A NONHOMOGENEOUS BACKWARD HEAT PROBLEM: REGULARIZATION AND ERROR ESTIMATES

Transcription

1 Elcronic Journal o Dirnial Equaions, Vol. 88, No. 33, pp ISSN: URL: hp://jd.mah.xsa.du or hp://jd.mah.un.du p jd.mah.xsa.du loin: p A NONHOMOGENEOUS BACKWARD HEA PROBLEM: REGULARIZAION AND ERROR ESIMAES DANG DUC RONG, NGUYEN HUY UAN Absrac. W considr h problm o indin h iniial mpraur, rom h inal mpraur, in h nonhomonous ha quaion u u xx = x,, x,, π,, u, = uπ, =, x,, π,. his problm is known as h backward ha problm and is svrly ill-posd. Our oal is o prsn a simpl and convnin rularizaion mhod, and sharp rror simas or is approxima soluions. W illusra our rsuls wih a numrical xampl. 1. Inroducion For a posiiv numbr, w considr h problm o indin h mpraur ux,, x,, π [, ], such ha u u xx = x,, x,, π,, 1.1 u, = uπ,, x,, π [, ], 1. ux, = x, x, π. 1.3 whr x, x, z ar ivn. h problm is calld h backward ha problm, h backward Cauchy problm, or h inal valu problm. As is known, h nonhomonous problm is svrly ill-posd; i.., soluions do no always xis, and in h cas o xisnc, hs do no dpnd coninuously on h ivn daa. In ac, rom small nois conaminad physical masurmns, h corrspondin soluions hav lar rrors. I maks diicul o do numrical calculaions. Hnc, a rularizaion is in ordr. Las and Lions, in [17], rularizd h problm by addin a corrcor o h main quaion. hy considrd h problm u + Au ɛa Au =, < <, u = ϕ. Mahmaics Subjc Classiicaion. 35K5, 35K99, 47J6, 47H1. Ky words and phrass. Backward ha problm; ill-posd problm; nonhomonous ha quaion; conracion principl. c 8 xas Sa Univrsiy - San Marcos. Submid Novmbr 9, 7. Publishd March 6, 8. Suppord by h Council or Naural Scincs o Vinam. 1

2 D. D. RONG, N. H. UAN EJDE-8/33 Gajwski and Zaccharias [1] considrd a similar problm. hir rror sima or h approxima soluions is u ɛ u u. No ha hs sima can no b usd a im =. In 1983, Showalr, prsnd a dirn mhod calld h quasiboundary valu QBV mhod o rulariz ha linar homonous problm which av a sabiliy sima br han h on o discussd mhod. h main idas o h mhod is o addin an appropria corrcor ino h inal daa. Usin h mhod, Clark and Oppnhimr, in [4], and Dnch-Bssila, vry rcnly in [5], rularizd h backward problm by rplacin h inal condiion by and u + ɛu = 1.4 u ɛu = 1.5 rspcivly. Alhouh hr ar many paprs on h linar homonous cas o h backward problm, w only ind a w paprs on h nonhomonous cas, such in [8, 9]. In 6, ron and uan [8], approximad h problm by h quasi-rvrsibiliy mhod. Howvr, h sabiliy maniud o h mhod is o ordr ɛ. Morovr, h rror bwn h approxima problm and h xac soluion is 8 ɛ 4 u., + 4 x, x 4 L, ;L,π, s [5, pa 5] which is vry lar whn ɛ ixd and is small nd o zro. Vry rcnly, in [9], h auhors usd an improvd vrsion o QBV mhod o rulariz problm in on dimnsional o in h nonlinar cas o uncion. Howvr, in [9], h auhors can only sima h rror in h cas which, h inal valu saisis h condiion k k < 1.6 k=1 s [9, pa 4]. h uncions saisyin his condiion ar qui scarc and so his mhod is no usul o considr many nonhomonous backward problm in h anohr cas o inal valu, which h condiion 1.6 is no saisid or uncions such as x = a, whr a is consan. W also no ha h rror bwn h approxima problm and h xac soluion is Cɛ, which is no nar o zro, i ɛ ixd and nd o zro. Hnc, h convrnc o h approxima soluion is vry slow whn is nar o h oriinal im. In h prsn papr, w shall rulariz his problm by prurbin h inal valu wih nw way, which is dirn h ways in 1.4and 1.5. W approxima problm by h ollowin problm

3 EJDE-8/33 REGULARIZAION AND ERROR ESIMAES 3 u ɛ u ɛ xx = whr < ɛ < 1, ɛp + p sinpx, x,, π,, 1.7 u ɛ, = u ɛ π, = x,, π [, ] 1.8 u ɛ x, = ɛp + p sinpx, x, π 1.9 p = π x, sinpxdx, p = π x sinpxdx 1.1 and, is h innr produc in L, π. W shall prov ha, h uniqu soluion u ɛ o saisis h ollowin qualiy u ɛ x, = ɛp + p s ɛp + p sds sinpx 1.11 whr. No ha our mhod iv a br approximaion han h quasi-rvrsibiliy mhod in [8], and h inal valu x is no ssnial o saisy h condiion *, which only in L, π. Espcially,h convrnc o h approxima soluion a = is also provd In [9], h rror u., u ɛ., is no ivn. his is an improvmn o many known rsuls in [1, 4, 5, 9, 1, 8, 9, 3, 31]. h rmaindr o h papr is dividd ino hr scions. In Scion 1, w shall show ha is wll posd and ha h soluion u ɛ x, saisis hn, in Scion, w sima h rror bwn an xac soluion u o Problm and h approximaion soluion u ɛ. In ac, w shall prov ha u ɛ., u., whr is norm in L, π and C dpnds on u and. Finally, a numrical xprimn will b ivn in Scion 3. C 1 + ln ɛ 1.1. Wll-posdnss o Problm In his scion, w shall sudy h xisnc, h uniqunss and h sabiliy o a wak soluion o Problm horm.1. L x, L, ; L, π and x L, π. L a ivn ɛ,. hn has a uniqu wak soluion u ɛ C[, ]; L, π L, ; H 1, π C 1, ; H 1, π saisyin h soluion dpnds coninuously on in C[, ]; L, π. Proo. h proo is dividd ino wo sps. In Sp 1, w prov h xisnc and h uniqunss o a soluion o In Sp, h sabiliy o h soluion is ivn. Sp 1. h xisnc and h uniqunss o a soluion o W divid his sp ino wo pars.

4 4 D. D. RONG, N. H. UAN EJDE-8/33 Par A I u ɛ C[, ]; L, π L, ; H 1, π C 1, ; H 1, π saisis 1.1 hn u ɛ is soluion o W hav u ɛ s x, = ɛp + p ɛp + p sds sinpx.1 or. W can vriy dircly ha u ɛ C[, ]; L, π C 1, ; H 1, π L, ; H 1, π. In ac, u ɛ C, ]; H 1, π. Morovr, on has u ɛ x, p = ɛp + p p s ɛp + p sds + ɛp + p sinpx = p u ɛ x,, sin px sinpx + π ɛp + p sinpx and = u ɛ xxx, + ɛp + p sinpx u ɛ x, = ɛp + p sinpx So u ɛ is h soluion o Par B h Problm has a mos on soluion C[, ]; H 1, π C 1, ; L, π. A proo o his samn can b ound in [3, horm 11]. Sinc Par A and Par B ar provd, w compl h proo o Sp 1. Sp. h soluion o h problm dpnds coninuously on in L, π. L u and v b wo soluions o corrspondin o h inal valus and h. From w hav s ux, = ɛp + p ɛp + p sds sinpx, vx, = whr h ɛp + p p = x sinpxdx, π For λ >, w din h uncion hn hλ =. s ɛp + p sds sinpx, h p = π 1 ɛλ + λ. hx sinpxdx. ln/ɛ hλ h = ɛ ɛ,. 1 + ln/ɛ.3

5 EJDE-8/33 REGULARIZAION AND ERROR ESIMAES 5 his ollows ha u., v., = π ɛp + p h p π p h p ɛ 1 + ln/ɛ = h. ɛ 1 + ln/ɛ Hnc u., v., h. ɛ 1 + ln/ɛ his compls h proo o Sp and h proo o our horm..4 Rmark.. In [9, 1, 8], h sabiliy maniud is ɛ s [5, horm.1], i is ɛ 1. On advana o his mhod o rularizaion is ha h ordr o h rror, inroducd by small chans in h inal valu, is lss han h ordr ivn in [8]. horm.3. For any x L, π, h approximaion u ɛ x, convrs o x in L, π as ɛ nds o zro. Proo. W hav x = p sinpx, whr p is dind in 1.1. L α >, choos som N or which π p=n+1 p < α/. W hav hn u ɛ x, x = π ɛ p 4 p ɛp +..5 N u ɛ x, x ɛ π p 4 p + α By akin ɛ such ha ɛ < α π N p4 p 1/, w u ɛ x, x < α which compls h proo. In h cas d dx L, π, w hav h rror sima u ɛ x, x = π 1 ɛp + p hn, w his compls h proo. = π π ɛ p 4 p ɛp ln ɛ ux, x p 4 p = 1 + ln/ɛ xx 1 + ln/ɛ xx.

6 6 D. D. RONG, N. H. UAN EJDE-8/33 horm.4. L x, ɛ L, π b as in horm.3, and l xx b in L, ; L, π. I h squnc u ɛ x, convrs in L, π, hn h problm has a uniqu soluion u. Furhrmor,w hn hav ha u ɛ x, convrs o u as ɛ nds o zro uniormly in. Proo. Assum ha lim ɛ u ɛ x, = u x xiss. L ux, = u p s p sds sinpx whr u p = π π u x sinpxdx. I is clar o s ha ux, saisis W hav h ormula o u ɛ x, whr u ɛ p = π w hav u ɛ x, = u ɛ p s ɛp + p sds sinpx uɛ x, sinpxdx. In viw o h inqualiy a + b a + b, u ɛ x, ux, π u ɛ p u p + π u ɛ x, u x ln/ɛ 4 = u ɛ x, u x ln/ɛ s ɛ p 4 ɛp + p ds p 4 p ds xx ds u ɛ x, u x ln/ɛ xx L, ;L,π 4 Hnc, lim ɛ u ɛ x, = ux,. hus lim ɛ u ɛ x, = ux,. Usin horm.3, w hav ux, = x. Hnc, ux, is h uniqu soluion o h problm W also s ha u ɛ x, convrs o ux, uniormly in. horm.5. L x,, x, ɛ b as horm.4. I h squnc u ɛ x, convrs in L, π, hn h problm has a uniqu soluion u. Furhrmor,w hav ha u ɛ x, convrs o ux, as ɛ nds o zro in C 1, ; L, π. Proo. Assum ha lim ɛ u ɛ x, = vx in L, π. L vx = v p sinpx whr v p = π π vx sinpxdx. Dno by w p = vp p and wx = w p sinpx. I is asy o show ha h uncion ux, dind by ux, = w p s p ds sinpx is a soluion o h problm u x, u xx = x,, ux, = wx

7 EJDE-8/33 REGULARIZAION AND ERROR ESIMAES 7 Sinc u ɛ x, is h soluion o , w hav whr So ha u ɛ p = π u ɛ p = π By a dirc compuaion, Hnc u ɛ p = p u ɛ p + ɛp + p, u p = p u p + p u ɛ x, sinpxdx, u ɛ x, sinpxdx, u p = π u p = π u ɛ p u p = 1 p uɛ p u p + u ɛ., u., = π u ɛ p u p ux, sinpxdx u x, sinpxdx ɛ ɛp + p.6 π u ɛ p u p ɛ + π ɛp + p u ɛ x, u x, +., 1 + ln/ɛ u ɛ., u., u ɛ x, u x, +., 1 + ln/ɛ Usin lim ɛ u ɛ x, = vx = u x,, w lim ɛ u ɛ x, ux, =. On h ohr hand, w hav u ɛ x, = u ɛ s p + ɛp + p sds sinpx I ollows ha ux, = u p + u ɛ., u., u ɛ p u p + s p sds sinpx u ɛ., u., ln/ɛ xx., ɛ p 4 ɛp + p sds Hnc, lim ɛ u ɛ x, ux, =. Usin h horm.3, w obain ux, = x. his implis ha ux, is h uniqu soluion o horm.6. I hr xiss m, so ha m m p convrs, hn u ɛ C1 ɛ x, x m m

8 8 D. D. RONG, N. H. UAN EJDE-8/33 whr C 1 = 4 m m p. Proo. L m b in, such ha m m p convrs, and l n b in,. Fix a naural inr p, and din p ɛ = ɛ n ɛp +. n I can b shown ha p ɛ p ɛ, or all ɛ > whr ɛ = np. Furhrmor, rom.5, w hav I ollows ha u ɛ x, x = ɛ p 4 p ɛp + = ɛ n p 4 p p ɛ.7 u ɛ x, x ɛ n n n n p 4 n p n.8 I w choos n = m, w obain u ɛ x, x C 1 ɛ m m. horm.7. L L, ; L, π and L, π and ɛ,. Suppos ha Problm has a uniqu soluion ux, in C[, ]; H 1, π C 1, ; L, π which saisis u xx., <. hn u., u ɛ., C 1 + ln/ɛ or vry [, ], whr C = sup [, ] u xx., and u ɛ is h uniqu soluion o Proo. Suppos has an xac soluion u in h spac C[, ]; H 1 I C 1, ; L I, w h ormula ux, = p From 1.11 and.9, w obain s p sds sinpx.9 u p u ɛ p = ɛp + p s p sds = ɛp ɛp + p s p sds p p p s p sds 1 + ln/ɛ.1

9 EJDE-8/33 REGULARIZAION AND ERROR ESIMAES 9 I ollows ha u.,., u ɛ.,., = π u p u ɛ p π 1 + ln/ɛ uxx =., 1 + ln/ɛ p p p s p sds C 1 + ln/ɛ Hnc u., u ɛ., C 1 + ln/ɛ whr C = sup [, ] u xx.,. his compls h proo Rmark.8. No ha in [8, horm 3.3], h xac soluion u saisis h condiion ux, L, π, whil h condiion o is in his horm is u L, π. So, his also implis ha h inal valu in our horm is only in L, π, no saisyin h condiion * ivn in [9] s Inroducion. Furhr mor, w also hav h rror sima u., u ɛ., which is no ivn in [8, 9]. Hnc, his rsul is an improvmn o known rsul in [8, 9]. horm.9. L L, ; L, π and L, π and ɛ,. Suppos ha Problm has a uniqu soluion ux, in C[, ]; H 1, π C 1, ; L, π which saisis u xxxx., <. hn or vry [, ], whr u., u ɛ., D 1 + ln/ɛ 1/ D = sup [, ] u xxxx., + xx., and u ɛ is h uniqu soluion o Proo. In viw o.6, w hav u ɛ p u p = p u ɛ ɛp p u p ɛp + p = ɛp4 ɛp + p ɛp 4 ɛp = u ɛp + p ɛp + p ɛp = p u ɛp + p p ɛp s p sds ɛp + p

10 1 D. D. RONG, N. H. UAN EJDE-8/33 Hnc, w u., u ɛ., = π his compls h proo. u ɛ p u p ɛ π ɛp + = p 8 u p + p 4 p 1 + ln/ɛ u xxxxx, + xx x, In h cas o nonxac daa, on has h ollowin rsul. horm.1. L,, ɛ b as in horm.7. Assum ha h xac soluion u o corrspondin o saisis u C[, ]; L, π L, ; H 1, π C 1, ; L, π, and u xx., <. L ɛ L, π b a masurd daa such ha hn hr xiss a uncion u ɛ saisyin ɛ ɛ. u., u ɛ., C ln/ɛ or vry [, ] and C is dind in horm.7. Proo. L v ɛ b h soluion o problm corrspondin o and l aain u ɛ b h soluion o problm corrspondin o ɛ whr, ɛ ar in rih hand sid o 1.7. Usin horm.7 and Sp in horm.1, w u ɛ., u., u ɛ., v ɛ., + v ɛ., u., ɛ1 + ln/ɛ ɛ ln/ɛ u xx., C ln/ɛ or vry, and whr C is dind in horm.7. his compld h proo. W considr h xac soluion o his problm is 3. A numrical xampl u u xx = x, sin x, ux, 1 = x sin x. ux, = sin x 3.1 No ha ux, 1/ = sinx sinx. L n b h masurd inal daa n x = sinx + 1 n sinnx.

11 EJDE-8/33 REGULARIZAION AND ERROR ESIMAES 11 So ha h daa rror, a h inal im, is 1 F n = n L,π = n sin nxdx = π n. h soluion o 3.4, corrspondin h inal valu n, is h rror a h oriinal im is u n x, = sinx + 1 n n 1 sinnx, On := u n., u., L,π = hn, w noic ha n n sin nx dx = n π n. lim F n = lim 1 π n L,π = lim =, n n n n 3. n π lim On = lim n n un., u., L,π = lim =. n n 3.3 From h wo qualiis abov, w s ha 3.1 is an ill-posd problm. Approximain h problm as in , h rularizd soluion is u ɛ 1 s 1 x, = ɛp p + ɛp p sds sinpx or 1. Hnc, w hav u ɛ x, = I ollows ha 1 1 ɛ + 1 sin x u ɛ x, 1 = 1/ ɛ s 1 n ɛ + 1 ds sin x + 1 sinnx. 3.5 n ɛn + n s 3 1 n ɛ + 1 ds sin x + 1 sinnx 3.6 n ɛn + n L a ɛ = u ɛ., 1 u., 1 b h rror bwn h rularizd soluion u ɛ and h xac soluion u in h im = 1. L n = 3 and ɛ = ɛ 1 = 1 π, ɛ = ɛ = 1 4 π, ɛ = ɛ 3 = 1 1 π, ɛ = ɛ 4 = 1 15 π. W no ha h nw mhod in his aricl iv a br approximaion han h prvious mhod in [8]. o prov his, w hav in viw o h rror abl in [8, p. 9]. Furhrmor, w coninu o approxima his problm by h mhod ivn in [8], which ivs rularizd soluion v ɛ x, = p p ɛ + 1 ɛ s p sds sinpx. + s

12 1 D. D. RONG, N. H. UAN EJDE-8/33 abl 1. ɛ u ɛ a ɛ ɛ 1 = 1 π sinx sin3x ɛ = 1 4 π sinx sin3x ɛ 3 = 1 1 π sin x sin3x ɛ 4 = 1 16 π sinx sin3x abl. ɛ u ɛ u u ɛ 1 π sinx sin x π sinx sin 1x π sinx sin1 1 x π sinx sin1 16 x π sinx sin1 3 x abl 3. ɛ v ɛ a ɛ ɛ 1 = 1 π sinx sin3x ɛ = 1 4 π sinx sin3x ɛ 3 = 1 1 π sin x sin3x ɛ 4 = 1 16 π sinx sin3x Hnc, w hav v ɛ x, = 1 1 ɛ + 1 sin x or 1. I ollows ha v ɛ x, 1 1 = 1/ ɛ + 1 sin x 1 s n ɛ s + s ds sin x + 1 sinnx 3.7 n ɛ + n s 1 1 n ɛ s + s ds sin x + 1 sinnx. 3.8 n ɛ + n Lookin a abls 1,,3, a comparison bwn h hr mhods, w can s h rror rsuls o in abl 1 ar smallr han h rrors in abls and 3. his shows ha our approach has a nic rularizin c and iv a br approximaion wih comparison o h prvious mhod in, or xampl [8, 9]. Acknowldmns. h auhors would lik o hank h rrs or hir valuabl criicisms ladin o h improvd vrsion o our papr.

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14 14 D. D. RONG, N. H. UAN EJDE-8/33 [8] ron, D. D. and uan,n. H., Rularizaion and rror simas or nonhomonous backward ha problms, Elcron. J. Di. Eqns., Vol. 6, No. 4, 6, pp [9] ron, D. D., Quan, P. H., Khanh,.V. and uan,n.h., A nonlinar cas o h 1-D backward ha problm: Rularizaion and rror sima, Zischri Analysis und ihr Anwndunn, Volum 6, Issu, 7, pp [3] B. Yildiz, M. Ozdmir, Sabiliy o h soluion o backwrad ha quaion on a wak conpacum, Appl. Mah. Compu [31] B.Yildiz, H. Yis, A.Svr, A sabiliy sima on h rularizd soluion o h backward ha problm, Appl. Mah. Compu [3] Campbll Hrick, Bh M. and Huhs, Rhonda J., Coninuous dpndnc rsuls or inhomonous ill-posd problms in Banach spac, J. Mah. Anal. Appl , no. 1, Dan Duc ron Dparmn o Mahmaics and Compur Scincs, Hochiminh Ciy Naional Univrsiy, 7 Nuyn Van Cu, Hochiminh Ciy, Vinam addrss: ddron@mahdp.hcmuns.du.vn Nuyn Huy uan Dparmn o Inormaion chnoloy and Applid Mahmaics, on Duc han Univrsiy, 98 No a o, Hochiminh Ciy, Vinam addrss: uanhuy bs@yahoo.com