a0 a1 star a2a3a4a5a6a7a8a9a10 4 13.1 5 ' §) U\5 12 13.1 5 ' §) U\5 r 5 '' § ? ?/“ L[u] = f; ￥ u …? L 5 ? f fi …?§? § g k g ' §? g ' §'XJf · 0§ § · g ' L 13.1 §a. § 5 ?L ˉ? § @ 2u @t2 ?a 2r2u = f L · @ 2 @t2 ?a 2r2 9D § @u@t ??r2u = f L · @@t ??r2 Poisson § r2u = f L ·r2 – ? ‰‰)^ § · 5 ' –r‰)^ ?aq ?/“' ‰′ XJ…?u? §L[u] = f ??§K?u· §L[u] = f )' 5 1 eu1 u2 · g §L[u] = 0 )§ L[u1] = 0; L[u2] = 0; K§ 5| c1u1 +c2u2 · g § )§ L[c1u1 +c2u2] = 0; ￥c1 c2·??~?' 5 2 eu1 u2 · g §L[u] = f )§ L[u1] = f; L[u2] = f; K§ u1 ?u2 ‰· A g § )§ L[u1 ?u2] = 0: § g § A)\ A g § )E· g § )' 5 3 eu1 u2'O v g § L[u1] = f1; L[u2] = f2; 13.1 5 ' §) U\5 13 K§ 5| c1u1 +c2u2 v g § L[c1u1 +c2u2] = c1f1 +c2f2: gC 5 ' § ?H/“ A0@ nu @xn +A1 @nu @xn?1@y +￠￠￠+An @nu @yn +B0@ n?1u @xn?1 +￠￠￠+M @u @x +N @u @y +Pu = f(x;y); ‰ L(Dx;Dy)u · h A0Dnx +A1Dn?1x Dy +￠￠￠+AnDny + B0Dn?1x +￠￠￠+MDx +NDy +P i u =f(x;y); ￥Dx · @=@x§Dy · @=@y?A0;A1;￠￠￠ ;An;B0;￠￠￠ ;M;N;P ·x;y fi …?§? § X?' 13.2 ~X? 5 g ' § ˇ) 14 13.2 ~X? 5 g ' § ˇ) ~X? 5 g ' § ?H/“· A0@ nu @xn +A1 @nu @xn?1@y +￠￠￠+An @nu @yn +B0@ n?1u @xn?1 +￠￠￠+M @u @x +N @u @y +Pu = 0; ‰ L(Dx;Dy)u · h A0Dnx +A1Dn?1x Dy +￠￠￠+AnDny + B0Dn?1x +￠￠￠+MDx +NDy +P i u =0; § X?A0;A1;￠￠￠ ;An;B0;￠￠￠ ;M;N;P ·~?' 1. L(Dx;Dy)·Dx;Dy g“ § h A0Dnx +A1Dn?1x Dy +A2Dn?2x D2y +￠￠￠+AnDny i u = 0: – 5 ?L(Dx;Dy)')? n 5 ? ?¨ L(Dx;Dy) = A0(Dx ?fi1Dy)(Dx ?fi2Dy)￠￠￠(Dx ?finDy); ￥fi1;fi2;￠￠￠ ;fin ·~?§ˇd?n ˇf gS –??N ' `&) u = `(y +fix),ˇ Dkxu = fik`(k)(y +fix); Dkyu = `(k)(y +fix); DrxDsyu = fir`(r+s)(y +fix); \ §= h A0fin +A1fin?1 +￠￠￠+An i `(n)(y +fix) = 0: ? §(? N\ §§auxiliary equation) A0fin +A1fin?1 +￠￠￠+An = 0 )·fi1;fi2;￠￠￠ ;fin§ p §K? ~X? 5 g ' § ˇ) u = `1(y +fi1x)+`2(y +fi2x)+￠￠￠+`n(y +finx); ￥`i; i = 1;2;￠￠￠ ;n·(p ? )??(ng )…?' ~1 ? §@ 2u @x2 ?a 2@ 2u @y2 = 0 ˇ)§a ~?' ) -u = `(y +fix)§KN\ § fi2 ?a2 = 0§ )fi = §a§ § ˇ) u = `1(y +ax)+`2(y ?ax): 13.2 ~X? 5 g ' § ˇ) 15 Fefi·? §~X· ? § (Dx ?fiDy)2u = 0; Kˇ) u = x`1(y +fix)+`2(y +fix): Fefi n? §= (Dx ?fiDy)nu = 0; K § ˇ) u = xn?1`1(y +fix)+xn?2`2(y +fix)+￠￠￠ +x`n?1(y +fix)+`n(y +fix): ~2 §(D2x ?2DxDy +D2y)u = 0 ˇ) u = x`(x+y)+?(x+y): 2. L(Dx;Dy) ·Dx;Dy g“ k ? ' § (Dx ?fiDy ?fl)z = 0: ( ) XJf(x;y;z) = 0· § )§K7k @f @xdx+ @f @ydy + @f @zdz = 0: (z) , ?§ Dxz = ?@f=@x@f=@z; Dyz = ?@f=@y@f=@z: \ §( )§qATk @f @x ?fi @f @y +flz @f @z = 0: (>) ’ (z) (>) “§ dx 1 = dy ?fi = dz flz: ? §|? Lagrange9ˇ §|'N·) y +fix = C; flx = lnz ?lnC0: ?– z = C0eflx = eflx`(y +fix): ˇd§ L(Dx;Dy) ·Dx Dy g“ §XJU L(Dx;Dy) ') n ˇf(z ˇf ·Dx Dy 5…?) ?¨§K –? § ˇ)' ~3 ? §@ 2u @x2 ? @2u @x@y ?2 @2u @y2 +2 @u @x +2 @u @y = 0 ˇ)' 13.2 ~X? 5 g ' § ˇ) 16 ) N·w § (D2x ?DxDy ?2D2y +2Dx +2Dy)u = (Dx +Dy)(Dx ?2Dy +2)u = 0: § ˇ) u = `(x?y)+e?2x?(y +2x): Fek?E5ˇf§X(Dx ?fiDy ?fl)2z = 0§Kˇ) z = xeflx`(y +fix)+eflx?(y +fix): 13.3 ~X? 5 g ' § ˇ) 17 13.3 ~X? 5 g ' § ˇ) g § ˇ) = g § ? A) + A g § ˇ)' § L(Dx;Dy)u = f(x;y) A)/“/L? u0 = 1L(D x;Dy) f(x;y); Ue {K? u0(x;y) 1. ef(x;y) = eax+by§ L(a;b) 6= 0§K 1 L(Dx;Dy)e ax+by = 1 L(a;b)e ax+by: F L(a;b) = 0 /' L(Dx;Dy) = bDx ?aDy§ (bDx ?aDy)u = eax+by: 13.2!￥ {§ Lagrange9ˇ §| dx b = dy ?a = du eax+by; = adx+bdy = 0; adu+eax+bydy = 0: d1 § ax+by = c: \1 §§ adu+ecdy = 0: ?– u = ?1ayec = ?1ayeax+by; = 1 bDx ?aDye ax+by = ?1 aye ax+by: 13.3 ~X? 5 g ' § ˇ) 18 a?)L§￥5? :'1 §3? 1 § )(?k¨'~?c) I \1 §§– x§ d· \ ¨'~?c?3? 1 § ) §qI L5 ¨'~?'1 §3?)1 § § 72 ?1 ¨'~?' 2. ef(x;y) = ei(ax+by)§w,k 1 L(Dx;Dy)e i(ax+by) = 1 F(ia;ib)e i(ax+by): ˇd§ a b ￠?§ L(Dx;Dy)￥ X? ￠? § 1 L(Dx;Dy) sin(ax+by) = Im ? 1 L(ia;ib)e i(ax+by) ? ; 1 L(Dx;Dy) cos(ax+by) = Re ? 1 L(ia;ib)e i(ax+by) ? : FXJL(Dx;Dy)·D2x;DxDy D2y { E …?§ L(Dx;Dy) = G(D2x;DxDy;D2y); K 1 G(D2x;DxDy;D2y) sin(ax+by) = 1G(?a2;?ab;?b2) sin(ax+by); 1 G(D2x;DxDy;D2y) cos(ax+by) = 1G(?a2;?ab;?b2) cos(ax+by): 3. ef(x;y) = eax+byg(x;y)§K 1 L(Dx;Dy)e ax+byg(x;y) = eax+by 1L(D x +a;Dy +b) g(x;y): y 5? Dx￡eax+byg(x;y)? = eax+by(Dx +a)g(x;y); Dy￡eax+byg(x;y)? = eax+by(Dy +b)g(x;y); ˇd L(Dx;Dy)eax+byg(x;y) = eax+byL(Dx +a;Dy +b)g(x;y): ? § k L(Dx;Dy) ‰ eax+by 1L(D x +a;Dy +b) g(x;y) = eax+byL(Dx +a;Dy +b) ‰ 1 L(Dx +a;Dy +b)g(x;y) = eax+byg(x;y): 13.3 ~X? 5 g ' § ˇ) 19 ?“ y' 4. ef(x;y) = xmyn§K 1=L(Dx;Dy)—m Dx; Dy ??§ ? A)' ~4 ? g §(D2x ?2DxDy +D2y)u = 12xy ˇ)' ) § A) u0 = 12D2 x ?2DxDy +D2y xy = 12(D x ?Dy)2 xy = 12D2 x 1? DyD x ??2 xy = 12D2 x ? 1+2DyD x +￠￠￠ ? xy = 12D2 x ? xy + 2D x x ? = 12 ? y 1D2 x x+ 2D3 x x ? = 12 ?1 6x 3y + 1 12x 4 ? = x4 +2x3y; ￥|^ 1 Dxx = 1 2x 2 ? * ddx x 2 2 = x · ; 1 D2xx = 1 6x 3 ? * d 2 dx2 x3 6 = x · ; 1 D3xx = 1 24x 4 ? * d 3 dx3 x4 24 = x · : A g § ˇ)fi3~2￥? § g § ˇ) u = x`(x+y)+?(x+y)+x4 +2x3y: 1=L(Dx;Dy)—m –k {§ˇ (J'~X§3 ? ~K￥§ – 1 (Dx ?Dy)2 = 1 D2y ? 1? DxD y ·?2 = 1D2 y h 1?2DxD y +￠￠￠ i : ˇd§ g § A) – u0 = 12(D x ?Dy)2 xy = 2xy3 +y4: ? ? { A) x4 +2x3y ?2xy3 ?y4 = (x?y)(x+y)3 = 2x(x+y)3 ?(x+y)4 · A g § )' 1 e g f(ax+by)§ L(Dx;Dy)·Dx;Dy (n)g“§K Drxg(ax+by) = arg(r)(ax+by); Dsyg(ax+by) = bsg(s)(ax+by): 13.3 ~X? 5 g ' § ˇ) 110 ?– L(Dx;Dy)g(ax+by) = L(a;b)g(n)(ax+by): ˇd§ L(a;b) 6= 0 § k 1 L(Dx;Dy)g (n)(ax+by) = 1 L(a;b)g(ax+by): ~5 ?) §@ 2v @x2 + @2v @y2 = 12(x+y)' ) k?A)' §w,? 1 ^ '?–A) v0 = 12D2 x +D2y (x+y) = 12?12 +12￠￠3!(x+y)3 = (x+y)3: N·? A g § ˇ)§l g § ˇ) v = (x+y)3 +`(x+iy)+?(x?iy): F L(a;b) = 0 /' k ? Aˇ g ' § (Dx ?fiDy)u = xr?(y +fix); A Lagrange9ˇ §| dx 1 = dy ?fi = du xr?(y +fix): u·y +fix = c§l ? u = 1r +1xr+1?(c) = 1r +1xr+1?(y +fix): ?–§k 1 Dx ?fiDyx r?(y +fix) = 1 r +1x r+1?(y +fix): E|^? (J§ –? 1 (Dx ?fiDy)kx r?(y +fix) = r! (r +k)!x r+k?(y +fix): ~6 ?)(D2x ?6DxDy +9D2y)u = 6x+2y§= (Dx ?3Dy)2u = 6x+2y: ) ~3fi? A g § ˇ)x`(y +3x)+?(y +3x)' g § A) u0 = 1(D x ?3Dy)2 (6x+2y) = 2(D x ?3Dy)2 (3x+y) = x2(y +3x): 13.3 ~X? 5 g ' § ˇ) 111 ˇd§ g § ˇ) u = x2(y +3x)+x`(y +3x)+?(y +3x): 2 ?u g f(x;y)§ –ˇL?) A Lagrange9ˇ § {) ?' ~X§ ? § (Dx ?fiDy)u = f(x;y); Lagrange9ˇ § dx 1 = dy ?fi = du f(x;y): ddx = dy=(?fi)§ y +fix = c' \dx = du=f(x;y)§ du = f(x;y)dx = f(x;c?fix)dx; u = Z f(x;c?fix)dx: O ¨' §2 c^y +fix ￡§= A) u0 = 1D x ?fiDy f(x;y) = ?Z f(x;c?fix)dx ? c=y+fix : ~7 ?) §(2Dx ?3Dy)(Dx +Dy)u = 5ex?y' ) A g § ˇ) `(y?x)+?(2y +3x)' g § A) u0 = 5(2D x ?3Dy)(Dx +Dy) ex?y = 1D x +Dy h 5 2?3(?1)e x?y i = 1D x +Dy ex?y = Z ex?(c+x)dx flfl flfl c=y?x = xe?c flfl fl c=y?x = xex?y: ?–§ g § ˇ) u = xex?y +`(y ?x)+?(2y +3x): 13.4 Aˇ CX? 5 g ' § 112 13.4 Aˇ CX? 5 g ' § k? xmyn @ m+nu @xm@yn /“ '- x = et; y = es; =t = lnx; s = lny§Kk Dt · @@t = x @@x; Ds · @@s = y @@y: u·§ x2 @ 2 @x2 = Dt(Dt ?1); y 2 @ 2 @y2 = Ds(Ds ?1); x3 @ 3 @x3 = Dt(Dt ?1)(Dt ?2); y 3 @ 3 @y3 = Ds(Ds ?1)(Ds ?2); ... ... /§ xmyn @ m+n @xm@yn = Dt(Dt ?1)￠￠￠(Dt ?m+1) ￡Ds(Ds ?1)￠￠￠(Ds ?n+1): ?–§?uL(Dx;Dy) d xmyn @ m+nu @xm@yn /“ |? §ˇLC x = et; y = es; = z ~X? ' §' ~8 ? §x2@ 2u @x2 ?y 2@ 2u @y2 +x @u @x ?y @u @y = 0 ˇ)' ) X C §K §z [Dt(Dt ?1)?Ds(Ds ?1)+Dt ?Ds]u = 0; =￡D2t ?D2s?u = 0'?–§ § ˇ) u = `1(t+s)+?1(t?s) = `1(lnx+lny)+?1(lnx?lny) = `1(ln(xy))+?1 ? ln xy · = `(xy)+? ?x y · : 13.5 ˉ? § 1ˉ) 113 13.5 ˉ? § 1ˉ) 3~1￥§Q?? Lˉ? § @2u @t2 ?a 2@2u @x2 = 0 ˇ)§?p ? u(x;t) = f(x?at)+g(x+at); ￥f g·??…?' ? )“L?§ˉ? § ˇ)§d ˉ|?' F f(x;t) L x? D′ ˉ§ t = 0 §ˉ/ f(x)§ – ‰ ˙a D′§ –ˉ/ C? F g(x;t)K L x? mD′ ˉ§ t = 0 §ˉ/ g(x)§ – ‰ ˙a mD′§ –ˉ/ C' § ?D′§p Z6'? ·ˇ ˉ? §· 5 g §§ k) U\5' l K ‘5§…?f gATd‰)^ (‰'XJrflK{z ?.u ˉ D′flK§@o§f g ,B d—'^ ?‰' ~9 ?‰)flK @2u @t2 ?a 2@2u @x2 = 0; ?1 < x < 1; t > 0; u(x;t)flflt=0 = `(x); ?1 < x < 1; @u @t flfl fl t=0 = ?(x); ?1 < x < 1 )' ) ?fi § ˇ) u(x;t) = f(x?at)+g(x+at): y3 —'^ (‰…?f g' )“ \—'^ § f(x)+g(x) = `(x); a￡f0(x)?g0(x)? = ??(x): 1 “¨'§ – f(x)?g(x) = ?1a Z x 0 ?(?)d? +C; 13.5 ˉ? § 1ˉ) 114 ￥C·¨'~?' ? (J ? 1 “??§= ? f(x) = 12`(x)? 12a Z x 0 ?(?)d? + C2 ; g(x) = 12`(x)+ 12a Z x 0 ?(?)d? ? C2 : 2 ￡ )“￥§ ? ?.?m ˉ? §‰)flK ) u(x;t) = f(x?at)+g(x+at) = 12`(x?at)? 12a Z x?at 0 ?(?)d? + C2 + 12`(x+at)+ 12a Z x+at 0 ?(?)d? ? C2 = 12￡`(x?at)+`(x+at)?+ 12a Z x+at x?at ?(?)d?: ? )? ˉ? §‰)flK 1ˉ)§‰d’Alembert)' ) n?′ F1 L?d— ￡-u 1ˉt = 0 ˉ/ `(x)§– '? '§ ?/ mD′§ ˙ a? F1 L?d— -u 1ˉ§t = 0 3x? ?(x)§3t §§ m?? /*— [x?at; x+at] §D′ ˙ ·a' ?.ugd ? ??D′A5§ – A */Ly 5'‰)flK @2u @t2 ?a 2@2u @x2 = 0; ?1 < x < 1; t > 0; u(x;t)flflt=0 = `(x); ?1 < x < 1; @u @t flfl fl t=0 = ?(x); ?1 < x < 1 ‰′ ·(x;t)?? ??'XJu :x03t = 0 (=?Au(x;t) ??x? :(x0;0)) -u§Kd ‰ ˉ9? x0 ?at ? x ? x0 +atS'? ? ·u x0: K ? §ˇ ? S?? : ‰￡ —'-u K § ? ? : ‰ ? —'-u K ' u x0: K ? 13.5 ˉ? § 1ˉ) 115 ?p x = x0 ?at x = x0 +at? ˉ? § (Lx0: )A 'aq/§dA x = x1 ?at x = x2 +at ? ??? x1 ?at ? x ? x2 +at; x2 > x1; t ? 0 ·(x? )?m[x1;x2] K ? ' ?m[x1;x2] K ? –l J flK ?? :(x;t)(=u x:3t ) ￡ . x? = : —'-uk’”)“ u(x;t) = 12￡`(x?at)+`(x+at)?+ 12a Z x+at x?at ?(?)d?: w ? §?u(x;t)?? ?? :(x;t)§ ￡= 6ux? [x?at; x + at]￥u — '-u§ ?m ? : —'-u?’'ˇd§x? ?m[x?at; x + at] ·(x;t): 6?m' §dx? x = x1 + at9x = x2 ? at? ? n /? § ·x? ? m[x1; x2] ?‰? ? S?? : ￡§ ‰d?m[x1; x2] —'-u ?‰' (x;t): 6?m ?p ?‰)flK § o" >.^ ' O(/‘§ ovk?( ?? ^ u(x;t)flflx!§1 ! 0 ‰ u(x;t)flflx!§1k.: ‘5§ (AT?( ?? ^ ' NflK §? ^ –d`(x) ?(x) N/“5 y'`(x) ?(x)o·? 3 k S' jxjO §`(x) ?(x) ?v fl/“u0'ˇd§3k m S§u(x;t)o ·3 k S 0' lVg ‘§?￠?? u , · n z ? '§TT ·L? 3 ? ? m m S§ ::: K – O' 13.6 ˉ 116 13.6 ˉ ˉ? §￡ ·– ‰ ˙aD′ P~ˉ'?·3 ?ˉ? § ? X {z b N' F￠S b‰3ˉ?L§￥ 3 §=u oU ˉ ' 3? t§u o?U· 1 2 Z l 0 ‰ @u @t ?2 dx: u o?UV(t) –' ? ? 'E3u dx§3 T @u@x flfl fl x+dx ?T @u@x flfl fl x = T @ 2u @x2dx ^e§u3dt mS￡? (@u=@t)dt§ ? ? T @ 2u @x2 @u @tdxdt: ? u¨'§ dt mS ? u? ? dW = ?Z l 0 T @ 2u @x2 @u @tdx ? dt = ? T @u@x @u@t flfl fl l 0 ? Z l 0 T @u@x @ 2u @x@tdx ? dt: ˇ u ‰§uflflx=0 = 0; uflflx=l = 0§?–k @u @t flfl fl x=0 = 0; @u@t flfl fl x=l = 0: \= dW = " ?12 ddt Z l 0 T @u @x ?2 dx # dt: d? = uu ?UV(t) ~ §dV(t) = ?dW'e5‰t = 0 u ? U 0§Ku3t ?U= V(t) = 12 Z l 0 T @u @x ?2 dx: ?U ?U \§ u oU E(t) = 12 Z 1 ?1 ? ‰ @u @t ?2 +T @u @x ?2? dx = 12 Z 1 ?1 ? @u @t ?2 +a2 @u @x ?2? ‰dx; ?– dE(t) dt = Z 1 ?1 ?@u @t @2u @t2 +a 2@u @x @2u @x@t ? ‰dx = a2‰@u@x @u@t flfl flfl 1 ?1 + Z 1 ?1 @u @t ?@2u @t2 ?a 2@2u @x2 ? ‰dx = 0; 13.6 ˉ 117 NX oU ˉ ' A/§ N3 § §§3 m t !?te· CC ' F? ? b §ˇd§ 9 ? ￡u(x;t) 5 ' d A ·§ D′ ˙ ~?§ 6u“˙ ˉ ' ?k? {zb § § /“ 0 N5 ?’'0 N5 ==Ny3 D′ ˙a ' !￥ w §ˉ? § ) –') !m ?D′ ˉ' – § ? ˉ' 3k ?m §3?Lv m §§ ? 'm ? ?U'u·§? – l ￥ ˉ§ , ˉ 3' y3§ mD′ ˉu(x;t) = f(x?at)§§· ˉ? § @u @t +a @u @x = 0 (z) )'ˇL? “? )?( § ?L§￥? ) p? §l Ly ˉ ' F3 §(z)￥\ @2u=@x2 §~X§ @u @t +a @u @x ?fi @2u @x2 = 0; (>) a fiE ~?'XJE, ˇ? ˉ/“ ) § u(x;t) = Z 1 ?1 A(k)ei(kx?!t)dk; @o§ \ §(>)§b‰ – ¨' gS§@o – ˉ?k / “˙0!?7L v ’X“ ! = ka?ifik2; ˇd§ §(>) ) · u(x;t) = Z 1 ?1 A(k)e?fik2teik(x?at)dk: ?‘?§ §(>)?￡ ˉ?L§§E,·– ‰ ˙a D′ ˉ§, % m ?/P~(b ~?fi > 0)'du P~ˇf ˉ?kk’§ ˉ? ' P~ §ˇd§ˉ CX?m – C§ ˉ/% X m Cz' F3 §(z)￥\ @3u=@x3 §~X§ ?p? ·E?/“ )' ￡·§ ￠ ‰J 'k7L·￠?§ˇ 3t = 0 ) 7L·x …?' 13.6 ˉ 118 @u @t +a @u @x +fl @3u @x3 = 0; (~) a flE ~?'XJE, ˇ? u(x;t) = Z 1 ?1 A(k)ei(kx?!t)dk; /“ )§Kˉ?k “˙!7L v ’X“ · ! = k?a?flk2￠; ?–§ kx?!t = k￡x??a?flk2￠t?; §(~) ) · u(x;t) = Z 1 ?1 A(k)eik[x?(a?flk2)t]dk: Fˉ D′ ˙(O(‘§· D′ ˙§= ) vp = !k = a?flk2 ·k …?'?‘?§ˉ? §D′ ˙ '$ ?uk2 < a=fl k2 > a=fl ' §D′ ˇd§3 m : §ˉ ?'( ˉ? ' ? ’~) X m Cz' ? y §? ˉ ' F‰′, ?D′ ˙§ vg = d!dk = a?3flk2; ? + §§·ˉ D′ ˙§ ·U D′ ˙' + §· ˉ q A:' + vg vp F 5 A'XJ3 ‰^ e§I 3ˉ? §￥ ? 5 §~X 13.6 ˉ 119 @u @t +a(1+ u) @u @x = 0; ? n) D′ ˙ · 0 n5 k’§ ￡ k’' – y§? § ) – ? …? /“ u(x;t) = f?x?a(1+ u)t￠; ￥fE·??…?§§ ·u(x;t)3t = 0 /Gf(x)§ u(x;t)flflt=0 = f(x): du §￥ y 5 u@u@x = 12 @(u 2) @x ; ˇd§) 2 k U\5'? §7,? ) J F § \J–?) F § )7,Ly # A:( n Ly # 5?5) ? 5 § ?)–9) A5§· 5 ? K' 13.7 9D § ‰5? 120 13.7 9D § ‰5? 9D § A:· 3 ' 9D §?u . §§3 §￥?k …?? mC ? ? mC ?§ˇd§ § k m C5' ?{‘§9D L§· _ ' ?? 0 9D flK§ @u @t ?? @2u @x2 = 0: /“ ! §(>) §?–9D § ) – ? u(x;t) = Z 1 ?1 A(k)e??k2teikxdk: ?x? /Ly § u(x;t) mt P~ D9 flK ? y3 9D §§ k, A:§? ·D9 ??' ? ?.?m Aˇ g9D § @g @t ?? @2g @x2 = –(x?x 0)–(t?t0); t0 > 0: ? §￡a ] ( 3ut0 ):(8￥3 mx0 :) 9 ? ) § |'b t = 0 0 § 0'? §39 y c(=t < t0 )§0 § w,E ‰ – 0' t > t0 § –? g(x;t) = 12p?…(t?t0) exp ? ?(x?x 0)2 4?(t?t0) ? : ? (JL?§ t > t0§K?u0 ? :x§g(x;t)o 0'? · ‘§ 9 y§K l? ?§o?=a § K ' ???? D9 § ,· ￠ 'ˇd§ k39 y v m §9 D § Uv —/￡ ￠S 9D L§' ?– y?? D9 §·du3 ?9D § §L'{z 9D L§ *ˉn'3oY0 ￥ y ? !5§~X 9D L§ 9 $? !5§ * L§ !5§o?du * f(' f! f!>f￠￠￠￠￠￠) $? -E –k ˉ D′m5'? §9D §BAT?U @2u @x2 = 1 ? @u @t + 1 a2 @2u @t2 : §m k ' FXJ1 ^§ § ) Ly ˉ?L§ ? A § 1 –w? · '? ?u !? SN' 13.7 9D § ‰5? 121 FXJ1 ^§1 · ? § § )–9D ? A ' 1 y§ ( – D9 k '?l‰)flK @2x47 @x2 ? 1 ? @x47 @t ? 1 a2 @2x47 @t2 = ?–(x?x 0)–(t?t0) (t;t0 > 0;?1 < x;x0 < 1); x47flflt=0 = 0; @x47@t flfl fl t=0 = 0 (?1 < x;x0 < 1) ) x47 (x;t) = a2J0 ? a 2? p (x?x0)2 ?a2(t?t0)2 · ￡exp h ? a 2 2?(t?t 0)i·?t?t0 ? jx?x0j a · –w ' a !1 §qkx47 (x;t) ! g(x;t)' 13.8 Laplace § ‰5? 122 13.8 Laplace § ‰5? Laplace § )§ r2u(x;y) = 0; ? N …?'§·) …? f(z) = f(x+iy) = u(x;y)+iv(x;y) ￠ u(x;y)(‰J v(x;y))' 3) …? ) ? S?? :a …? § ‰ u–T: % – :…? ? f(a) = 12… I jz?aj=R f(z)d : ‰n?) …? ￠ (‰J ) ??§ u(x0;y0) = 12…R I C u(x;y)dl; C :(x?x0)2 +(y?y0)2 = R2: ˇd§ u(x;y) 6= ~?§u(x;y) § ‰ U y33 – ' ‰n 4 n · N …? ? A5' – ?n N …? Vg'XJ3? VS…? ? 3§ vn Laplace §§K?T…? VS n N …?' ‰n?un N …? E,??§ˇd§4 n E,??' n N …? ￠~ n Laplace §(3 IX￥) ? “)' ˇ Laplace §￥§? gC ? ? §?–§? ? “ ?) ‰· gC g…?' F~?(0g g“) F g…?( g g“)x; y; z F g…?§ ?)k? § 2z2 ?x2 ?y2; xz; yz; xy x2 ?y2: Fng…? ?)k § (4z2 ?x2 ?y2)x; (4z2 ?x2 ?y2)y; (2z2 ?3x2 ?3y2)z; xyz; (x2 ?y2)z; (y2 ?3x2)y (x2 ?3y2)x: F /§lg…? ?)k2l +1 '