一类非线性偏积分微分方程二阶差分全离散格式new

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1、

J.Sys.Sci.&Math.Scis.28(1)(2008,1),51–70∗ (


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410081) 
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Lubich
 

 MR(2000)!"65M60,45K05,65D321#$%

2、
'
tut(x,t)+u(x,t)ux(x,t)−β(t−s)uxx(x,s)ds=f(x,t),(1.1)0−1t2β(t)=1,)t=0*0

3、x,s),0≤s≤t,4*)(1.1)+5#!+,/6#%,"**--7%./(10271046)0
&.81./9:2'/(0)Æ12005-03-31,(2;30)Æ12006-02-13.52*+<=<28,-44%.)/5,50"6>@A167&(1.1)27&$)883994:'6#%,:;;B&<5Burgers27&$)#.67=&88399"#%,:;;B",-(1.1)–(1.3)4%.):'6#%50;B>:'+5[1]

4、C+Sanz–Serna#&:?D?E8=9(<9)"@A(>)F@$12"6A1&GIB$)(CB)Ef(t)(t>0)H@AE
t1−1(I2f)(t)=(t−s)2f(s)ds,(1.5)0[1]+F%
t11(I2(I2f))(t)=πf(s)ds.(1.6)0−11[2]224*πI!CDGBF&CHI0>IJK,!Ad12GD=dtBFD
t11−111D2D2f(s)ds=f(t),π2D2I2=LG.012)(1.1);>

5、MI0DH11D2Du=π2uxx,(1.7)24*(1.1);>+/?DDEDu=auxx,Du=buxx(a,bAJ4)F@$"NF6&IGH'OJI#&>";>JP'K=KLThom´ee"Wahlbin[3]M0KPEuler&Q@KM0%'L&#CHGLRMSM[4]TN&HNJN%!>OOI"Mclean,Thom´eeP0&Euler"PUKP[1]&Q@K0Galerkin'LQ&R$%&,-(1.1)–(1.3)JN

6、%OI"Sanz–SernaO#&>,-&)"@K&QM0&KPEuler"$U
LSN#&T[5]UW,UW
9%&RPOOI"QV)&Euler"Crank-Nicolson[6]"$ULPU
L&HN&:SOOI"L´opez-Marcos#&*%&M0&$U"@TTN"1D"@TN(2WXUV'
&X[7]RLVYGY&A&ZZ>S[W&LX2%&$&]Y&VYI[8]"GYRL"Sloan,Thom´ee^T]YLZ@&P

7、0]UL^"L´opez-Marcos[6]#&*%&"@KP0KPEulerLQ@KP0$U
L^"_[[[6]U&_[I0PUKPSM"@TN&Q@KM0'PU&T#M0PU
L&)%#`[6]J1;]L´opez-MarcosR^"1D)t=0`DUW&9OOI)3)HaD!bN"@PU_c&dVH`eRTD[6]'&a_2]&'4P0WaQ,[[6]`TDa"T[&bb(c[[6](1.7)),

8、T0≤t≤T,0≤x≤1,−1−3

9、utt(x,t)

10、≤ct2,

11、uttt(x,t)

12、≤ct2,(1.8)−1

13、uxxt(x,0)

14、≤c,

15、uxxtt(x,t)

16、≤ct2,1cfdgi53[1]1TjUWv(x)>f(x,t),(1.1)–(1.