x (t) = f(t, x(t)), t [0, T ], x(0) = x 0 R n, x k x(kh),

Transcript

1 Á Á "!$# %'&)(+*-,/.34.5 LNM OPOQMSRUTWVTOPOQXSX R XYZY\[^]QX _$àx Ob]Q[cRUR XZdO$e]$]fXeUO$gWe:h[i_$[cj]Qe X X RUek)l]QM RUme:g[ijSjTY\[i[c_$eUOPe Xn_$X]QàX e:o$ty\[ o j[ ]Q[iR:YZeUM ( p R:T`O$eq]QX _$àx Ob]Q[cRU[iY\Y\[rk`O$eUTWRUd ]P]Q[ie:OP]$[ijs[iR:et]fXuVX R:RUeUOb]Q[cjvgeh[c_$[ij]$e XX R:e:kl]fM R:meUgW[ijw_$[iTWj XaoxX_PVdao ]Q[clW]fMuVeU[cjyjTY\[i[i_PeUOb]fXz_$X]QàX e:o$[cyze:op]fx ( { e:]p]q[ij}]fx_$àx OP]$[iR:R XX jwàx `O$e:geUY\[ijOPeUde:O$[cjwkl]fM R:mjwgeUOxo `_$[ ]Qde:j]$e X~geh[c_$[ijWO$O$e:YZ[cj[c]$[iR:YnM R:R M $X\gWeUO$`_P[c]Qd eugtj kl]fm R:mjS_QX]$àX euop[iy\euop]qxn ƒ" $ ƒ ˆ UŠ fš ƒ ( %'&)(%(N rœ8 Ž a ) LNMO$OQM do$x O$O$X ]QX _$àx Ob]Q[cRU[iY\Y\[nle:[iYZX j ]fxuvx RUR:eUOb]Q[ij ge:h[i_p[ij]qeuxx RUekl]QMao RUm eug[cjsx RU`T)o$X~_$[cTj Xao"X _PV d ]Q[cl]fMuVeU[ij jty\[i[i_peuope X~_QX]$àX euoptyz[cj[c]$[ir:yzeum ( %'&)(%(%(š q œaž ˆ Ÿaƒ" fƒe â q rte:op]$[iruruxx j+ C[i_PTO$`T_PO$OPeURURUX Nd RURU[ce:]QẌ k`o$eux O$`[iRUY\[ijW[c]Q[cRUY\e M X R:`T X _bo Vd]Q[il]QM VM j x (t) = f(t, x(t)), t [, T ], x() = x R n, _QXa]QàX euop[iy\euop[i`o$e ( M e:o$oqml M jvyªtd)gwdop]$[c]$]$eue:j«pd jd OPe:]Q[cj [c]p]fm Y\eUO$O$M x, x, x... h > d jsvx R:e:]P]QTSX OP` [ir: e:]$tto ( x k x(kh), c / ± i ± ²u²' iž²ª³ Œ8µ 3Ž ŗœ²u²u z zž)µž² Ž ± z¹w² º»X eu``[ieujsk`o$eujw` [i_b]fx e:o$e:j¼x R:`T X _PV d ]Q[cl]fMuVM jrjty\[i[c_$e:j[ij _QX]$àX euoptyz[cj[c]$[ir:ynm½dj ¾ TR:[i_$e:jSgeUOxo `_$[ ]Qde:j]$e ÀBR:[ie:j[ijS[i`OP RUe:O$eUe]$]$eUj[cj x k+ = x k + h f(t k, x k ), t k = kh, k =,,,... sá VX e:l[ie:j[ij ÂNTjÃ[ v = f(t k, x k ), ºBTW]$]QX~YZ[cj[c]$[iRUYZMªVd eug XXj `e:_äpde]$]fx XWÅ v i = f(t k + c i h, x k + h i j= a ij v j ), i =,..., s, x k+ = x k + h s i= b i v i. LNM OPOQM?â U f ƒ v,..., v RUX O$`[c]QXX j?_$[c`t_$opeue:ve:o$[iob]qe4$x«twto$e s k+ j M e:op]fm ( LNM YZM«[iOPe:]$[c]fM M j kru[c[ijo$mz]fx TWRUT`` d j XW PdO$O$XZ[iOPeUe:j]"kk Y\[ij[ ]Q[iR:YnMjr`[i_P]$deUY\[c] x a ij, b i, c i. ¾ `O$ R:eUO$e:e:]P]QeUOPeUR:RU[qY\[ o j[ ]Q[iR:YZe:RUR:[qO$[»djSY½TWd ]QdX)Å ¾ O$e:YZ[c_$`eU`O$e¾ TR:[i_c šTWjÃ[ Á) ((( (((... c a... c 3 a 3 a 3... c s a s a s a s3... a s,s b b b 3... b s b s $X~`R X OPO$e:j[ij ÂNTjÃ[ ºBTW]$]QX½oÇYZ[cj[c]$[iR:YnM] Vde:g XX js[iope:]p]fmm

2 Á ( "! UC ¾ TR:[i_ šTWjÃ[ MeURURUM Ÿ œ /ˆ Q ^dj VX Ob]fXXuVX OP]$e ` [c_p]fxrut`t X Á) ºBRUX O$OPeUj[cjrÂNTjà [ p =,, 4 [crue ºBTW]P]fX x k+ = φ h (x k ) + O(h p+ ), Y\eUO$O$M φ h (x k ) dj ]$[il]fmuvm j x (t) = f(t, x(t)), x(t k ) = x k _QXa]QàX euoptsl[c]$` [ir:r M t k + h. M e:op]qm R:dàXX R:eU[ij\VeU_$lW[ieUgW[ij [iop]$euyzxx]$[ieuob]fxªo$xx]$eueujå Œ8 cž - œ Ÿ Ÿ x œaž ˆ Ÿaƒ" fƒç â x (t) = f(t, x(t)), x() = x P ƒ œ UŠ ž f ƒ ƒe â Š $ c 8 Šf ƒc ƒ œaž â ~â œ EŸ Ÿ Šn ƒ" \ Ÿ œ 8ˆ Q Ÿ C E!C L œaž " Ch p+ "¼ ƒ ƒe â C E, h W ƒ" f x(tk ) x k h p C L (elt ) œ œ t k = kh [, T ].

3 Á ( "! / u ± ²u²' iž²^³ Œ4µ4 3Ž ¼Œ ²u² Žµ4Ž² Ž ¹W² ¾ `O$ R:eUO$e:e:]P]QeUOP[c] YZ[cj[c]$[iRUYZM] do$de]$]qx TW]QT)V Xa]-l X jw`xrueu`opee `Tj ]$[il]fmuvm OPOQM~[iOPeUeUj]"kḱ Y\[ij[ V)eUX~eURUY\eUm e:]fm»[ir:erue:j[ux_$euopdeugwtj OPkOb]Q[i[cYZe:j jwd [ix OP]$eVX eo u (t) = f x (t, x(t)) u(t) ` [c_$_pdeujyzx]$_$eue:o$e:rur X-d j½]"kk e:rur:euo$[cop]$ed YZe:j X euo$x _PV da$xw $Pde:g[ij~_P[uXX R:eUdO$X]4d V Xa] e:]$o$[ie:oqx _bvdr]fxxjªo$twt)o _$eux-j[iãx]qe:e:veuope X R:T`Ta$X ( LNM YZM4Pdl]QTWT Š ƒe i UŠ ž)ž ½Y\[ij[ ]$]QM YZe:O$[cOP]fM ( { e:e:]fmhdjã[cruyzx OP]QX- MM Oxo ]fm M j "! UŠ ƒ±ƒ UŠ ZY\[ij[ ]Q[iR:YnMRUR M ( ¾ O$e:YZ[c_$`eU`O$ekl]fMRUmj x = x ¾ TWRU[i_xo"ge:O$`_$[ ]Qde:j]Qe X j]qxx x k+ = x k h x k, [ir:e x k = ( h) k x. eu`[uxq_$x]qàx e:o$t x(t) = e t x RUM l[cop]"kk Å jª`xopvx [iopoqx+jd [ix OP]$e)jd RUR X X ( d ]$]QXtjTY\[i[c_$eUjW[ij _QXa]QàX euoptw]$[i`euopeto$x Y\deUj <VXXge:]QXX j t h <. MeUjwOP]fXeUe:RUeUOPTTO½VXX]$eUe± <[c]p]fm h /5, L4deUO$XX R:]QX "! UŠ ƒ±ƒ " «¾ TR:[i_PeUjSYZ[cj[c]$[iR:YnM x k+ = x k + h f(t k+, x k+ ), O$d V [irur:[c]p]qtj X~]Q[cl]fMuVMM j x = x X j]qxx x k+ = x k h x k+ [irue x k = ( + h) k x, Y\eUàM d j OP]QXeUe:RUe± [ir:e x k àx eu`eur:r X h >. OPe:]$]QX euopge:h[i_p[ij]qeuxx R:e:kl]fM R:meUg[cj«ge:O$`_$[ ]Qde:j)o j[ceuo$o$x ]$TRU[cYZY\[ PTTW_$e]fMYnM j àx R]fX e:o$e:eujz]$eurux j]q[ie:o$e:euj ( ÀBR:[ie:j[ij sá [ir:e]fxtrut``dj X V XeUl[ceUj[cjSšTjWÃ[ ºBTW]P]fX~Y\[ij[ ]Q[iR:YnMªV deugxx js`eu_äpd e:]$]qxx v i = f(t k + c i h, x k + h s j= a ij v j ), i =,..., s, x k+ = x k + h s i= b i v i, kl]fm R:m_PklYZM j M ((( ((( c a a... a s c a a... a s ( c s a s a s... a ss b b... b s L8X TRUTW``dzd jwopeue:o½oqx YZX jrux eujw[ij `Te:jw[c`O$ R:eUO$e:e:]P]QeUOPeUR:R Ḿ Y\[ij[ ]Q[iR:YZe:RURUM ( e:eururum VX e:j X e:gdop]$e X RUX ` dr:yze:dyzx]q_peueo$ee PdRUR:deUj\k)l]QM RUm_bklYnM~_$X]Q`[iO$e _P[i`T_PO$eUeVeUO$[cOP]$e L4Tjj[ ]QTe:YZ euxªeuy\ R:eUO$e:e:]P]QeUOPe MBÂNTjÃ[oxºBTW]P]fX½Y\[ij[ ]Q[iR:YZeUM ƒ " œ fš œ C UŠ ƒx fši ƒa '$XN ž ŠfŠ a i,j o"do$x drue d VX]B "! UŠ ƒ ƒ± " ž) "! UŠ ƒ e:yz R ( ¾ TR:[i_ euy\ R ( ` )o"o$mm j]qm»x TOPObo

4 "!P! UC MeURURUMªRUd `X X RUe]HVeU_Pl[i[ ]Nd V Xa]t`[i_b]fX R:T`T X p =,, 4. "Y\ RUe:O$eUe]$]$eUj[cjn` [co$`eu WeUOP]$[iO$MM j]qmzdj X T`eU`eU_Pde:]$[c]P]QTj X)Å v = f(t k + h, xk + h v), xk+ = x k + h v. v Å js[irue:yze:jde:j]$e Pdl]fXX½kl]fM R:m)mjSOPTd_QX X j k+ Å RUR:[ ( x x k+ = x k + h f ( t k + h, (xk + x k+ ) ) LNM Ob]fM Y½TdgdOP]QX~O$[iRVe MM½Yªk monopkksy\[ij[ ]Q[iR:YnMj jweuy\[i[ij ( "Y\ RUe:O$eUe]$]$eUO$eUM YZ[cj[c]$[iRUY\e M àm k]$[c]$]qm [iopoqmno$[ct_qxxuvx k+ o"x _PV d dj»_qx]$àx euob]fxuvx x n oçgeuy\[ijwo$eud euo$[cop]qx [i M RUe:j[uXX_$eUOP[iOb]fX½kl]QM RUm _PklYZM OP]QM Ä`TWj dj [i MRUeUjW[uXX _PeUj[cj ( º^Muk]fMjjmOPOQM\]QM YZM~]$[ilgMM j f [ ]QdjWeUj YZ[cj[c]$[iRUYZM R:R M àmuk]$]fm[ij XRU`T X _bvdj X¼Pdj`Tj [i`op RUe:O$eUe]$]$eUO$[cjsYZ[cj[c]$[iRUYZM j [co$e:y ( ¾ TRU[c_$e:j -X j]fx YZXX~X W _$d`opeuyzxx]qe:d ]fx ( %'&)(%( ( ž C b ˆ Ÿaƒ" fƒç â TY\[i[c_$e:O$e XyY\[ijW[c]Q[cRUY\e Mv[cjO$e:YZYZM e:o$[ij `[i_p]qx RUT)V)TWj Obk)Ob]Q[c[iY\eUj _$[ctj Xao"X _PV d ]Q[cl]fMuVM RUR:[zO$XX g XXj àm k]p]fm YZM RURUM?O$d evxx geh[c_$[cj]$e XXRUe:kl]fMRUmjsge:O$`_$[ ]Qde:j]$e X¼$X `eu_pde:]p]fx YZX RURUX X _Pd`O$e:YnXXa]QeUd eurur:[ x k kl]fm R:m_PklYZM ( L/X _PàX OP]$[iRURUXX j X RUT`OPe-RUe:j[uXX_$eUOb]fXS]Q[ilo ]fmuvmmwå { x (t) = A(t) x(t) + g(t), t (, T ), a x() = α, b x(t ) = β, Y\eUO$O$M A(t) R n n, g(t) R n, a R p n, α R p, b R (n p) n, β R n p. ºBTj MM gwky\yz[ PdàX?]QX X TW`OP[iOPOQXwkl]fMRUm_bk)lWYnMMj H]QM O$O$Mw[ceBYZ[cj[c]$[iRUYZM j [c`op RUe:O$e:e:]$]$eUObkko g[cop]fmsdr:[½yze]fmmj¼[c]$t X $XW `do$àx euy\ RUe:O$e:e:]$]$eUOP[c]NY\[ij[ ]Q[iR:YnMa]»X j]fxuvx]b`d_$`[ux jz` [i_b]fx R:TWVTjnPd Y½TT)]fX Y\eUR:R X^VX e:l[ie:ruruxw d'vx]njw[^_p[itjxaoxx _bvd ]$[il]fmuveu[cj¼ge:o$`_$[ ]Qde:jj[ceUO$O$X½k)R:[ie:O$eUY\ eum ( qe:o$`_$[ ]Qde:g XX j ]fmynm½e:yz R:eUOPeUe:]P]Qe:O$[iR:R M+`[iO$`e: euob]q[io$mm jjwmrurum\xo$` [cru e]qttwg[ir:r X h = T/m x k = x k + h ( ) A k (xk + x k ) + g k, Y\eUO$O$M A k = A((k )h), g k = g((k { XX gxx jsrue:j[uxx_$eujw[ij kl]fm R:m_PklYZM )h). a x α I h A I h A x I h A I h A hg ( ( ( ( x ( ( hg ((( = (((. I h A m I h A m x m hg m b x m β LNM OPOQMydj ]QTj]Q[cYnXa]Qdj]fX $XvOQX YZX j V [i_p_qx j kl]fm R:me:]QM ( LNM YZM j `[i_$_pdeujwynx]$_$e:euo$e dj j XTl X YZX eujw[ij Pd ]Q[cj n(m + ) k)l]qm RUmj _$X]QàX e:o$[cyze:j[ij [irue:yze:jde:jjeur:r X d j O$Tl]Q[c[iR:RUeUOP[ij [igwtrur:euop]qxwå O(mn 3 ) R XO$`TW]Qd euy\e:]qtwop]fx ( ¾ M R:eUj[iXX _$e:o$[cruru[t_$[ctj Xao"X _PV d ]Q[cl]QMuV MRURU[ { x (t) = f(t, x(t)), t (, T ), a(x()) =, b(x(t )) =,

5 "! OQX X YZY\[BàMuk]P]fM [cj~$mruru[c[ij e:yz R:eUOPeUe:]P]Qe:OP]fMB`[iO$`e: euob]q[io$mm j]qmm\kl]fm R:m_PklYZM jå % &)(:% a(x ) = x = x + hf ( h, (x + x ) ) x = x + hf ( 3 h, (x + x ) ) ((( x m = x m + hf ( m h, (xm + x m ) ) b(x m ) =. do Obk)Ob]Q[c[iY\eUj~ge:YZ[cjO$e:d+dj n, ]QM O$O$M»d j (m+)n ]QTj]$[iYZX]Qdj]QX $X+OQX YZX jªv [i_$_$x j~kl]qm RUm e:]fm ( r[c_$`e:]qmm j u = (x, x,..., x m ) $Xq`eU_ÄPd e:]q[ ]fxxj^krurumtdr:[cvxškl]fmrum_bk)lwynmšy½tdgdo$o$x F(u) =. d O a, b $X f [ce:vm]+d RU[½R:eUj[iXX _$e:o$euxw ]fm YZMndj [c M R:eUj[iXX _$e:j[ijrk)l]qm RUm PdàX àm k]qm jjmopoqm _QXa]QàX euob]fxxj [ ]$dje:jry\[ij[ ]Q[cRUYZM RURUM ( 4Mlg[c]QMM jrx RU`T X_PVX T`O$[cOP]QX $X e]q[c_$de:g XX jå u F u (u j ) δ j = F(u j ), u j+ = u j + δ j. LNM R:RUme:j [ ]$dje:j¼y\[ij[ ]Q[iR:YnMj½Pd `XeUO$[cRURUX~X O$`[iRU[cRURUX d jr_$x]qàx e:op]fxuvx RUeUjW[uXX _PeUOP[iOP]QX½k)l]QM RUmo _PklYZM Ob]fM δ ( LNM YZM js` [c_$_pdeujyzx]$_$eue:o$ecdjª$m RUR:[i[cj j X Tl XYnX e:j[ijå j a x (x ) I h f x F u (u) = I h f x I h f x I h f x ( ( ( ( ( ( I h f m x I h f m x b x (x m ), Y\eUO$O$M f k x = f x((k )h, (xk + x k )). u Ž ' L8X _P`XOP]Q[cRURUXX j [iopeuy\[i_p`e:j % &)( ( & _P[iTj X'oxX _bvd ]$[il]fmuvmm u = v, u() = a, u(c) = b, v = µ + v, µ =, s = + v, s() =, s(c) = L, Y\eUO$O$M a =, b = 4, c = 4, L =.

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10 UC L djnopkyzy\[c]$_$e:j[ij ipd]q[ijnj M YZM dy\euj XeUOPV [i`]qd _$e:] d VX] X TW]Qd YnXXa]$]Qe:O$[cOP]Qe3`dl]QeUOPTd_$X O$O$Xª]Qde:O$eo XX j VX Ob]fXXj ( d]$]fx½jm euob]fm O$XX]QX euopeueujzd_p]$djd_pynx X RUe àx j]fx) )dj jw[qveu[irumªo$àxxr X]P]fXuVX e]qttgw[iro ]fx X j k)``mo$e:`ope (C k] Y\eUO$O$M v k = m j= sin (α k j) = m j= ( cos(α ( k j)) = ) m Re m j= wj, w = e i α k] k. w m+ =, Pd]Q[ij = w m+ = ( w) ( + w + w + + w m ). ºtd O$àX w, dj dr]fxuvx m j= wj = [irue m j= wj = $X v k = LNMa]Q[ij (m + ). d_b]qdjd _$YZXX RUe] d YZe:j X euobv[c`]$d_$e]-o$xx g XXjÅ q k = v m+ k. L Å js eu[cjeujsdy\euj XeUOQX_PV d½dj λ = 4 sin π ( ) >, Pd ]Q[cj L djs CdO$e]Qe:e:VeUOP[iOP]$eg[ je:e:]$]$e ( m+ ºtd O$àXq[igW[iRURUM A h = h +q I = L+q I, jm lg MMj [ ]$]QM Å RURUXšd j½oqx YZX]8dY\eUj X e:opv [i`]qdo A _$e]-`te:j h h L Å RUR M-$X Å j dyze:j X e:oqx _bvd ] d VX] 4 A h sin (kπh/) + q, [cruecàx eu``ed VX]- CdOPe:]Qe:e:VeUOPe X ( LNMO$OQM ]Qde:j[ij ]$d)ge:op]$to RU[cYZYZX RUR:[ % &)( W( h Ž ² ¹ Ĺ %( re:rur MªX O$`[iRU We:]QTWTg[ij h X _PV deur:r X X `RUX O$OPeUj[cj¼ÂNTjÃ[oxºBTW]P]fXªo"Y\[ij[ ]Q[cRUYZM e:yz WRUeUOPeUe]$]Qe:j[ij\` [co$`eu WeUOP]$[iO$MM j]qm ]QTWd ]$]QXX½OP]QXeUe:RU["$X½_QX]$`XeUO$T'$Xªk)l]QM RUmR:RU[ x = m x, Y\eUOPOQM m > ( re:rur XeUO$[c`O$eaOQXX] d ]$[ij]qeuxx R:e:kl]fM R:mjqge:O$`_$[c[c]$eUjB]Q[cl]QMuV Mj u`tj^_$[itwj X [ilwgd ]d VX] u() = α, u () = β? º»Muk]fḾ X W _$d`opeuyzxx]qe:d ]fx u () ( u() u( h) ) /h. re:]qmzd VX] VX OP]QXXuVX js` [i_p_$de:jyzx]q_peueuopeuj d YZe:j X euo$x _PV d ]4$XªoEV[i`]$d_$e] ( ¾<]QOPe m m oçynx]$_$e:euo$e:j L = [ dy\euj XeUOQX_PVd]4$X~d_P]$djd_PYnX X RUe] dy\eujx euobv[i`]$d_$e] ( ( dlgx~]$deuop[ij X Ob]Q[i[cjr CdRk)jWdYZe:O$d Ve:]$T`O$[iR:R X ]Qe:d ] _$[ctj X [clgdr:ru[ u () = β X _Pd`O$e:YnXX'o ) u () h( u( h) 4u( h) + 3u() = β. ºBTeUjW`Xª]QX _$`àx O$[»dj½ eururux euop[i`o$ec Cd ]Q[cj]$e XXRUe:kl]fMRUmj gweuo$`_p[c]qd euj]qe]qm RURUMªY½TT)]$]QTWT &)( [i_b]fxx~ge:o$`_$[ ]Qde:gTjSd C[i_$XX]$]$d_$e:j d YZe:j X euo$x _PV da$xš$xªoçv [i`]qd_p[ie]fx½]q[cl]qmuv Mj A h { u (x) + q u(x) = λ u(x), x (, ), u() = u() =, dy\euj XeUOQX_PVd eule:j»$xªo ÄTjW`]$eUde:leUj ( X]$` dopoqxnj M[iY\YZ[BVeU[cR M X _$e YªTTjRUX euob]fx~eug[ixx [irur:eu W]$eUOP[ij kl]qm RUm jrjty\[i[c_$euop[i`o$e8_qx]$àx euop[iy\eo O$[c`O$eEÅ e:j]q[iã _QXX R:e:kl]fM R:mjš`XTW]$]QX/$X-jWeUeUjq`T)]QO$T)]$]QT <¾ jwe:]q[/[iru[cyz[cj]yz[ ]Qldg 3o"X _Pd`O$eUY\de:j]$e (

11 Á Á %'&)( ( 8 Žµ 3 Ž a W W²u² 3 a ±µ 8 i 'Ž² C ±µc²' b $# RU`d)dj v C 4 (, ), O$e]Q[cj 8[ ]$]QM v() = v() =. º»Muk]Q[ ]fmm j [ijopeuy\ynm e:o$[cruru[n$xr]qde:o$[cruru[ g[c_$e:vxxa]fx R:RU[+X _$d `OPeUYZXX]$eUde]fX Ä`O ( O$eVT %( )% ( ) v x (x) = h v(x + h) v(x h) + O(h ), ( ) v xx (x) = h v(x + h) v(x) + v(x h) + O(h ). X R:e:]fX X j n $X X OP[c]Q[ ]fxxj h = /(n + ) ( LNM RUR:meUj v xx ( h )... v( h ) v xx ( h)... v( h) v xx (3 h) (((... v(3 h) h (((, v xx ((n )h)... v((n )h) v xx (n h) v(n h) ` d O$àX v() = v((n + )h) =. R:` ddj ]fx e<]qm YZM j X _Pd`O$eUYZXX]$eUd /$X Y\[i_xo `e:]qmm j v i = v(ih) v h = (v,..., v n ). X Ob]fXXuVX OP]$e LNM RUR:meUj krur MrdRU[o VX«X W_$d`O$e:YnX X]Qe:d ]Qde:O$eUR:RU[~g[c_$eV X X]Qde:RUR:[ªVde:g XX jz`eu_äpd e:]$]qxx (v xx ) h = (v xx (h),.. `O., v xx (nh)). ( % &)( ºBTj l X RUTX YZY\[½`d_$dOb]fXX) YZe:leUj _P[iTj X[il]Qde:leUj ]QM YnM YZX]Q_PeUe:O$e4RUeUe]$]"kk `eu_äpd e:]fxyzy\[ (v xx ) h ]QM YnMj h v h. Y\dRU[cYZY\eUOPOQX M euopoqm qeu_pe flr:[c] js_$[ctj X [clgd ] ( h,d D { X Y\de:j [ijopeuy\ynm e:o$e:ruru[ng[c_$ev X X]Qde:RUR:[tO$XX g XXjrX _$d `OPeUYZXX]$eUd (v x ) h D h,d D v h, YZe:O$OQM D h,d D = h do v Å jrr:d TW[il]Qd dj Vde:g XX jsàmuk]p]fmm\x W_$d`O$e:YnX X]Qe:d ]fx `O ( ]Q[cl]fMuVM %'&(:%( v x () =, ( ) v x () = h v( h) 4 v( h) + 3 v() + O(h ). r[c_$`e:]$o$[iyzm R:R M~X W_$d`O$e:YnX X]Qe:d½jdR:R X `O$eOQXXg XX j v n+ = (4 v 3 n v n ), PdRURUX v xx (nh) (v h n+ v n + v n ) = ( v h 3 n v 3 n). { e:euoiå qe:_$ë flru[ ] [ityzx jj»_p[itj X[ilgd eururux v() = v x () = OQXXg XX j ]$deuop[ij}g[i_pe:vxx]qx j X W_$d`O$e:YnX X]Qe:d`O$e (v xx ) h = h,d N v h, YZe:O$O$M h,d N =.. h Œ C Œ ²'Œ4 TWVTO$O$X R X OP` [ ]$]Qe:eUj YZX]Q_PeUe:O$eUj dy\eujx euo$x _PV d ] `O ( %'&)( h,d D λ k = 4 sin ( kπ ), k =,..., n. h n+ YZe:j X e:oqx _bvd ]d'vx]o$eue:ovm RUe:RURUM ( 4, ). { X YZX M]$[i[ YZX]Q_PeUeUOPeUR:RU[ Ãd_PeUje:j R X TWO$[i[cRURUXN$X½]fX _P`e:OP]QX [ij _$[ctj X e:op]$[i[c]š[c_$eu`op[i[ij h ( h,d N. [co$euy\[i_p`eu`o$e B[i_PO$l)o

12 UC ŗ W u Œ8 ²u² Ž ±µžµv² 4 Œ4 R:` ddj¼jk] = (, ) (, ), v C (), $XZ]$d ]Q[cTW]$]QX ` ddj v _$[ctj X [clgd ] v(x) =, x. ºBTW]Q[cj [ig[crur M) VX RUe]fXXj¼jk] n, n N $X X OP[c]Q[ ]fxxj r[c_$`e:]qmm j h = /(n + ), h = /(n + ). v(ih, jh ) Å j½x _$d `OPeUYZXX]$eUd ]QX Å RURUM (' MYnMNdj½Y½TW`XuVXt`e:_ÄPde]$]fX XNYZX]$_$eUe:O$e:eUj v (i,j) v (,)... v (,n ) V h = ((( ((( R n n. v (n,)... v (n,n ) º»Muk]P]fM YZM R:R Mª[cg[iR:R M½dR:[cVXX~X W_$d`O$e:YnX X]Qe:d ]fxªop[iàm Å jr[ ]$]fm Å j O$TTWjj X OPOQXZO$XX g X X j x x v x x (ih, jh ) (v h (i+,j) v (i,j) + v (i,j) ), v x x (ih, jh ) (v h (i,j+) v (i,j) + v (i,j ) ), ]P]fX [cj ltwdyze:d)d j«_$[ctj X [clgd ]i j M YZMnVde:g XX j YZX]Q_PeUe:O$eU`e:[iRU[cRURUM `eu_pde:]p]fxxªyªtàx VX Ob]Qe i n, j n. (V x x ) h h V h, (V x x ) h V h T h. LNM OPOQMNYZdR:[iY\YnXa]d [c_qxx]p]qd_pe:]d'vx] šo oç]"kk e:o$eum ( MeUjBOQX X g XX jbàx `OPeUge:YZ[cjO$e:deUOP[ijB X RUX i[cj d C[i_$XX]$]$d_$e:j ge:o$`_$[c[c]$]$ev[c_$o$e:då v = v x x + v x x r[c_$`e:]qmm j ( v) h h,h (V h ) = h V h + V h T h. v h = T (V h ) Å R:R X V [i`]qd _$e X v h = T (V h ) = [ T v (,), v (,),..., v (n,), v (,),..., v (n,n )] R n n. Å O$O$XZdjSOPeUeUO YZX]$_$eUe:O$e:j O$X _QX ``[i[c]t MMRURU[c``MeUj ( L8d euo$e:jsoqx jd[ijå v h (v h ) i+(j )n = v (i,j). V h RU`d)d ] V R n n $X v = T (V ) R n n `TW]$[ij [cg[ir:r M^$X p OP[c]$[c]fX X j¼r:euj[ixx _$e:`twvx TO L : R n n R n n A R n n, B R n n. L(V ) = AV B T. Ž) Zâ aš ƒç â " $ œaž â ž Š L R n n R n n Ši ~ UŠfŠi v (B A) v, B A Ÿ œ ^ QŸ œ "«ƒ ž) Ÿ B, A B, A... B,n A B B A =, A B, A... B,n A Rn n n n. Ÿa UŠ " Ši Ÿ Ÿ B n,a B n,a... B n,n A T (AV B T ) = B A T (V ). j j Muk]Q[c]P]fMuVMW [c]p]fm (AV B T ) i,j = ( (B A) v ) i+(j )n, i =,..., n, j =,..., n. X OP[iY\YnX R]fXª WTdRU[cjrO$XX g X X j n n (AV B T ) i,j = A i,k (V B T ) k,j = k= k= A i,k n n n V k,l (B T ) l,j = A i,k V k,l B j,l, l= k= l=

13 $X~deU`[uX R]fX½ TWdRU[cj ( (B A) v ) n i+(j )n = l= n B j,l (A = B j,l (AV ) i,l = l= [ v+(l )n ] v ) +(l )n. v i n +(l )n n n B j,l A i,k V k,l, Pd ]$[ij j M YZM½d'VX]Nkl]fM O$TTW_$[c] ( LNM]$[ijzV [i`]qd _$[ie:leuj O$d V[cRURU[ ]$]$Tj XS`X`OPeUge:YZ[cjO$e:deUjW[ij X R X i[ij d C[i_$XX]$]$d_$e8Vde:g XX j `eu_bo Pde]$]fX XWÅ v h h v h = ( ) I n h + h I n vh. ºBTj h = h = h OQXXg XX j [ 4 ] I 4 4 [ 4 ] 4 I I 4 h = 4 [ 4 ] ( h 4 ( (. I ( 4 ( ( ( ( ( I l= k= I [ ] do v Å R:R M»dj Å j _P[iTjX RURUX»]$deUOP[ijRUX euop[c] _$[ctj X [clgd ]i WY½TT)]$]QTWT VX Ob]fXXuVX Ob]Qe ( ¾ O$e:YZ[c_$`eU`O$e _$[ctj X [clgde:rurux h OQX X g XX j Ž ² ¹ Ĺ v(, x ) = v x (, x ) = v x (x, ) = v x (x, ) = h = I n h,d N + h,n N I n. %( ¾<]QOPe]Qde:O$[cj ` [i_b]fx R:TWVTjrX _Pd`O$eUYZXX]$eUd^_$[cTj X [clgdr:ru[ v() + α v x () =. re:rur XeUO$[cj k`o$eugweuy\[ijopeude:o$[ij Å j O$XX]i `TWj¼X R:`T MM js_p[itj X[il]QdZdj h v() = ( ºBeU_Pde:]QXªX T`e àx `O$e:geUY\[ijWO$eUd euj[cj h, `Tj _$[ctj X [clgd ]td VX] v(, x ) = v x (, x ) = v x (x, ) = v x (x, ) = $X h = h. reur:r X e:o$[c`opeop[»]$tru[c[ `Tj h h ( Muk]fM) W[c]P]fMªYZX]$_$eUe:O$e:j B A dyze:j X e:oqx _bvd ] d VX] Y½Td]QdX λ µ, Y\eUO$O$M λ dj A Å j $X µ dj B Å jrdy\euj X e:oqx _bvd ( de:]šdr:[c]$]qxxw [ ]$]fm\o$[c`m A Å R:R X~[c]P]fM B Å R:R M~dj¼gWeUY\[ijOPeUdjO$X V[c_$_$X jn[c_$euoptt_pe X»dYZe:j X e:oqx _bvda$x ( re:]$àm^d VX] àx `O$eUgWeUY\[ijOPeUde:O$[ij Å jndy\euj XeUOQX_PVd]u `Tj `XeU``e X R:R X d j qe:_$ë flru[ ] j _$[itwj X [il]$d h

14 P! UC %'&)( (N u Œ8 ²u² Ž µ4žµ ² Žµ² ± ² ¹ L/X _PàX OP]$[iRURUXX j qe:_$ë flru[ ] js]$[il]fmuvmm % &)(:% { u(x) = a(x) u(x) + g(x), u(x) = f(x), x x Y\eUO$O$M = (, ) (, ) $X a dj $Xa]Q`TWVXB$X\[ieoÇj[iÃX]Qe:e:VeUj[cj O$[c`M $X]Q`TWVeUX ( X RUe]fXXj ]fx X O f, g n, n N $X~XO$[c]$[c]QXX j h = /(n + ) h = /(n + ). r[c_$`e:]qmm j u(ih, jh ) Å j X W_$d`O$e:YnX X]Qe:d ]fx Å RUR M~$X u (i,j) a (i,j) = a(ih, jh ) O$[càM g (i,j) = g(ih, jh ). º»e:_ÄPde]Q[ ]fxx j j M YZM~ e:]$`eu`o$e o HV[i`]$d_$[ceU`O$e n n u h, a h, g h. do\_$[ctj X [clgd ]nd'vx]\ld YZdà [i[ijweuo$[ ] geuop`_p[c]$de:]$tv X RUX i[cj f = d [c_qxx]p]qd_pe OQXXg XX j `TW]Q[cj¼[cjj[ijÅ $X~geUOP`_$[c]$deUgWT`O$ekl]fM R:m`O$eEÅ h = I n h + h I n h u h = A h u h + g h, Y\eUO$O$M~YnX]$_$e:eUO$e djrgwe X Ãd j XX R:eUj[cj»$X~O$[cj¼geUX Ãdj X X RUeCY½TdgdOb]QTT V[c`]Qd_PeUOP]QX A h a h. ¾ M ld YZdà [i[ijweuj[cj^_p[itjx [il]qdndj»ltdy\eud e:]fxuvxhàmuk]q[ ]"k)opoqmšge:h[i_p[ijopo$e X _$d `OPeUYZXX]$eUdOPOQX ( ºtdY~o Cdj[ij]$[ie:]P]fX e:js`e:_äpde]q[ ]$]QTWj X½O$[»dj ( ) ( ) h u(i+,j) u (i,j) + u (i,j) + h u(i,j+) u (i,j) + u (i,j ) = a(i,j) u (i,j) + g (i,j), i n, j n. ºBTj?]fM OPOQM i = ]fx e ]QX e _$[ctj Xao"X _PV da$x n ( ºBeU_Pde:]$[c]QXX j j MeUg[cj X euàx R:RU[ j = ]fx e n, ]fx _bve:]fx X j Yªk mo % ( u (,j) = f(, jh ), u (n +,j) = f(, jh ), u (i,) = f(ih, ), u (i,n +) = f(ih, ). ºB[i_$M]fMMj j M euob]fm O$XX]QX VX]n]QTjjW[c]QT)] ]Q[i_PYZe]nkl]fM R:mj de:` [uxruru[ Td RU[iR:RU[SV[c`]$d_$e:`O$e l X _Pde:]$TOP]$[il]fMuVM LNMRURUm euj ge:o$`_$[ ]Qde]QT kl]fm R:m OQXX gxx j Y½Td ]$d)dj φ h `O ( % &)(:% % ( h A h ) u h = g h + φ h. LNM YZM j `[i_$_pdeujwynx]$_$e:euo$e djnj[iãxa]qeueveuo$[cop]$ecgw[ je:e:]p]qe `O ( ]Q[il]QM VM dr:[iyzx O$O$X~k`O$e:àM O$e]$]Q[ceUj[cjS_QX]$`XeUO$T h A h ( Pd]Q[ijZ]fM R:R M»dj u Ž ' R:` ddjs_$[itwj X [ilwgd ]rldy\dã[i[cjeuop[c]i a =. $X g(x) =, `Tj x p. $XzY½TWTW]Q[cj g(x) =, p = (.,.6) LNM YZMz`TWVXX RUM Y\ m ]$eurux j ]fxoqx Xeo jda$x`xtw]qtyzxx\j[ir:eumjry½td ]$deuop[io$o$xzr:[cvko$oqm) PdàXZdj«X O$[ ]Q[c]P]QT l[crurux jrr:[cvkj MM R:RU[ $XZ_P[iTjX] e:g[c]qmm j¼jwdrurux R M Y\ Cm ]Qe:RUX OPOQX ( M Y\ m VeU_b]fX TWOšR:[cVkj¼ e:jj X R]fX½kYZ M _$e:op]qmmj djrv [i_p_qx jjwdrur:euj[cj R MYZ Cm ]Qe:R X [c_$ddj ({ [ct_qx X VX OPOQX\`T)V XO$OQX dj jtyz[c[i_peuj[cjr_$x]qàx e:o$t )`TWj n = n = 9 (

15 ¾ O$eUY\[i_P`eUj.6.4. %'&)( jtwyz[c[i_$e:j[ij _QXa]QàX euopt ¾ O$eUY\[i_P`eUj.6.4. %'&)( jtwyz[c[i_$e:j[ij _QXa]QàX euopt.8

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20 Ž ² ¹ Ĺ UC %( eue]$]qd Ã_QX gweu[ij]p]qeuy\[ijw[c]q[cruyzm O$O$Mn`eU_Pde:]$[c]$]$eUe:j α k = r k / p k, A p k [ie:`m `TW]Q[cj?X eo [iy\y\euj R XO$` [ ]$]$eueuj α k = p k, r k / p k, A p k. Muk]fMW 3[ ]$]fm %'&(:% Å OP]QXZO$[cT_QX XW 3[c]$]QM j M YZM~X j]fxuvx]šoqxynx j Å j ( LNM RURUMª]fXuVX RURUXªOQMMOP]Q[ ]fmmj k`o$eo$e:oqm]$trud j RUX O$`[iY\eUj[cj ( α k ( Muk]fM) [c]$]qm~rueue]$]$dã_qxgeu[cj]p]qe:yz[cj[c]$[iruyzm RR:[ M]Q[c[ Å Pd`d φ(x k ) = ]fxe φ(x k+ ) < φ(x k ).

21 ((( # % ( /=8A C>9<;4.=87)><}I85KaF/D4.,<5.5 ( ( ( ( LNM OPOQḾ RUT)V)TWO$OQXn]QX _$àx Ob]Q[cRU[iY\Y\[^ X _QXCdR:eUO$[cjrkl]fM R:mj¼gWeUO$`_P[c]Qd euj]qeux º»TWj«dO$e]$]QX euopge:h[i_p[ij]qeuxxao RUekl]QM RUm jry½twtw]$]$tapeuj X d VX]BX e:àx+$x\ X eu`àxw 3`TW]Q[cj [iopeuy\[i_$`e:`ope8rum Y\ mo$xzxx R]Qd kl]qm RUm eururumw dj RUTWdjjdR:RUe:OP]fX) C[ ]$]QM X $X jzoptl]q[c[ijzkl]fmrumr:r MZd j X R:`T[ilgWd ] ]Qe:R X jjw[ªl[ ]Q` [crurum t = `Tj ]fx X O½ X e:àx j O$TWl]$[i[cj OPeURURUM d j ]fxuvx RUR:eUO$[cOP]$e _P[iTjX [ilgwd ]uå<e:ruy\eumj àmuk]p]fmuk]"kyze:j[ij ]QX _$àx Ob]Q[iR]fXao VX j X RUTW[i[ij _$[ctj X R:R X do»ge:o$`_$[ ]Qde:YZY\[ kl]fm R:mj Pde:RURUX `euj¼gw[i_$evxx]fxj X _Pd`O$eUY\de:jj[ie:RURUX [ijwo$eujz X e:`àx YªTTW]$]$TaPeU[cj O$Tl]Q[c[ij d]$]fx[ij\ltwdyze:d)d j\_p[itjx [ilgwd ]u i$m R P[iRUR:[4$MM»ge:h[i_P[ij]QeUXX R:e:kl)o ]fmrumªklg[cjsy½ttw]p]qta$x j ±X'$X j HO$Tl]$[i[ijª$X½OPeURUR:[šX RU`T X _bvd ]i W[cRUe3]fXuVX RUR:eUO$[cjngeh[c_$[cj]$e XXRUe:kl]fMRUmj X R:`TX _PV d ]Q[cl]QMuV M) Pdldj Vd euy\yz[+opd V[iR]fXX [iopeuy ÂNTjÃ[oxºBTW]$]QX½]fX ey\djeux O$`[iR:YZ[cj[c]$[iRUY\e M { d V[cRUR:[c]$]QX [iopoqxngwe:h[i_$[cjo$opeuy\[ij[ ]Q[cRUYZMM~R M Y\ m kl]fm R:m)m j X eu`àx ge:o$`_$[ ]Qde:j]$e O$TaPTWT«OQX YZXX j ]fxao XXj `Te:j [crur:eu W]$eUO$[cRUR:[Skl]fMRUmR:RU[ ( p $X jsoptl]q[i[cj X_QXCdRUe:O$[ ] kl]qm RUm]Sd'VX] œ œ ƒe Å-jeUe:O$O$M [iopeue:j]"kk jwd [ix OP]$e/VX euy\[ijw[cve X `dyz Cdj[cj]$]Q["$X ( LNM]$[ij OP]QXeUe:RUe:O$TTOb]fX _PàX OP]$[iRUT)] $X euy\ R:eUO$e:e:]P]Qeo O$[ ]šyz[cj[c]$[iruyzm] d O$de]$]fXTW]QTWVX]Nd RU[i[cRUR:eUO$e:`O$e ( L8X _$àx Ob]Q[iR:R XXjSX RUT`OPek)`OPeUTR:d ]$]$[ieuob]fx½rum YZ Cm kl]qm RUmM % (:% u t (x, t) = α u xx (x, t) + g(x, t), < x <, u(, t) = u(, t) =, < t < T, u(x, ) = f(x). qe:o$`_$[ ]Qde:g XX jr[ijopeuj¼ X euàx j x OPTl]Q[i[cj `TW]Q[cj X eu[cyzy\euj ( X R:e:]fX X jso$eue:o n $X\X OP[c]Q[ ]fxxj h = /(n + ). RU`d)d j u i (t) X W_$d`O$e:YnX X]Qe:d u(ih, t) Å RUR:[ $X~YZ[c_$`e:]QMM j u (t) g(h, t) f(h) u h (t) =, g h (t) = (((, f h = (((. u n (t) g(nh, t) f(nh) { XXg XX js Td RUeU`OPegeUOP`_$[c]$de:]$T ]Q[il]QM VM % ( Y\eUO$O$M { u h (t) = α hu h (t) + g h (t), t [, T ] u h () = f h, h = h % ( dj ]fxuvx RUR:eUO$[cj}ge:h[i_P[ij]QeUXX RUekl]QM RUm OPkOP]$[i[iY\eUjwX R:`T X _PV d ]Q[cl]fMuVMW Pd ldj V deugxx jyopd'v [ir o ]fx Xz`T_$OPO$e:RUR X zd We:]$]$Ta$X YZ[cj[c]$[iR:YZeUM ( X RUe]fXXjrPd`eUjyXeUàXaoxXO$` [cr δ $XzY\[i_P`e]fMMj u k Å RUR X Å j X W _$d`opeuyzxx]qe:d ]fx ( dobàmuk]q[ ]fmm j ž) " n ƒ" \ O$XX g XXj ]QMuk)OPeUj ge:o$`_$[c[ o h u ]Qe:`O$e]Q[cl]fMuVM `O$e h (kδ) { u k+ h = u k h + δ (α hu k h + gk h ) u h = f h. do g =, jweueuj RUM Y\ m kl]fm R:mjy_QXa]QàX euoptsr M lw[iop]"kk jwdruruxxw `Tj t ( /M]$[i[i`mvO$X YnX jty\[i[i_peuop[irur:[b_qx]$àx euoptrur:[ "jgt`]$eudr:r XªO$XX g XXj Ä`TWj α = u k h = (I + δ h ) k f h..

22 ( Á Á rte:op]$[c]fx X j [c]p]fm UC lim k Bk = λ < àx eu`eur:r X λ σ(b). k] 4d ]Q[cj]$e XXRUe:kl]fMRUmj geuop`_$[c]$deujwjeuj kl]q[ kg[iopoqm ]$d)gw[c]$]$eue:j [c]$]qm B = I + δ h. σ( h ) ( 4/h, ) <Pd ]$[ij σ(b) ( 4δ/h (+{ ]QXe:eURUe:O$TTWO$[ilWgd`O$e»OQX X g XX j OPeUe:O, ) δ < h / /YZe:OP]fM OP[iT_$XXW [c]p]fm δ Å jwdjwdr:]qxuv X«lkVeUj eu[cje eu[cjeur:r M Å j X _PV deur:r X ( L4deUO$XX Ro ]fx R M Y\ Cm'kl]fMRUmj _QXa]QàX euopt ]$TRU[c[nX $X j `TR:T[iOPOQXrkl M¼OPeUR:[uM Y\YnM`OPeE Pd ]Q[cjSPdO^OP]QXeUe:RUe:O$TTOPdj)o h Ã[cRUY\e X+[ie d RUeUOPeE avde]fx e:o$eue:j½d ]P]fXXBlkV)e:j`eUjZ e:]$`eumbx euàxao"x O$`[iR:[ie:]QX ( LNMYnM j ]fx`euxbr M Y\ m kl]fm R:mRUR:[ $XªkR:[i[ijWOQM~ X _$Xd RUeUOPeUR:RU[ ]Q[cl]QMuV)e:RUR:[ àmuk]q[ ]fmm j X e:àx geuop`_$[c]$deuj]$eueuj "! UŠ ƒ ƒ± UŠ n ƒx ~ ( "Y\ RUe:O$eUe]$]$eUj[cj ¾ TRU[c_$e:j Y\[ijW[c]Q[cRUYZM~X j]fxx~ge:o$`_$[c[c]$euj ]Q[cl]QMuV Mj LNM OPOQMBPd TgTW]QXX jªpdàx { u k+ h = u k h + δ (α hu k+ h + g k+ h ) u h = f h. X O$`[iRU[cRURUX½_QX]$`XeUO$[cYnX X j RUe:j[uXX_$eUjW[ijnkl]fM R:m_PklYZMW (I δα h ) u k+ h = u k h + δg k+ h, Pd ]$[ijr]"k myzmm _QM\X OP` [ir]fx\` dlgw[ij«djr X R PdjrOPTT_P[iY\ e`tweuj«¾ TR:[i_PeUj YZ[cj[c]$[iRUYZM R:R M ( L8de:OQXXR:]fX Å ( ( ( t j O$TWl]$[i[cj V X`e:d` [c_p]$deuy\euop[io$o$xs]qx X TW`OP[iOPOQX _PeUe]$]fM Mn]Q[ilWg M¼YZX]Q_PeUeUOPeUj l X'Pd]Q[iR:YnX?VX euj ` [c_$_$x jw$x}`muk]$]qmm OPe:]QM PdàXwX O$`[iRU[cRURUX "YZ WRUeUOPeUe]$]Qe:O$[cRURU[ Y\[ij[ ]Q[iR:YnMRURU[¼OP]fX'o I δα h e:eurue:o$ttwo [iebkru[i[cjoqmvx eul[ctw]fxwx OP` [ir: e:]$ttop_qx'pde]qt`ope X ¾ O$e:Y e:yz WRUeUOPeUe]$]Qe:O$[cRUR M ¾ TR:[i_$e:RURUX `Tj g =, α = $X u k h = (I δ h ) k f h σ((i δ h ) ) = {( δλ) λ σ( h )} (, ) àx e:`eururux δ >. L8X VX R:RUe:O$[iOb]Qe XO$` [cru e]qttwo V de drurux+jeue:j X R Pd j\optt_p[iy\ ee [ ]$]fm»` d ` dj XeUOP]"k myzmm _QMH$MM»eUY\ R:eUO$e:e:]bo ]Qe:O$[cRUR M YZ[cj[c]$[iRUYZM R:R M [c`o$ R:eUO$e:e:]P]QeUOb]fMª WeU[ijW[iY\YnM `OPe ( top[ie:jnàmuk]$[c]p]"kny\[ij[ ]Q[cRUYZM^ X_QXCdRUe:OP]$[ij~kl]QM RUm eug[cjnx e:àx geuop`_$[c]$deuj]$eueuj~d jnj` ( "! UŠ ƒ±ƒ " œ fš œ C UŠ ƒ" fši ƒ $Pd ]QXN]QM O$O$Mškl]Q[ k)gw[io$o$mb`t)]qo$t)]fxx jªyªkmo < P œ + fÿ Š Ÿ ge:o$`_$[ ]Qde:jje:`OPe ( { [^X j]fx X gweuo$`_p[i[c]$eujs]q[cl]qmuv Mj { ( % ( I δ α ) h u k+ h = ( I + δ α h) u k h + δ g h (t k+/ ) u h = f h, Y\eUO$O$M t k+/ = (k + ) δ. LNM YnM~dj Yªk mo δ Å j O$Tl]$[i[ijr` [c_p]fxrut`t X u Ž ' RU`d)d j g(x, t) =, `Tj x <.4 $X g(x, t) =, `Tj $X x >.4 f =. º»T)V Xj kr M _Pe:VeUR:R MZd jzjty\[i[c_$eujw[ij _$X]QàX e:o$t `TWj n = 9 V XO$[iY\YZX RURUX ¾ TWRU[i_PeUj Y\[ij[ ]Q[iR:YnMRUR M~XO$` [cru e]qttwg[ir:r X δ =. d eu` [ix RURUX~OQX YZXW `Tj δ =.3. M lwg MM j C[c]P]fM [i`op RUe:O$eUe]$]$eUO$[cjv¾ TR:[i_PeUj X OP` [cru e]qtto δ =.3 kr:e:]$]qmmnpd OP]QXeUe:RUe:O$TTOP_QX'$Xj ÄjTYZ[c[i_PeUj[cj _QXa]QàX euopt\x RUàXX+VM _QMlg[iR:R M ( p R X _Pe:VeUR:R Mtd jzo$x YnX+]$[il]fMuVM _$X]QàX e:op]qt e:yz R:eUOPeUe:]P]Qe:O$[iR:RUMN¾ TR:[i_$e:RURUẌ $X euy\ R:eUO$e:e:]P]QeUOP[iR:R MB` [co$`eu e:op]$[ioqm M jjmr:r M\X O$`[iR: e:]$ttg[crur X δ =.. re:]qmm jsop]qxe:eurue:o$ttwo$djã [iruy\e X [ie d RU[ ( J8J>`N>H:*:<D<4)I:<:<I V%D IEJIK8)D$>H>H:*D<5768J9/:<;H:<:<I x (t) = f(t, x(t)) =G34"8J> x k+ = x k + δf(t k+/, (x k+ + x k )/ ), 4"I?S9H9HE)D*8)IP:*:$9H9HK B>H>`N>H:*:$> M%D*8J939WEJ>/8)B>/D$4G8>%Y x k+

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