ? qf(x;y;z;t)5 ^?ê HW =?ê8??? s0¥á q x12.4×?ù5?9: x12.4×?ù5 ×??s?`B?Hq/t8¥?r?×?a'? ? HWM? H5?s? ? @PoissonZ? r2u = ?f?: +Y ^ ?Tf = 05μLaplaceZ? r2u = 0: ? ?Z? í¥ ^r?×?¥t ? ? áè?¥è ]` ?To?Z??u9? ? HWM? è ?áè?¥è ]u(x;y;z)9 ? @PoissonZ? r2u = ? ‰" 0 ; ?‰ ^è? á"0?1? bè ? q(? boèè ?) ?Tè? በ· 05áè?¥è ] ?@LaplaceZ? r2u(x;y;z) = 0: ? ?o` ?To?Z? @2u @t2 ?a 2r2u = 0 ?u(x;y;z;t) ? HW? ù1M? ? q1! u(x;y;z;t) = v(x;y;z)e?i!t; 5v(x;y;z) ?@HelmholtzZ? r2v(x;y;z)+k2v(x;y;z) = 0; ?k = !=a?1o ? [ ¥+?'¥ ê±sZ?Vt ?  Aμ FQ?o?V?¥o?Z? FQ? ? ?V?¥ £.?Z? FQ?×? ?¥PoissonZ??LaplaceZ? V ?D  A?9?zM?1s1 ? ? Fo?Z? ?D  ?? ? w?Z? F £.?Z? ?D  ?? ?t?Z? F PoissonZ??LaplaceZ? ?D  ?????Z? ? ? ?Z?¥ p3ù5| ^' [?¥?? ? x12.5H?HqD SHq?10: x12.5H?HqD SHq Fo? Newton?=? p ¥? ?DZ?i? ?·B1 ??é?¥? F1? ? ??B?é?¥?" ±sZ?-??A?μ SHq?5=¨è±s Z?¥Y3?cμ ? ?iè ?y73? ^·B ??¥ Foμ ê±sZ?9? ?·B1a ??1 í B? 8¥t ?V? F=¨ ê±sZ?¥Y3cμ ? ?if ? è ? ê±sZ? @2u(x;y) @x2 = 0 ¥Y3ü ^ u(x;y) = c1(y)+xc2(y); ?c1(y)?c2(y) ^y¥ ?if ? ?μZ?73i?·BVt ?  ? A9 ^1 ?¥y1 Fw?Z? Ho I n oé¥ =?i àμ I noéYVV ????¥MoT¨y N?ì a ?Z?o a¨?oé =? F ?Tù5D HWμ1¥?w?Z? H9i àμ I noé¥ ? N ? ?Tá ì a? ê |9 H¥ ,? * 1ü V[ aZ?9o a¨?t > 0¥ ?B H Y 1 ? ? íB? μ ??3¥t ?ù5 ?D ü ^1?B??3ù5?" ± sZ?-??A?μH?Hq? SHq  SHq` SHq??? ? í S H Y(t = 0 H)oé =?#H?  ?iB?¥ ? F?o?Z? ? aü ^??ó S H Y¥êM? ?( ?T ^ ?Dù5¥?) uflflt=0 = `(x;y;z); @u@t flfl flfl t=0 = ?(x;y;z); (x;y;z) 2 V: F? £.?Z???Z??oC??f ?u(x;y;z;t)t¥B¨ ê±  ?[o3ó  S H Y¥? uflflt=0 = `(x;y;z); (x;y;z) 2 V: H?Hq¥? T1?"?1? 8ù5? í ?¥ 8 ? %?9¥e5 ^H?H q??? ? íH? ò? ?B H Y(t ? 0)¥ ? ?¥???` ?T?¥o  %?p5H?Hqü ^ uflflx=0 = 0; uflflx=l = 0; t ? 0: x12.5H?HqD SHq?11: ?¥:??` ?x = 0 %?5x = 0 ¥H?Hq ˉ ^ uflflx=0 = 0: m12.4` ? ? s? ?D? ? ü? ? 6B (x = l) sxZ_ ¥? ?T¨?ê ? ¥ ? ^F(t) _w?Z?¥÷E  ?x = l)? |Bl voéé1"? Newton?=? p V??Bl oé ? s¥? ?(? ?F =? ?)????oé¥é e[oé? B?¥F ? ‰"S @ 2u @t2 flfl flfl x=l?fi" = F(t)S ?P(l?";t)S; ?0 ? fi ? 1 7" ! 0i} ? P = E@u@x; 5μ @u @x flfl flfl x=l = 1EF(t): ? ?T? ?10'x = l ^1?¥5 @u @x flfl flfl x=l = 0: ? ?T? ?F(t)? ^B? ??¥X?f ?7 ^??T4¥?? ?5 F(t) = ?k£u(l;t)?u0?; k ^?T¥?" ?? ^ ?@u @x + k Eu ? x=l = kEu0: £.?ù5`èn¥H?Hqμ/ +? ?? FH? ò?¥?X? uflfl§ = `(§;t): §V UH? ¥M?] H9V U?t?¥US F?ê HW =aYV?ê ?¥H? ? @ ?¥ £ X? x12.5H?HqD SHq?12: m12.5`H? ?)¥ £ @ ?? H? =§? |Bl aoéB?? ?oé¥V ? 6B?? ?oé =??8 ¥ ? ?M?¨ t?0? ?  o?? p V?oéV ?? ?#§ ? @ ?¥ £ - ??????B voé? 6ú ?31¥ £ ? ^?o饨 t?0 HYV§ ? @ ?¥ £ ?? t?0 (y1§ ? t?0)oé¥ £ ?  t?0 (y1oé¥é  t?0)y NYVoéV ? @ ?¥ £ ?? ??YV a¥ 6B? ? @_oé =?? ^ V[ H?Hq @u @n flfl flfl § = 1k?(§;t); ?@@n?1E_±  ? ^0 O ?ELZ_n ¥g? @ @n =n¢r' @u @n =n¢(ru): ?H? ' £5? · 0 @u @n flfl fl § = 0: FoéYVH??Newton ? ?? p ? £?ê HWYV?ê ?V ?????D¥ £ D oéV ??uj§????u0-μ??1 !1 è" ?1H ?k @u@n flfl flfl § = H?uflfl§ ?u0¢; ??? @u @n +hu ? § = hu0:  ?C¥H?HqμB? ]¥+?ü??f ?7y ? ì? ^L?¥éB?%s ? V[s1 ? ? F?B ?H?HqóH? ò?¥f ?′ F?= ?H?HqóH? ò?f ?¥E_± ′ F? ? ?H?HqóH? ò?¥f ?′DE_± ′-W¥L?1" ??H? ? ò?¥H?Hqi?B? ?μdB¥Vr T9?n¤] ??B ? ??  L  ?) ?¥???¥H?Hqü ^ ?N x12.5H?HqD SHq?13: Fí? bW¥ù5 ? H¥H?Hqü??ó??f ?í kù)¥K?1 è ? f ? .à ?¥? ?í kù)μ? μ? bW¥ù5?μ H91Cμ?Hq è ??á ì?¨US"a?U S" oUS" H ê± @u=@rUSe? > ?ily731? 8 f ? USe?? μ?Hq  eHq x12.6 =?? ? ¥ ?¤Hq?14: x12.6 =?? ? ¥ ?¤Hq ?T u×¥ =?C2 ¥M * 1?tM?(La ?) ±sZ?9? ?? ??t?(La ?) ?1? M?¥HqYè?1 ?¤Hq Ke?¥ è0 ^? ?÷?é F?¥oé è ?? ??]? ¥??% 7 ?¤7 ?¥?Vo?Z?¥w?V? V[ A??é ?]  ? ¥??.l ??]± sZ?? ?9ü?] ??B ?@ 2u1(x;t) @t2 ?a 2 1 @2u1(x;t) @x2 = 0; ??= ?@ 2u2(x;t) @t2 ?a 2 2 @2u2(x;t) @x2 = 0:  ?¤??'íE±sZ?7??}-[ ?¤Hq ?¤Hq? ?D ?¤¥Z T? ?μ1T1K ?X¥í ?L !?? ?¤('M)o? ì1? 3?B?7 O ?¤¤dè 1%;á * 1 ?¤?( !US1x0)¥ ?¤Hqü ^ u1(x;t)flflx=x 0?0 = u2(x;t)flflx=x 0+0 ('êMM?); @u1(x;t) @x flfl fl x=x0?0 = @u2(x;t)@x flfl fl x=x0+0 ('f ?M?); ? u(x;t)flflx=x 0§0 = lim"!+0u(x0 §";t): ' P ^B?? ?¥oé9 V ?C ?¤Hq ˉ[?¥???1 è ?T ( ?¥ B?  s?μKvl¥ ?(o"?? ?p)f(t)¥ T¨Z_uà¥μ_ * 1?B?( ˉ:1x = x0)Z?9?? ?9??}-[ ?¤ Hq u(x;t)flflx=x 0?0 = u(x;t)flflx=x 0+0 ; T ?@u(x;t) @x flfl fl x=x0+0 ? @u(x;t)@x flfl fl x=x0?0 ? = f(t); ? ìsYV UêMM??f ?D? ? ü? ÷éB? ?T??"?? ? ^?B?×tM4¥7 O??×t??]?1? 3 ?×t??-W àμMêM? H ?¤Hq???M1 u(x;t)flflx=x 0?0 = u(x;t)flflx=x 0+0 ; T ?@u(x;t) @x flfl fl x=x0+0 ? @u(x;t)@x flfl fl x=x0?0 ? = Mg +M@ 2u @t2 flfl fl x=x0 : áè?¥ù5?÷èn? ?¤Hq x12.6 =?? ? ¥ ?¤Hq?15: è ? ?èoé¥? ?§0 è ] ??(? <¥ M_s  ??)?èêM O ¥E_ s  ?? u1flfl§0 = u2flfl§0; "1@u1@n flfl flfl §0 = "2@u2@n flfl flfl §0 : ? ú¥u1?u2sYV Uèoé1?èoé2?¥è ]"1?"2sY ^? ?èoé¥è ? q x12.7?3ù5¥ a???16: x12.7?3ù5¥ a??  I 1Hq/?3ù5¥3 ^i¥·B¥i O ^×?¥$ F3¥i??3ù5μ3 ?T?3HqVoM ±5?3ù5í3 è ? ?TBZ ?1 p?¥  %? 6BZ ??1 p ?¥ ? s? ??¥? ?T¨? ?1 pü ^oM ±¥ F3¥·B??3ù5¥3 ^·B¥ ?T?3Hq?@?3ù5¥3ü? ^·B¥ ?[1 p?3ù5¥3ii O·Bü ^1 p?3ù5`¤o? ?p?3 Hq1??  ?z) F3¥×?? ?T?3ù5?¥X?Hq( è ?Z??``3Hq?¥X?f ?)μ ±l?M H39oμ±l¥?M /?3ù5 H? VE ?191Te??í ?A ?oμ×?? ?? V¥K = ?T¥e??í ?? ^μil¥ ?ì?3ù53¥i?a·B??×??d? a?? o1 L=t ?ù5¥` ^? ?¥ SHq¥ ? ^? ?1a ??1 í  S H Y(Yè |1t = 0)8" =?[#H? ?  ?iB?¥ ?H?Hq¥ ? ^? ?7 O ??1 í  H? ?  ?iB?t ? 0¥ ? * 1?"?¥?3ù5üB? ^ a?¥9ü ^ a3B? ^i¥a·B¥i O ^×?¥ DNM1¥ù5 ^ SHq?H?Hq?C¥X?f ?A? ?@B?¥ ???1 p [ £.?ù51 è ?TH?Hq ^ u(x;y;z;t)flfl§ = f(§;t); 7 SHq ^ u(x;y;z;t)flflt=0 = `(x;y;z); * 1ü??μ f(§;t)flflt=0 = `(x;y;z)flfl§: μt?3ù5?B? ?@??1 p V[ !Xü S?s?1`(x;y;z)¥B v oéb?B???ìa( è ???1u0)?V7 PoéV ?¥?9á ?r?? ?u0 ?T1 p¥ú? VoéV ? ? ? 6?V?¥?Y V[- { * 1ü V[e?1|H?Hq? u(x;y;z;t)flfl§ = u0: x12.7?3ù5¥ a???17: ?"S¥2TD5?ú ?¥H?Hq?μμY?o1??μY@l * 1 3¥×??üá ?á ì?N? ¥3¥μs9 ^@l¥? ? ?Tá ìü ^ 1ù??? ? ? 6?V?¥?Y??í ?ü ^? V |¥