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

Á Á "!$# %'&)(+*-,/.34.5 64687)5:9<;4.=87)><?4@4ABC3D5E5F/F/686HG4I/5JJ3D38K'K'5AB38.37A+= 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

Á ( "! UC ¾ TR:[i_ šTWjÃ[ MeURURUM Ÿ œ /ˆ Q ^dj VX Ob]fXXuVX OP]$e ` [c_p]fxrut`t X Á) 6 6 6 6 ºBRUX O$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 ].

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 3 6 + 3 6 4 + 3 4 6 e:yz R ( ¾ TR:[i_ euy\ R ( ` )o"o$mm j]qm»x TOPObo 3 4 6 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 )) =,

"! 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 =.

UC X R:e:]fX X j\x O$`[iRU We:]QTWTg[i`OPe `T X _bv XT`O$[i`OPe8 X_QXXC[iR:e h =. $X XRo 4 u(t) = a + (b a)t/c + L c t(t c). c [ ]Qdj O$TW [cj[i[ _$[ceu X X OP]$e$X½V)e:_$l[c[ij jd_pyze`e:[i_p_$d`o$e]$]qx euj dj %( )% % W( %% W( ( [ o ( )% [ o %'& LNM YZM~OQXX]$eUe:jSX euàxx j OP[iT_$XXuVeURURUX o" Ma]Q`eUR:R MWÅ 3.5.5.5 3 3.5 4!!"# $ $% & %!' "() & *"#$% $ & %!+,-' $.), %/-"# $ ' "- ")3-" #)" "( 4"# $ $#4" 34"#!$# %)%-5 76 8 9 $%;:$ < ' %4" %=$%$ $( & -# & &;>& %) 4" ' "( >@? BAC/-" > > 9.) 5D% ' "($ E3 &-' " F G6 > H@? IA ' "($)%J3 &-' $ %G/$ ", )!F? KMLONPK QRNTSKQRNVU)KMLWXNPK)Y WZN[.K!S\@NVK.M]^A_WRà*F^N b Kc$!' #^ACQRde!fMLF^Nhg KZACU-+S)F\iAjY] S)FkN b ALOd`_FDK) fzal4+f]\)s f g!]? ]^AmM+ S)FkN = "# & $ b AjYXd`_FDKZAn4+!F\S)+Dg!] S f)y]g!]don D &, & M# $ b ACpRd`_FDK)Y!]g!] ' %$#^ALqdr!fMLF^N b AmQod`_FDKU-]\)SkN S ' ' #KXs 9 t )) os N 3- ' RAnod b ALqd`_Fkd sr4+ +ksuf^nt/ <% ' <onv. ' / ' %wn x Kc$!' #^ACQ]!fQodyLF^N x & K!#(3 # $zacc$!' #kamq]!f Q-F F^N {/)K)$ 9 $zamq-fknv%4"(-#k}zn >!' KMLk`~%4"(-#d :$ < ' % "$ M" ' x AL^`mY!FDKo b ALqdyLF)+BN b AmQRdyLF RN D & $.) '6 x & ALkǹYZdyL^`lQ-FDKo ulww=wznƒwww7l RN # $%G/$ ", x AmQ](!f po`rq-]!fq-fko b ALOde!fMLF)+ N b AmQode!fMLF)+U RN ' 3 3 & $.) '6 x & AmQ-]!f po`lq]!fqodnq]!f4l^`rq] fqmfdkw uljww=wznƒwww7l RN # $%G/$ ", >!'G6 KML^`~Id 3K)W?m ]^A b A!`d 6 fmlfdf b A`d 6 F F^N $#M"(34"#$ x AmQ] 6 +L^`rQ] 6 f)y!fk b A`d 6 fmlf + b A`d 6 F +.M] > An34F^N /-" > > 9.) 5D% #$ K)W?n ](.M] > H An34F^N G6 G#$% x & AmQ] 6 +L^`rQ-] 6 f)yzdmq] 6 +Dpw`rQ] 6 f Q-FKo + {/M+ dh{/-+ on /$ ", $%/ <K x &Mˆ x N "# %Š:$ < ' %Š!#$ u u u

>!'G6 KML^`Ar!fMLFkd b A`d 6 FDK b A`d 6 F +(<oamq] 6 +po`rq] 6 F^N $%/ G6 E!$./ %J '6 & #? 3- ' oanod b A Lqd`_F^duS ' ' #^AnMF FkNT/ <% ' <on % ' @A x F!!" )%, ".$ R? $%/onp. ' / '> > =>w? : & $#)"-$ M" %; &!" # $%;/-" > >)9.) )5% ' "($)E3 &-' " >& %4SM" ' %G<K > A H F #g K#g RA L(f H AjYF Y!F^N <Ko H AjYF N H ACpF ] #g NƒWNh# g8 RN => Hw? : & $#)"-$ M" %; &!" # $% > `~%G/$ ",!D " "#)" >& %4SM" ' % K > H A H F #g K#g RA L(f H AjYF Y!F^N #g H K H AjY!F\)#gXN Ko VW L W W N W H ACpF ] #g H # g W N W W W W N W #g H W WE RN ºtd _$` [ix Y\YnX j ` [c_p]qx RUTWVTj kl]qm RUm j V de)pd` dz`eu_äpd e:]$]qxx½[ijwo$euy\yzm euop[ij ` [c_p]qx RUTWVTjrOPkOP]$[i[cYZe:`OPe $X}O$d V[cR:]fX X [igw[irurum?[io$e]q[ ]$]"kmwr M lw[iop]"ky\euob]fx X X?]QX e^x _Pd`O$e:YZde:g X O$TWd_QXXj `d_$`[ux Y\ e X`e:j g[c_$e:vxxa]$]qd'$x½[i_$d]qtopdoqx YZMM _PeU[cjrXuVTR:R X½O$[cT_QX X VXXj ]QX XX j ( L8X _$àx Ob]Q[iR:R XXj ]Q[il]QM VMM % &)( L8X kr:d_$e:j RUX TOP[i[iOb]fX { u (t) + p u (t) + q u(t) = f(t), t [, T ] u() = α, u(t ) = β. u(t + h) = u(t) + u (t)h + u (t)h + 6 u (t)h 3 + O(h 4 ) u(t h) = u(t) u (t)h + u (t)h 6 u (t)h 3 + O(h 4 ). X O$`[iYZX RURUX½ TdR:e:]P]fX e:jnkl]$[i[cj¼oqx X g XX j % &)( u (t) = (u(t h) u(t) + u(t + h)) + O(h ), h $X~ TdR:e:]P]fX e:jnvm l[ij]qm YnMRUR M % &)( & u (t) = (u(t + h) u(t h)) + h O(h ). X [ ]fxx jzv MRUe e:op]$[ieuopeue:j [, T ] t j = jh, h = T/(m + ), j =,,..., m +. º^Muk]$]QM [ij j M e]fm O(h ) o"x _Pd`O$eUYZXX]$eUde]fX\g[i_Pe:VXX]$deUR:RU[½O$XX g XXj % &)( Å j _QXa]QàX euoptrur:[ u jty\[i[c_$eujw[ij

( Y!F] X W_$d`O$e:YnX X]Qe:d u j u(t j ) kl]fm R:m_PklYZM OP]QM (u h u + u ) + p UC ( u h + u ) + q u = f(t ) (u h u + u 3 ) + p ( u h + u 3 ) + q u = f(t ) ((( (u h m u m + u m+ ) + p ( u h m + u m+ ) + q u m = f(t m ) u = α, u m+ = β, [ir:e m m YnX]$_$e:eUO$e:Y½TdgdOPOQX ( h + p D h + q I) u h = b, YZe:O$OQM % &)( h = h......... ÂN[iTj X[ilgd]td j ]QM O$O$M+Pd, D h = h...... ltdy\eude]qt^$x½]$tjj[ ]$]QT do$x\opeueu_p_$[ ]$]"k u Œ4 ²u² Ž µ4žµ} ² Žµ²' Ä C² ¹ d jry½twd ]QdX, b = % &)( { u (x) + q u(x) = f(x), x (, ), u() = u() =, Y\eUO$O$M q qe:o$`_$[c[c]p]qe]q[cl]fmuvm V deug X X jsjk]t`eu_äpd e:]$]qxx % &)( A h u h = f h, Y\eUO$O$M A h = h + q I, $X -)euopoqx\]qm YZM j YnXa]Q_$e:eUOPeUj O$XX~`Ma]Q[cVM Ob]Qe]$[il]"kM\O$[iTW_QXXuVX RURUXWÅ f h = (f(h),..., f(mh)). KZAm/M" t A A_Yf g!](. ' %!$#^Ar dylf F)+?)?? /-" t A ' %!$#^Al*+LqdyLF^dyLF +D/-" t A ' %$#karq+lodylf^d +LF F\. b oçv[c`]qd_peueuj ( YXN f(t )+α( p h h ) f(t ) f(t 3 ). f(t m) β( p h + h ). u Ž ' R:` ddj f(x) = x 7 x $X q =. ºBTWVX O$O$X d jyo$[c`m ]fx _P`àX _QXa]QàX euopt [c]p]fmªjty\[i[c_$e:j[ij _$X]QàX e:o$t ( TdYZX]fXXj [c]$]qm½x OP` [ir:[ij h TWdRUe:eUj]QT[cO$O$X^VeU_Pl[qY\[ij[c[ jde:jsj[ir $M O$dO$XX j ( LNM YnM~OPd euelkveuj ge:o$`_$[ ]Qde:j]$e:VeU_Pl[i[cjSO$TT_PTTW]$[i[ij ( X TOP[i[ij C[i_PTOP]$[i[cRUR X½]$[il]fMuVM RURUM %'&)( dj dr:[iyzx O$O$X~k`O$e:àM O$e]$]Q[ceUj[cjr_$X]QàX e:o$t ( ¾ j]qm\djo ` d\ge:o$`_$[ ]Qde:gTRURUXª]Q[cl]QMuV MRUR M %'&)( -X e:j X½_QX]$àX euopt TdYZX]$]QX ` ddjs[ijopeujjm `euj [ ]$]fm dj OPkY\YZ[ ]Q_PeUj[cj Pd ]Q[cj Å j ge X à dj XX R:e X R:`eUd ]^d VX] A h σ(a h ) R. A h /h + q $X `TRURUX `euj}_$e o VeUR:R M ge X à dj XX R:eUj TRU`d Td RU[iR:R X~dR:[cVeU[cj X RU`e:deUg[cj¼e]QO$[ceUO$X _PV dap[ij OPTYZYZXZd j /h. LNM RUR:meUj B[i_$OPlÃd_PeUje:jSR X TOP[i[cOP]fX\O$[cT_QXX) C[ ]$]QM Å j¼àx e:``e d YZe:j X euo$x _PV d ] d VX] A h /h ` [iop`euo$[co$o$m O$M]Q[ceUO$[cO$O$M\`eU[c` dopoqxw Pd]Q[ij + q Á /h Á σ(a h ) [q, 4/h + q]. ¾ _$e:]"ke:o$[iob]qe σ(a h ) apdo q > Y\eUOP]QM½O$e:eUO O$[cT_QX XW [c]p]fm~geuop`_$[i[ ]Qe:RUR Mª]$[il]fMuVM R:R M½dj k`o$euàm OPe:]P]Q[ie:j[ij _QXa]QàX euopt ( Ž) fƒç â % &)( ½Ÿ œ Š œ aš ƒ ƒx " «P ƒ œ UŠ ž Š <œaž q = Ÿ <ÂNeUe]$]QMM dopde:]p]fxxw H[c]P]fM kl]qm RUm RUR M %'&)( ndj VX e:j jwdrurux _QX]$àX euopt `TWj f h = ( qeh[o _$[cjo$opeuàxxuvx OP]QX?OQXX gxx jy]fmrurum euj u i+ = u i u i, PdOP]QX?X RU`T[clgdR:R X u = OQXX gxx j

7 6 h=.5 5 h=.5 4 h=. 3 h=....3.4.5.6.7.8.9 @ )6 L4[il]fMuVM j %'&( _QX]$`XeUO$T $X jty\[i[c_$eujw[ij _QX]$`XeUO$T ]QX X TW`OP[iOPOQX f(x) = x + x 7 q =. $X e:jgt`]qe:do"d RU[c]$T`O$[cOP]fX u = u, u 3 = 4u u = 3u u j = ju, j k, [cg[ir:ru[c[ij u k+ = (ku ) (k )u = (k + )u. M e:j = u m+ = (m + )u, Pd ]$[ij u = $X u j = àx eu`e:rur X j =,,..., m +. Œ C Œ ²'Œ4 ¾ g[ir:r M~[iOPeUe:j]"kjk]šYnX]$_$e:eUO$e L = = [......... dj $Xa]Q` d O$OQXrO$[ij V [i_p_qx j ]QM _$`[umw 8[c]P]fMrRUX O$`[c]QXX j OP[ij?dY\eUj XeUOQX_PVd] $XSoÇV [i`]qd _$e:] ( R:` ddj OP[ij ` d ` d m m. ºšdO$àX k)l]qm RUmM %'&( VX Ob]fXXuVX j«dy\eujx euo$x _PV d ]Q[cl]QMuV MjrdY\eUjX euoätj`]qe:d ] d VX] u n (x) = sin(nπx), [ ]QOPe:]fM M j L Å jrd YZe:j X euobv[c`]$d_$[ce:]qxªyªkm OšYªTd)gWdO$O$X ] ºtd O$àX v = (sin(α), sin(α),..., sin(αm)). sin(α(j )) + sin(αj) sin(α(j + )) = ( cos α) sin(αj), dy\euj XeUOPV [i`]qd _$e:kl]fmrum L v = λ v M]$[i[ dy\euj XeUOQX_PVd RUR X λ = cos α ` dy\ Cdj[ij]P]QeU[cj do$x R:]QX ( d ]P]fX½VeUeUY\[ie:O$[cj`eUj `dyz Cdj[cj]QeUjSkl]fM R:m ]$d ]Q[cTW]QTWeUO$e± )VXX gwe:]fx X j,..., m [crue sin(α(m + )) = α k = kπ/(m + ). ºtd O$àXz`dO$e:je dj X eugd OP]Qe-VM lw[ij[ V M VM RUe:RUR M [i_$e:o$ttw_$e X dy\eujx euo$x _PV da$x«oqxxg XX jwj M e:op]fm) `Tj [, π] k =,..., m, YªTW]$]QX OPeUe:j M l M jsdjw`e:jªpdzàx e:``e ( LNM]$[ij L Å R:RU[BO$XX g XXjrdY\eUjX euo$x _PV d ]4$X dy\euj XeUOPV [i`]qd _$e:] % &)( λ k = cos( kπ m+ ) = 4 sin ( kπ m+ ), v k = sin( kπ m+ ) sin( kπ m+ ). sin( m kπ m+ ), k =,..., m.

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 (

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........ 3 3 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

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=

$X~deU`[uX R]fX½ TWdRU[cj ( (B A) v ) n i+(j )n = l= n B j,l (A = B j,l (AV ) i,l = l= [ v+(l )n ] v ) +(l )n. v i n +(l )n n n B j,l A i,k V k,l, Pd ]$[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]$]fX XWÅ v h h v h = ( ) I n h + h I n vh. ºBTj h = h = h OQXXg XX j [ 4 ] 4......... I 4 4 [ 4 ] 4 I......... I 4 h = 4 [ 4 ] ( h 4 ( (. I........ 4 ( 4 ( ( ( ( ( I l= k= I [ 4 4......... 4 4. ] do v Å R:R M»dj Å j _P[iTjX RURUX»]$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

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 (

.8.6.4..8.6.4. ¾ O$eUY\[i_P`eUj.6.4. %'&)( jtwyz[c[i_$e:j[ij _QXa]QàX euopt.8.8.6.4..8.6.4. ¾ O$eUY\[i_P`eUj.6.4. %'&)( jtwyz[c[i_$e:j[ij _QXa]QàX euopt.8

YZN YZN Á C UC u Ž ' R:` ddj jk] kl]fmrum ldy\dã[c[ije:j[ij g = a = $X _P[iT)o j X [clgd ] u(x) = x, x. LNM YnM `TWVXX?l X_Ä$X]$[iR:]P]fXX) H[cRUe ÄRUe o `euyzmm _PeUj Y\eUje:YZe: euj]qxxhj MeURURUM _$[ctj X [cl)o gde:rurux (.8 ¾ g[ir:r Mnd RURU[c[iOPOQXn`TWVX OPOQX.6 n = n = 9. do<]qx _$``TTObV X X]Qe:Y½TO [iewdru[-`d V)e:j O$TT_Pe.4 `TW]$[ijy_P[c]$`e:dRUd O$Tl]Q[ceUOPOQX O$XX g X X j lko. Ve M X _$d `OPeUYZXX]$eUde]fX Pd Y½TWTW]fX Y\eUR:R X le:r X e:op]$[ieur:r M ( e:]fm M j OP]QXeUe:RUe:O$TTOPdjÃ[cRo.8 Y\e X}[ie^dRU[ ( qeu[c_$[ie:o$[co$oqmw`twvx O$O$X.6.4 n =. 5, n = 3...4.6.8 MYnM~OQX X]Qe:eUj X e:àxx j X euàxxj - ]QeU[cgdOb]QdR:R X "Y /?!!"# $ $ 9.) 5D% U3-S $ & A H FK)zA H F & A H FDf t A H F =9 -#)"()5%!$"5)# # d $ & % $./ ' & A H FK > A H Fkdƒ & % HG' %=%!$"5% $ & %!? & -# & &;>& %) 4" ' "(GXd > d 6 tw? %ql(ky ZNv%!YK)Y ZNv.qL(KML\iAl%qL(fMLF^Nr.!YK4L\kAl%!Y fmlf^n L(K#(3! #$za +/-" t AjY] ' %!$#^An%qLOdyLF Ff/M" t A ' %$#kal%ol+lqdlfkdylfdf /-" t A ' %!$!#^An%qL+LOdyLF^d +LF F\.OL YZN YK#(3! #$za +/-" t AjY] ' %!$#^An%!YXdyLF Ff/M" t A ' %$#kal%y!+lqdlfkdylfdf /-" t A ' %!$!#^An%!Y+LOdyLF^d +LF F\.Y YZN K!#3! # $ZAn!' %wa YZdC$ 9 $zal%qlf Ff!' %oa_$ 9 $zan%!yf^d LF FkNh!Kc$!' #^Al%qL ]%!YZdyLFkN >!' "DKMLk`~%qL d >'J6 KML^`~%!Y dvk"df %OL ]^A 6 +LF^N AnodnMFDK AmodlMF + za _"].qlqd 6 ].!Y -FkN `j#e3 ' " # /M" t!' % "- " " #)" oanmfk t A _" ](.qlqd 6 ](.!Y -FkNT$%/oN $ %/RN $3. ' 't $ $ %4" %!$%7$ *" >!' "DKMLk`~%qL d K"^N #K > A _"](.OLqdCW -FkN $ & %!!+,' oanmfk oanmf)+#\.!y b A "DfMLqdLFDK!#kN "# ) % K"Df%qL ]^Al%Y!+LF^N #K > A "](.qlqdl -F^N `~.!$% oanmfk oanmf)+#\.!y b A "DfMLqdl%!Yf)Y!FK!#kNh$%/ J6 & ' #$ ), % >!'G6 KMLk`~%!Y d - " " #)" " % b KML(f%qL ]^A 6 +LF^N # K > A nwxd 6 ](.!Y -F^N oanmfk oanmf)+#\.ql YZN b ALqd 6 f4lfdk!#kn K %qlf %ql ]^A 6 +LF^N #K > A Lqd 6 ](.Y -F^N oanmfk oanmf)+#\.ql YZN b Al%qL(f)YXd 6 fmlfk!#knh$%/ & K ˆ wn!" # % 9.) 5

Á >!'G6 KMLk`~%!Y d,' b AjYR`Al%qL(fMLF^d 6 fmlfdk & A A 6 +LF]%qLfMLk` 6 ](%OLFkN $%/RN - " " #)" " % b b ALOdyLFDK > A nwzdcw -F^N b Al%qL(f)YXdyLFDK > A ulqdcw -F^N, "$ b ALOdr%!Yf)YFDK > A mwzdyl -FkN b Al%OL(f)YZdr%Yf)Y!FDK > A ulqdl -F^N % & )M" % $#.oa " %4#(3!S $ZAjWZdyLqdl%!Yf)Y!Fkd_" %4#(3S$zA_WZdLqdl%qLf)YF^d b FkN Ž ² ¹ Ĺ %( ºBeU_Pde:]QX [i M ldy\dã[c[ije:o$[ij qe:_$e flwru[c] j _P[iTjX [ilgwdj«x e:l[it)]$]fxynxªv [i`]qd_pe ( ( Muk]fM) [ ]$]fm kl]qm RUm O$OQM %'&)(%% YZX]Q_PeUeUOPe g[ jeue]$]$e φ h XT`e ( A h h dj¼obk)y\y\[c]q_peuj[cj $Xn CdOPe:]Qe:e:VeUOP[iOb]Qe ( RU`d)dj ]$[il]fmuvm OPOQM %'&)(%( a = $X g =. M k]qmw <[c]p]fm ge:o$`_$[ ]Qde:gTj?]Q[cl]QMuV Mj % &)(:% % H_QXa]QàX euopt ]Qd ]$[itw]p]fxx UŠ œ Q f ƒ± " \ œ Š "~ ƒ ƒ" f min f(x) u (i,j) max f(x). x x +e:lap[ Å d jrjk]šj[ir $M j V)e:[i_P[ieUOP[ijrX _PV dj ` [iop`e X _bvd ( u (i,j) ( re:rur XeUO$[c`O$eNOQXXa] ge:o$`_$[c[c]qe:jy]q[cl]qmuv Mj `Tj _P[iTj XRUR X d j [ctynxjjeujv[il]qd x = u x (x) = η(x) $X~Y½TeUR:R X^dO$e:RUR X qeu_pe flr:[c] js[ilgwd ] u(x) = f(x) %'&)( ( ±µž a ižµ ² ¹ 3 8 z¹ µy ² Ž a )²' ± µ4žµ? ' W² ) ± cž) ±µžµ 8[c_$TOP`T_$OPO$[ie:RURUXªdj»Pd^]QTR:RUTW] [co$eur:ru[»x TOPO$eUjZ[iR:eUY\eUjde:j]QeUY\[ijW[c]Q[cRUYZMW $PdàX^d j`eujnlkvm kru[ceuo$y\[ o j[ ]Q[iR:YnM RUeUjW[uXX _PeUOPeURUR:[ kl]fm R:m_PklY\eUR:RU[ A x = b _$X]QàX e:o$t euopoqxwå H K \ ( qe:o$`_$[ ]Qde:gTj kl]fm R:mj %'&)(%% ]fx X euopeurur:[q]q[cl]fmuveurur:[»`muk]q[ ]fmmj¼kru[i[cjoqm\`te:]$[ij`euj ƒx P ƒ ˆ UŠ ^Y\[ij[ ]Q[iR:YZeUM ( TWdYnXa]fXX j [c]p]fm~k)l]qm RUmjS`[i_$_PdeUjWYnX]$_$e:eUO$e:O$O$X djsvx euj VM l Mjrjd RUR XOP]fX½ Cde:`o ` [ixuv)euxsx RU`eUd e:]fx CO$X jd ]QXX j [c]p]fm YnXa]Q_$e:eUOPe4d j W â A = A h ( ^XTO$OPeUj Y\[ij[ ]Q[iR:YnMj djã[cruyzxzdj [c]$]qm h O$[ R X OP` [cjj X j«`tr:t[iopoqx k_$`eue ]QM k]p]fm YZMM jrl X _PVX jw`e:j¼yzx]q_peue:o$euj 'PdR:RUde:j ]QeU[ ]Qd`dj[c[ij«Y½TWeUOP]$eo ]fx_pv[»$x¼r X OP`TW]Qde:YZe]QTOb]Q[cjzYZMM _$M àx ObV X X¼RUe:e X j O$TWT_$[c`OPeE <`do$àx¼àmuk]fm jwjmo$o$m«_qx]$àx euob]fxuveuo$o$x ]Q[cl]fMuVeUO$O$M YZX]Q_PeUeUOPe:]ªd'VX]\R:Td`àXX 5 ]fx enoptt_$[cyz We X 5 ( ºBTj dj l X _bv X) <]Qde:YZe]QTO djw[igwtrur:euj[cj ÄVM l M jwr XO$`TW]Qd euy\e:]qtw`ope X A (Ç]$[i_QXa]QeUeVeUO$[ ]~Y\[ij[ ]Q[iR:YnMa] [ie:vm] YªTd`o v àxxs` [i_p_$de:jyzx]q_peueuope XW 3VXX j A v àm k]p]fmuvm]^j M e:]qm YnX]$_$e:eUO$e V[c`]Qd_Pe4oÇ]$TRUd'$X ( M eug[cjzy\[ijw[c]q[cruy\eu[cj eugw[ux j X djvrum l]q[im PdOb]fX `euj X R:`TX _PVX T`OP[iOP]QX x $Xz X _QX j]qxx ]QM]fM O$e]Q[ij H[ ]$]fm OQXXa]QT Pd jd, x O$TW [cj[i[~`dl]qe8deu`[uxx\_qx]$`xeuo$tx k x. ºtdOPàXndO$e]$]QX euopge:h[i_p[ij]qeuxx RUekl]QM RUm e:]fm^jtyz[c[i_peuo$[cop]$e _QXa]QàX euob]fx [co$o$xnpd kl]fm R:m_PklYZMM»Y½TdgdOb]Q[c]P]fX [co$o$xªdj OPTd_$e]Q[ ]$]QTnX _Pd`O$e:YnXXa]QeUd e:]fx Äg[i_Pe:VXXao ]fxa]h`d_pvx]p]qtzge:h[i_p[ijopo$[ie:rurum /[ie3o$[cj X ev Xj ]QX _$`àx _$X]QàX e:o$[cyze:j[ijàxxj\d RU[ Y\eU[iR:[iàM Ob]fM ( ÂNeUe]$]QMMW `Tjl Xjrkl]fM R:m ]N_QX]$` d ]QXX jsge:o$`_$[ ]Qde:j]$e:VeU_Pl[c]P]fM½ X_$[iY\YZX RURUX^]fX_$``TTg[cRUR X ( ]Q[ ]fxx j^[iopeur:ru[/vx e:j»k`o$e± X R Pdj»`Muk]Q[ ]$]"k»e]q[i_$x]qe:e:veujw[ijby\[ij[ ]Q[iR:YnM)Å ƒ ƒxÿ P E ƒ ƒ± "n ƒ" \ cdjaptãx]q[ Ã_QXgeU[cj]ZY\[c]QlWd)g ( { [ O$d V[cR:]QTWTv]Q[cl]QMuV)e:RUR:[ Pde:O$OQX YZX]Q_PeUeUOPe-dj OPkYZY\[c]$_$e:j[ij«$X

Á Ÿ Š ƒ ˆ ƒ±ƒ ( ºBTW]Q[cjz[cg[iR:R M dj YZX]Q_PeUeUOP[ie:leUj ( UC j M l]"k Cd ]Q[cj]Qe X X RUek)l]QM RUmj¼geUOP`_P[c]$deUj]QePd l]qxx ]fm R:R X e:o$e:euj RU`d)djwO$e:eUO X jwj[c]p]qt ]$[il]fmuvm A x = b Y\eUOPOQM djwopky\yz[ ]Q_PeUj[cj $X d O$e:]$eUeV)e:g[ jeue]$]qe ( LNM R:RUme:j A d j Y kmon CdO$e]QeUeVeUg[ jeue]$]$e$x½]$[il]fmuvm j _QX]$`XeUO$TSdj ÄTjW`]$eUdj A φ(x) = b A x, A (b A x) Y\eUje:YZe:` dl]fx OPeUR:R M φ(x) >, `TWj b A x ( e:e:]$]$dã_$x ge:[ij]$]$euy\[ij[ ]Q[iR:YnM+ k_$`e:erum k]fmao YZMM j ]fm YZM j Y\eUje:YZe:` dlwg X j ]Qe:[c]Q[cjàMM jsyzx]q_peueuope X Y½TWd)gd OP]fXYnX]P]fX ( A RU`d)dj x jk`keuj[cj X W _$d`opeuyzxx]qe:dq_$x]qàx e:o$tr:ru[$x k r k k O$[ij VeU_Pl[ [ir:e Q fš ž ( = b A x RU`d)dj p Pd`eUj V[c`]$d_$eW$X~[c]$O$e]fMM j k φ Å j Y\eUjWeUY\e M OPTd_$X RURUX x k + α p k. LNM R:RUme:j ` dop`x φ(x k + α p k ) = r k α A p k, A (r k α A p k ) = φ(x k ) α p k, r k + α p k, A p k, α Å jªx _PV d α k = p k, r k / p k, A p k Xj]QXX ]fm YZM j^yze:jeuy\euj ( p OP[c]$[c]fX X j x k+ = x k +α k p k, PdR:RUde:j φ(x k+ ) < φ(x k ) [cruru[ce α k = ( X R:e:]QXX j TTWg[i`O$e4Y\eUje:YZe:jl X `TOPTTjj X`OPe p k+ = r k+ + β k p k Y\eUOPOQM, β k = r k+ / r k. LNM YZM+V XRUeUj]QX PdjW`X+ C[i_PTOP]$[iR:[iY\eUO$[c[ij\[iY\YZ[H]fM OPOQMB_PklgWk do$d e:]$]qx TW]$TT½]Q[cld`àXX `O$e ( LNM RUR:meUj je:yze]$]fmeujn M]Q[c[ ]QdgeUOb]QTO We:]QàM lw` m Å % &)(:% r k, r j = p k, Ap j =, MeUjSOQXXg XX j `Tj k j. ± ²u² 3 3 ' 8 ÄŽµ²u²' ŽµŽ²uŽ) ¹ X R:e:]QXX j x $X X O$[ ]Q[c]QXX j p = r = b Ax. X OP` [c]qxx j `TjWj[iOtO$d evx½o$t C[ijW[iY\eUO$`_Pe:]Q[c[i_Pe]Qd ]$[itw]$tt α k = r k / p k, A p k x k+ = x k + α k p k r k+ = r k α k A p k β k = r k+ / r k r k djs eu[cje Å p k+ = r k+ + β k p k LNM OPOQM PdàX X O$`[iR:[iRURUX\]QX _PVe:]$O$[c[~R X OP` [uxnk`o$e YZX]$_$eUe:O$e V[c`]$d_$eoE]QTR:dÅ A p k $X Y½TTW]QX YZeUX\V [i`o ]Qd _$[ie:g[ijsr:euj[ixx _$e:` dy euj X X]Qe:de:]QXqO$[c`M O$e:OQM]$TRUd'$X ( LNMa]Q[ijS]"km YnMM_QM½X OP` [cr:]fx~`dlg[cjrdjslkveuj e:[ije ( YZe:j X e:o$ttgw[ij %'&)(% t C[i_PTOP]$[i[iR:R Xn_P[iOPeUgT X X RUeV[i`]$d_$e]+d VX]B` [co$`[ij MMj d_b]qdãd j XX R:eUO$eUX ( LNM Oxo ]fm O$[cT_QX XW [c]p]fm m o"ge:yz[cjo$e:deuop[ir:ru[ ]Q[il]QM VM R:RU[ r m { eueuoc [crur:[ie- km _$euob]"kopveu_pl[ie]fm d ]Q[ ]fx =. x, y = x T y

ltdy\euddj <_QXa]QàX euoptno$xx g XXjZVeUe:YZ[ceUOP]QMM j\x OP` [ir:[ir:r X TOP[ieUj½Pd X R PdjSX eu[cyzy\euj ( m. L kkgwk]$]qm VMMj _$X]QàX e:o$ttwjn MMOP]fM M j { [ct_qxxuvxx j«]qx TR:T`` ddj«dj¼]$eurux OP]$de:]$TS]"k myzmm _$eumz[co$euy\[i_p`euo$o$m %'&)( 3`Tj n = n = n. À+l)o ]fmrum_bk)lwynmw 'PdO$O$X\OPeUeUO d j ]$Tj]Q[cYnX]$dj]fXW d jr_$x]qàx e:op]qtsop[iàm»x TOPO$eUjS[cRUeUY\eUjWdeUj]Qe:YZ[cj[c]$[iR o YZM RURUMN[ ]$]QMqRUe:e:]$]$dÃ_$X geu[cj]qeur:rux n (c M e:]qmnvx Ob]fXXuVX]/RUeUTW`TWRUT`Td C[i_$XX]$eUde:g[ijBYZMM _$M]4d'VX]8]fX TWRUT`dj ]Qd euo$[co$o$ẍ $Xt`dRUYZX jj[co$o$xtoqx_qx `` [c[iopoqx ( À+l]fM R:m_PklYZM j n n oç` [c_$_$d eujyzx]q_peue:o$e d jªj X TWl X YZX eo j[cjåjwdrurux OP]QXš CdeU``[uXuVe XšX RU`eUd e:]fx dj^vx euj^[c]qm euobkkg[crurum geux Ãdj X X RUe:OP]fX ( LNM Ob]fMtOP[iT_$XX a[c]$]qm [ir:euy\eujd euj]qe:yz[cj[c]$[iruyzm j ]"kmyzmm _$Mrdj R:Td`àXX n O(n 4 ) [ieuàm O(n 6 ), `TW]$[ij?OP[ndR:eUO$e ]QM kg[crurum YZX]Q_PeUeUOPeUR:R X ±X R:R X fl el /n 4 ( n fl el fl lg "($ fl lg /fl el u el u lg %( & & %( % ( &)% W( W% % % & )( W( & W% &%( & )( & W( W% W% % )( W( 3 3 3 3 eue]$]qd Ã_QX gweu[ij]p]qeuy\[ijw[c]q[cruyzm j~rud [ ]$]fxyze:o$`_$e]q[i[c_$e:j MB`Muk]Q[ ]fmmj OP[iT_$XXuVXXWÅW`dO$àX½ge:O$`_$[ ]Qde:j]Qeo Y\[ij[ ]Q[iR:YnMrdj?]Qde:OP]fX«`[i_b]fX R:T`T XW <l XRUTW]QXX j _$X]QàX e:op]fx¼kl]fmrum_bk)lwynm¼ope:]$[ij <[ ]$]QM VX Ob]fXXuVeUR:RU[ ÄTj`]$eUdR:RU[qVeU_Pl[ oçjd_$y\euopoqx½dr:euope L /n. k] u = u(x) dx h u(ih, jh)) = h u h, i,j [ir:e u u n h. XRUTW]QXX j O$eUe:ON_QX]$`XeUOP]QX ]QX _$``TTgW[iRURUX LNM l Mj¼ MMOP]fM M j u h O(/n). `Tj«XO$[c]$[c]QXX j RUd C[c]$TO$[clgd`O$e r k L8X TRUTW` dj~veue:yz[ceuo$[co$o$m+oqx _$X `` [i[co$o$x dj < ÄTjW`]$eUde:g[ij / log(n). o"jd_pyze:j½y½tàx e:j[ij½kl]fmrum_bk)lwynm j~_$x]po àx euoptwveu_pl[ PdàXZ]Qdg[cRUR X dj L /n [cr $M j]$[i[ij¼oqx _$X `` [i[co$[c[ij d j«`eu_ä$xa]$]qt¼rueue]$]$dã_qxgeu[cj]p]qe o Y\[ij[ ]Q[iR:YnMj ]fx _bv)e]qop[iyzx]šx O$`[iR:YnMM_QM]i 'Pd ]$`X~j Muk]$]QMuV Ma]td RU[cVX j. n. L8X TRUTW` dob]fxzjm lg MMj [c]p]fm RUeUe]$]$dÃ_QXgeU[cj]$eUj\]fX _bv)e]qop[iyzx~]"k myzmm _QM~djr eu[cj[iy\ epdw `TWj n = $X~O$TlgW[ Y½TWTW]$]$TT ]fm R:RU[qkl M [igtwrurue:o$[cyzyzx `O$e n Å j `XOPVX [iopoqx ( ºtd RUY\[ige:YZ[cjO$e:deUOPeUO$O$X^]$[il]fMuVeUOPOQMZR:eUe]$]Qd Ã_QX gweu[ij]$euj\[c]qt dj VeU[cR M½ X R PdjSO$TTW_$[iY\ e ( eue]$]qd Ã_QX gweu[ij]p]qeuy\[ijw[c]q[cruyzmmt$x Y½Te]fX e:]$[i_$x]qe:e:veuope X YZ[cj[c]$[iRUY\e MZO$[iàMSjeUe:g[ij OPT C[ij[cYZe:O$jdao C[iT`O$eUX\àM OPe:]$[iRURUMM jn]qx _$`[iy\yze:j `T_$OPO$eUR:R X "!$#%&#'( )!*!,+!$- &+/. # ( "334576389/:<;/8=">@?A%A 8)8)A=">/8CB7D*EF=GHD*8)I8J>H>HKL4)HM%D<N>H>HKO="9/EF=GIP4"8G57B 4"IPIKRQ I:<D B9H4)D8JI:<:<939/K?S>/8)EJD*D$4"D$>TM IPE)?LA 8J>3>8)D<HD<:<:<>&Q 8J9/8J9U4J>H>HV&>H>/KN7D*IP:<9WN9/6&9HKM&D*IPK%IK%K&IX8"8G59%Y%ZT9/D<KS[\3]&^\/_U8JIBHIPIHQ`=G4a?U>/8)EJD*D$4)DHK(?S9H9/EJD*8)I:*8G5S6&>HE)N>/B 4)DFbc`d&\3e chf gh9/?s9hki4j>h>t=ghb3lb3hb3id<:*ip?u>h:*:$>8j>/dm&d8jb`9h6%bh;3:*:$9u>/k&>h:*575 4)D*:<:<9&Y

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 ).

((( # % ( /=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..

( 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+

((( Á ((( ((( (((.4.3....4.3..... 5 5 3 4 5 5 5 3 4.4.4.3.3....... 5 7 6 5 4 5 3. 5 6 5 4 5 3 @ )6 L4[il]fMuVM j % (:%»_$X]QàX e:o$t ( À+R M _Pe:VeUR:R Ḿ ¾ TR:[i_$e:j Y\[ij[ ]Q[cRUYZM RURUM X O$`[iRo e]qttw`opeuj δ =. $X p RUX _$ev)e:ruruḿ euy\ RUe:e:]$]$eUOP[iRURUḾ ¾ TRU[c_$e:RUR Xª$X e:yz R:eUOPeUe:]P]Qe:O$[iR:RUM»` [co$`eu WeUOP]$[iO$MM jjm RUR M δ =.3 X OP` [cru e]qttgw[irurux. δ =.. M R:`eUY\YnM e:j[ij dj le:[iyzx j ]fx _P` [cyz e ( 7 qe:o$`_$[ ]Qde:j]$eTWO$[uXYZ e:geuy\[ijwo$eud euo$e:rur:[ _Pd RU[c[iY\deUR:RU[ OPTaPTT O$X YnX X jn]fx XX j ( [iopeuy\[i_p`e:`o$e % ( RU`d)d jz]q[cl]fmuvm j M u t (x, t) = α u(x, t) + g(x, t), < x, x < u(, x, t) = u(, x, t) = u(x,, t) =, u x (x,, t) =, u(x, ) = f(x). X RUX i[cj\d [c_qxx]p]qd_peuj~ge:o$`_$[ ]Qde:j]QeW]$[ilg M M jz_p[itjx [ilgwd ] ltwdyze:deugw[ij~`tw]q[cjnàx `O$e:geUY\[ijWO$eUd eo O$[cj d ]$[ij]qeuxx R:e:kl]fM R:mj^]QX X TW`OP[iOPOQX $X+X euàx gweuo$`_p[c]qd euj]qe [co$euy ( euy\ RUe:O$eUe]$]$eUj[cjB` [iop`eu e:op]q[coqmmj)o ]Qm `T)]Q[ij krur M ( RU`d)dj ge:o$`_$[ ]Qde]QT d [c_qxxa]$]qd _$e _$[ctj X [clgde:j h u(, x, t) = u(, x, t) = $X h geuop`_p[c]$de:]$t x h = h x _P[iTj X[ilgd eururux...... ( ( ( ( ( ( ( ( (...... u(x,, t) = u x (x,, t) = [crue, h = h...... ( ( ( ( ( ( ( ( (...... 3 3.

Á! UC.8.8.7.7.6.6.5.5.4.4.3.3.... 3 3 5 5 5 5 5 5 3 5 5 5 5 5 5 3.8.8.7.7.6.6.5.5.4.4.3.3.... 3 3 5 5 3 3 5 5 5 5 5 5 5 5 5 5 @ )6 L8[cl]fMuVM j % ( jtyz[c[i_peuj[cj _$X]QàX e:o$t X'$X jlw[c]q`e:rur M.5,.,. $X.. t = qe:o$`_$[ ]Qde]QTSàX `O$eUgWeUY\[ijOPeUde:j[ijS X RUX i[cjsd [c_qxxa]$]qd _$eo$xx g XXjS]QXX ONºB_$d j[ f`[i_$e:jr]$trude:ruruxwå h = I h + h I. donx e:àx geuop`_$[c]$deuj]$ec]$[ilg M M jseuy\ RUe:O$eUe]$]$eUO$[cRUR:M+` [co$`eu e:op]$[ioqm M jjmr:r MW )OQXXg XX j ]QM kopeujsgeuop`_$[c]$deo gt`ope]$[il]fmuvm `O$eW$M R:RU[c[ij { ( % ( & I δ α ) h u k+ h = ( I + δ α ) h u k h + δ g k+/ h u h = f h, Y\eUO$O$Mtjk] u k h = ( u k (,), uk (,),..., uk (n,), uk (,),..., ) $XšV XOP]fX X VX Ob]Qe uk (n,n ) u Ž ' R:` ddj g k+/ h α =, g(x) =, `Tj x p. $XzY½TWTW]Q[cj g(x) =, p = (.,.6) $XvX R:`T[il]Qd ºBTWVX O$O$X dj jty\[i[c_$eujw[ij _$X]QàX e:o$t `Tj u(x, ) =. n = n = 8 X e:àx geuop`_$[c]$deuj]$euj X¼e:YZ WRUeUOPeUe]$]Qe:j[ij ` [co$`eu WeUOP]$[iO$MM j]qm δ =.5, X euàx WeUOP]$[ie:Obo OQM t =.5,.,.,.. +eue:yz[ceuj[cjzd j Pd¼lkVeUj?R Ml[iR:R Ḿ ]fx O$X X e:jd ]$eur X X [iruë VX OP]QXXuVX j Cd ]Q[cj]$e XXRUe:kl]fMRUmj u + g(x) = _$X]QàX e:o$t X ( LNM YZM~OQXX]$eUe:jSX euàxx jsxeuàxx j Y % {? ]$eu[igwdop]$drurux O$[càM f h.

D!!"# $ $GY+/-"$ %M#)" ' " # $%;$., % & K U3-!S $za & F^A H FDf t A H dlmf =9 -#)"() ' %!$" ' # # $ & %!)$./ ' " & AjWZd H YZdlMFK & ALqd H YZdlMFK & A H LOdCWXdl4FDKW d & H Y za H LOdyLqdlMFK)W D & $./ ' & A H dcwfdk > A H F? %ql(k)y ZNv%!YK)Y XN[.OL(KML\kAn%qL(fMLFkNr.!YKML \kal%!yf4lf^n > KW? LqN L(K!#3! # $ZA A +/-" t AjY] ' %!$#^An%qLOdyLF Ff?)?? /-" t A ' %!$#^An%qL+LqdLF^dyLFf/-" t A ' %!$!#^An%qL+LOdyLF^d +LF F\.OL Y!FkN YK!#3! # $ZA A +/-" t AjY] ' %!$#^An%!YXdyLF Ff?)?? /-" t A ' %!$#^An%!Y!+LqdLF^dyLFf/-" t A ' %!$!#^An%!Y+LOdyLF^d +LF F\.Y Y!FkN YzAl%YZdr%!Y!+L^`~%!Y!FKo ny\ prdu+y\ p!\d.!y YXN K!#(3 # $zam!' %oa YZdC$ 9 $zal%qlf)fdf ' %waj$ 9 $ZAl%Y!Fkd LF)F^N K)W] A`dyLF^N /K)W? W Nv%)K QWZN & K)W]wN >!' "DKML^è%qL d >!'G6 KMLk`~%!Y d & A "Df %OL ]^A 6 +LF FDK > YzA _"].qlqd 6 ].!Y -FkN $ %/RN $%/RN rurd b )K & A_$ 9 $zal%ol ](%!Y!F)+W?n ]/M] > ] FkN U b +.! 6' $(- KU b $./ %,!"(%J!$ % & & Kc$!' #^An%qL(f)YZdl%!Yf)Y!FkN KE 'D, " $!" %oal%mf^n U$ > > 84"$ $% %7$!", "$ >!' KML^è%) d % 9 3 &!6 & "# # >!' "DKML^è%qL d >!'G6 KMLk`~%!Y d oa "Df %OL ]^A 6 +LF FDK t YzA _"].qlqd 6 ].!Y Rd Anf)W?n ]D/ MF)F^N $%/on $ %/RN <KU ˆ A & f/-]kajw?n ] > ]ka ] & Ff *F FkN & K b ˆ <wn >!'G6 KML^`~%Y d & & AjYo` Al%qL(f4LF^d 6 f4lfdk & A)A 6 +LF ](%OL(f4L^` 6 ]%qlf^n $%/on & & A`dl%!Yf)Y!FKZAmQ] &)& A`dr%YfMLF + & & A`dr%Y!F)F\pXN $#(.oa & & F^N H "#^A! nwzdr%!y f)yzdcwzdl%ql(f)yzdjwzdcw? W MF^NT/ <% ' <on A`dlMF K t $ > D$ZN $%/RN 'D, " $za F^N Ž ² ¹ Ĺ %( RU`d)dj _QX'Pde]Q[ ]$]QT ( OPde:]QXW [c]p]fm % ( ª]QTWd ]$]QXX _$X'Pde]Q[ ]$]QT'$X _$X]QàX e:o$ta$x àx eu`e:rur X g δ > ( ( ºBTjªk`O$e:geUY\[ijOPeUde:j[ijªR M Y\ m kl]fm R:m dj½ge:o$`_$[ ]Qde]QT½ XeU`àX Y½TTW]P]QTa$XjªO$TWl)o ]Q[c[ij d j O$XX]$T ]QX VX R:RUe:j[ij«gWe:h[i_$[cj]Qe X X RUek)l]QM RUm u t = u xx u h (t) = hu h (t), Y\eUO$O$M Å j dy\eo j X e:oqx _bvdap[cjn]qe:[ig[ ]fmmjndr:[cvx jzvm R:eUR:R M h ( 4/h, ). R:` ddj θ [, ] $X»]fX _P`XOP]Q[cRURUXX j X e:`xgeuop`_p[c]$deuj]qeux re:rur M u k+ h = u k h + δ h (( θ)u k h + θu k+ h ). X _PV deur:r XªjM eujsge:o$`_$[ ]Qde]QTSOPkOP]$[i[cYZe dj OP]QXe:eURUe δ >, θ [, ], h > Á

((( UC %)( ^@4AB3D5E5F<F/646 4646/7c.,9<;.)=87)> o ( LNM OPOQMBRUTWVTOPOQXBO$d V [irurux Y\YZ[HOQX YZX j]fx X euope Xš]$[i`je:eU` d e:]fxqxx R:]$d k)l]qm RUmR:RU[ `TeUj XeU[iY\Y\eUjª`[ile]Qe:Y Y\[BR M Y\ m kl]fm R:mRUR:[ %)(%( i 4 ± zž)µ4 u C ±µžµw IJ ² ¹ L8X _$àx Ob]Q[iR:R XXj jk]šk`opeutr:d ]$]$[ie:o$[ij XXR:]Qd kl]fm R:mj C[i_$TWOP]Q[cl]fMuVMM u tt = c u xx, x (, ), t >, u(, t) = u(, t) =, u(x, ) = f(x), u t (x, ) = g(x). qe:o$`_$[ ]Qde:g XX js[ijwo$euj X e:``xy½ttw]p]qta$x j x O$TWl]$[i[cj 3`T)]Q[ij¼X eu[cyzy\euj ( ( ) u xx (x, t) = h u(x + h, t) u(x, t) + u(x h, t) + O(h ). X R:e:]fX X jso$eue:o n $XnX OP[c]Q[ ]fxxj h = /(n + ) ( R:` ddj u i (t) X _Pd`O$e:YnXXa]QeUd u(ih, t) Å R:RU[H$X Y\[i_$`e]fMM j u (t) f(h) g(h) u h (t) =, f h = (((, g h = (((, u n (t) f(nh) g(nh) PdR:RUde:j (u xx (, t)) h h u h (t). { XX g XXj WTdRUe:`O$ege:O$`_$[ ]Qde]QT ]Q[cl]fMuVM { %()(:% u h (t) = c h u h (t), t >, u h () = f h, u h () = g h. %()(:% 8dj~]fXuVX RUR:eUOP[ijª]$deUOP[ij ` [c_p]fxrutwvtj geh[c_$[ij]$e XX R:e:kl]fM R:mOPkOb]Q[i[cYZe:j½X R:`T X _PV d ]Q[cl]fMuVM ({ e Pdeo ]QTW`OP[iR:R X w h = (u h, u h ) OP[ O$XX g X X js[ijopeuy\ynm e:o$[cj ` [i_b]fx R:TWVTj OPkOP]$[i[iY\eU`OPe [ ] %()( w h = M Y\eUO$O$M I h w h, M h = c. h LNM YZMNV deugxx j»_qx]$àx euob]fx àm k]p]fm [cjª[iopeuy\[i_p`e:`o$eâštwjã[ o"ºbtw]$]qxny\[ijw[c]q[cruy\e M ( "YZ WRUeUOPeUe]$]Qe:j[ijq` [ioxo `eu e:op]$[ioqm M j]qm\x j]fxx\x OP` [ir: e:]$ttg[crurux δ Å %()( w k+ h = (I δ M h) (I + δ M h) w h, w = [ ] f h g h. ºtd O$àXªgeh[c_$[ij]$e XX R:e:kl]fM R:mO$O$MB[ieC[iO$e:eUjjk\[ijWO$eUY\YZM euob]fm»g[i_pe:vxx]p]fxx u h, àx jjx]$]qxx TOP[ie:j `eu_bo Pde]$]fX X?X _Pd`O$eUYZXX]$eUd OPTd_$XX j u Å R:RU[ ( X RUe]fXXj X euàxao"x O$`[iR h δ $X?dRU`d)dj u k X _Pd`O$e o YZXX]Qe:d Å RURU[ ({ d V[iR]fX YZX R:R X h u h (kδ) X euàx gw[i_$evxx]$]qxx j OQX YZXXngWe:h[i_$[cjO$OPeUàXXuVXXZ`Te:j X eu[cyzy\euj X e:``xg[i_pe:vxx]p]fxx j WOQX X g XX jsge:o$`_$[c[c]$eu`o$e ]Q[cl]fMuVM `O$e u k+ h u k h + uk h = c δ h u k h. X R:`T X _PV d ] LNM YZMqàX `O$eUX O$`[iRUY\[ijW[c]Q[cRUYZMš]QX _PVe:]$O$[c[-àM kjjweuop]qm Y\eUO$[c[ij u h = f OQXXg XX jsl8x kr:d_$e:j RUX TOP[i[iR:R X h. u h u h u h (δ) = u h () + u h()δ + u h()δ + O(δ 3 ), $X u h. TWdjjdR:RUe:O$[iOb]Qe