FD-RBF for Partial Integro-Differential Equations with a Weakly Singular Kernel

Transcription

1 Apple a Compuaoal Mahemacs 5; 4(6): Publshe ole Ocober 5 (hp:// o:.648/.acm ISS: (Pr); ISS: (Ole) FD-RBF for Paral Iegro-Dffereal Equaos wh a Weakly Sgular Kerel Jafar Bazar * Mohamma Al Asa Deparme of Apple Mahemacs Faculy of Mahemacal Scece Uversy of Gula Rash Ira Emal aress: bazar@gula.ac.r (J. Bazar) eg.asa@gmal.com (M. A. Asa) o ce hs arcle: Jafar Bazar Mohamma Al Asa. FD-RBF for Paral Iegro-Dffereal Equaos wh a Weakly Sgular Kerel. Apple a Compuaoal Mahemacs. Vol. 4 o. 6 5 pp o:.648/.acm Absrac: Fe Dfferece Meho a Raal Bass Fucos are apple o solve paral egro-ffereal equaos wh a weakly sgular kerel. he prouc rapezoal meho s use o compue sgular egrals ha appear he screzao process. Dffere RBFs are mplemee a sasfacory resuls are show he ably a he usefuless of he propose meho. Keywors: Paral Iegro-Dffereal Equaos (PIDE) Weakly Sgular Kerel Raal Bass Fucos (RBF) Fe Dfferece Meho (FDM) Prouc rapezoal Meho. Irouco Mahemacal moelg of some scefc a egeerg problems lea o paral egro-ffereal equaos (PIDEs). I hs paper he followg PIDE wh a weakly sgular kerel s cosere. ( ) ( ) ( ) u ( ) () = µ u + s u s s < < µ a s subec o he followg bouary a al coos: u( ) = u( ) = u( ) = g( ). hs ype of egro-ffereal equaos appears some pheomea such as hea couco maerals wh memory [ ] populao yamcs a vscoelascy [5 6]. he umercal soluo of PIDEs s cosere by may auhors [ ]. PIDEs wh weakly sgular kerels have bee sue some papers. umercal soluo of a parabolc egro-ffereal equao wh a weakly sgular kerel by meas of he Galerk fe eleme meho s scusse [5]. A fe fferece scheme a a compac fferece scheme are presee for PIDEs wh a weakly sgular kerel [8] a [] respecvely. A specral collocao meho s cosere [7] for weakly sgular PIDEs. Also Quc B-sple collocao meho [3] a Crak colso/quas- waveles meho [3] are use for solvg fourh orer paral egro-ffereal equao wh a weakly sgular kerel a some ohers [ 4]. I rece years meshless mehos as a class of umercal mehos are use for solvg fucoal equaos. Meshless mehos us use a scaere se of collocao pos regarless ay relaoshp bewee he collocao pos. hs propery s he ma avaage of hese echques comparso wh he mesh epee mehos such as fe fferece a fe eleme. Sce 99 raal bass fuco mehos (RBF) [3] are use as a well-kow famly of meshless mehos o appromae he soluos of varous ypes of lear a olear fucoal equaos such as Paral Dffereal Equaos (PDEs) Orary Dffereal Equaos (ODEs) Iegral Equaos (IEs) a Iegro-Dffereal Equaos (IDEs) [ ]. I he prese work for he frs me ervaves we use he Fe Dfferece (FD) scheme o screze he equao whch makes a sysem of paral egro-ffereal equaos. he we use raal bass fucos (RBFs) o solve hs sysem. Recely FD-RBF meho s use o solve some problems lke olear parabolc-ype Volerra paral egro-ffereal equaos [] fracoal-ffuso verse hea couco problems [33] a wave equao wh a egral coo [34]. I hs paper FD-RBF mehos are apple for umercal soluo of PIDEs wh a weakly sgular kerel. Sgular egrals whch appear he meho are compue by he prouc rapezoal egrao rule. he paper s orgaze as follows. I Seco he RBFs are rouce. Seco 3 as he ma par s evoe o

2 446 Jafar Bazar a Mohamma Al Asa: FD-RBF for Paral Iegro-Dffereal Equaos wh a Weakly Sgular Kerel solvg weakly sgular PIDEs by fe fferece a RBFs. A llusrave eample s clue Seco 4. A cocluso s presee Seco 5.. Raal Bass Fucos Ierpolao of a fuco u : R R by RBF ca be presee as he followg [4] Where φ :[ ) R s a fe uvarae fuco he ( ) coeffces λ = are real umbers ( ) = () s a se of erpolao pos R a s he Euclea orm. Eq. () ca be wre as follows Coser φ ( ) = φ( ) Φ( ) = [ φ ( ) φ ( ) φ ( )] Λ = [ λ λ λ ]. (3) + sc suppor pos ( u( )) =.... Oe ca use erpolao coos o f λ s by solvg he followg lear sysem whch a ame of he RBF Gaussa AΛ = u A = [ ( )] φ u = [u( ) u( ) ( )] able. Some well-kow RBFs. Defo φ( r) = e φ( r) = ( cr ) Iverse Quarc r + c φ ( r) = r + c φ( r) = Hary Mulquarc Iverse Mulquarc r + c Cubc ( ) := λφ ( ) R. = s s ( ) := λφ ( ) = Φ ( ) Λ = φ ( r) = r h Plae Sple φ ( r) = r log( r) Hyperbolc Seca r φ ( r) = sech( ) 3 c Some well-kow RBFs are lse able he Eucla sace s real a o-egave a s a posve scalar calle he shape parameer. Also he geeralze h Plae Sples (PS) are efe as he followg: (r) = r log(r) m = Some of RBFs are ucooally posve efe (e.g. Gaussa or Iverse Mulquarcs) o guaraee ha he resulg sysem s solvable a some of hem are cooally posve efe. Alhough some of RBFs are cooally posve efe fucos polyomals are augmee o Eq. () o guaraee ha he oucome erpolao mar s verble. Such a appromao ca be epresse as follows () = are polyomals o a mos m a m R of egree l =. Here l s he meso of he lear space Π m of polyomals of oal egree less ha or equal o m wh varables. Collocao meho s use o eermae he coeffces λ λ λ λ λ λ. hs wll ( ) a ( ) l prouce + equaos a equaos s usually wre he followg form + pos. l aoal = (5) 3. Applcao of FD-RBF Meho I hs seco we epla he process of solvg PIDEs wh a weakly sgular kerel he followg form ( ) ( ) s = λφ + λ + p ( ) R. (4) = = u u λ p ( ) = = = u u bouary a al coos (6) = µ wh he followg A frs we rouce gr pos = a + h =... M M s a eger a h = ( ) / M. Coserg (6) a po ( ) we have ( ) ( ) ( ) u ( ) =! (9) = µ u + s u s s As he fe fferece echque we have l ( ) µ ( ) ( ) u ( ) = u + s u s s < < u( ) = u( ) = u( ) = g( ). (7) (8)

3 Apple a Compuaoal Mahemacs 5; 4(6): u( + ) u( ) u ( ). h Dscrezg (9) by he θ -weghe meho leas o λ φ ( ) λ + λ + Φ Λ u ( ) = + + = ( ) = (4) u( + ) u( ) = µ θ + θ h ( u ( + ) ( ) u ( )) ( ) θ ( ) θ + ( s) ( u s + h + ( ) u s ) s = "# a θ [ ]. By usg he oao u ( ) = u( ) we have ( u ) + + u ( ) = u ( ) + hµ θu ( ) + ( θ ) ( ) ( θ ( ) θ ( )) + h ( s) u s + h + ( ) u s s. () We ow use he prouc rapezoal egrao echque well aresse [9] o appromae he egrals + ( ) ( s) u s + h s wu ( ) () φ ( ) = φ( ) = a + k =... are ceer pos k = ( b a) / a s a eger $ % & (5) Λ = [ λ λ λ λ ] s a ukow a + + vecor. ( ) = Collocag Eq. (4) a sysem leas o he followg wo aoal coos as meoe seco ca be wre as a ( ) ( s) u s s w u ( ) () λ = () λ =. () Eqs. (4) (6) a (7) ca be wre he followg mar form - u AΛ = ' ' ' % & Λ = [ λ λ λ λ ] + + a or Subsug () a () o () resuls u ( ) = u ( ) + hµ ( θu ( ) + ( θ ) u ( )) hθ wu ( ) h( θ ) wu ( ) + + u ( ) hθ ( µ + w ) u = u ( ) + hµ ( θ ) u ( ) h( θ ) w u ( ) + h ( θw + ( θ ) w ) u ( ) (3) for = m-. Le s appromae he fuco u ( ) erms of RBFs as follows + ) % % + * % % % % % % / (8) By wo mes ffereao from Eq. (4) wh respec o we oba Where = λ φ ( ) Ψ ( ) Λ (9) u = = % 5& Subsug Eqs. (4) a (9) Eq. (3) leas o

4 448 Jafar Bazar a Mohamma Al Asa: FD-RBF for Paral Iegro-Dffereal Equaos wh a Weakly Sgular Kerel Φ ( ) Λ hθ ( µ + w ) Ψ ( ) Λ + + = Φ ( ) Λ + hµ ( θ ) Ψ ( ) Λ + h( θ ) w Ψ ( ) Λ + h ( θw + ( θ ) w ) Ψ ( ) Λ for!. So we coser collocao pos 6 = 7 o oba he eres of he vecors of he coeffces Λ =... M Eq. (). hs leas o =! a + ( ) % ( ) + 8 = + ( %; ) % ( %; ) + + * / As he frs sep we mus eerme Λ a Λ. Obvously Λ ca be obae by he al coo u u ( ) = ( ) = g( ) AΛ = u = o appromae [ g( ) g( ) g( ) ]. Λ we use Subsug () o (9) smlarly leas o a AΛ hθ ( µ + w ) Λ = AΛ + hµ ( θ ) Λ + + Λ + h( θ ) w + h ( θw + ( θ ) w ) Λ u( ) u( ) u ( ). h ( A v hµ ) = ( + w ) v = s s. s w = Λ A Λ s s. s (3) () () (3) he covergece of RBF erpolao has bee aresse by Buhma [3 4] a oher researchers [5 6 9]. 4. umercal Eample I hs seco a eample s prove o llusrae he effcecy of hs approach. For he sake of comparso purposes we use he wo orm a fy orm of errors. Coser he followg weakly sgular PIDE [ 8] ( ) ( ) u = ( s) u s s < < wh he bouary a al coos u( ) = u( ) = he eac soluo s u( ) = s( ). M eoes he fuco 5 3 u( ) M ( π )s( π) 3 M z z ( ) = ( ) Γ ( + ). = = We wll use θ = h =. = a = 5. Errors of he umercal soluos for PS-RBF ( m = 4 ) IMQ-RBF ( c =.) a Sech-RBF ( c =.) are show he able a are ploe Fgures a 3 respecvely. able. Errors. PS-RBF IMQ-RBF Sech-RBF ǁE ǁ E ǁ ǁ E ǁ ǁ E ǁ ǁ E ǁ ǁ. 5.98e-4.6e-3 8.9e-3.7e- 5.3e-3.6e-..83e e e-3.48e- 3.6e-3.5e-.3.47e e e-3.7e-.9e-3 7.e-3.4.7e-3 7.5e e e-3.6e e-3.5.e-3 4.e-3 4.6e-3.7e-.5e-3.8e-.6.33e e e-3.3e- 3.5e-3.4e e-4.3e e-3.9e- 3.5e-3.8e e-4 3.4e e-3.34e-.9e e e-4 3.e-3.4e e e-4.7e e-4.8e e-4.7e e-4.9e-3 5. Cocluso hree ffere RBFs are mplemee a FD meho for solvg a PIDE wh a weakly sgular kerel successfully. he resuls of applyg he meho o he llusrave eample cofrm he ably a he usefuless of he propose app roach. I comparso wh hose resuls repore [ 8] hs meho acheve more accurae resuls wh less aa gr pos. ǁ E ǁ

5 Apple a Compuaoal Mahemacs 5; 4(6): Error Fgure. PS-RBF Error. Error Fgure. IMQ-RBF Error.

6 45 Jafar Bazar a Mohamma Al Asa: FD-RBF for Paral Iegro-Dffereal Equaos wh a Weakly Sgular Kerel Error Fgure 3. Sech-RBF Error. Refereces [] Z. Avazzaeh Z. Beyg Rz F.M. Maalek Gha a G.B. Loghma. A umercal soluo of olear parabolc-ype volerra paral egro-ffereal equaos usg raal bass fucos. Eg. Aal. Bou. Elem. Vol. 36 o. 5 pp [].Yu. Bakaev. O he galerk fe eleme appromaos o mul-mesoal ffereal a egro-ffereal parabolc equaos. BI umer. Mah. Vol. 37 o. 997 pp [3] M.D. Buhma. Mulvarable erpolao usg raal bass fucos. PhD hess Uversy of Cambrge 989. [4] M.D. Buhma. Raal bass fucos: heory a mplemeaos. Cambrge: Cambrge Uversy Press; 3. [5] C. Che V. homée a L.B. Wahlb. Fe eleme appromao of a parabolc egro-ffereal equao wh a weakly sgular kerel. Mah. Compu. Vol. 58 o pp [6] R.M. Chrsese a L.B. Freu. heory of vscoelascy. J. Appl. Mech. Vol. 38 o pp. 7. [7] M. Dehgha a A. Shokr. A umercal meho for wo-mesoal schroger equao usg collocao a raal bass fucos. Compu. Mah. Appl. Vol. 54 o. 7 pp he wo-mesoal se-goro equao usg he raal bass fucos. Mah. Compu. Smul. Vol. 79 o. 3 8 pp [9] L.M. Delves a J.L. Mohame. Compuaoal Mehos for Iegral Equaos. ew York: Cambrge Uversy Press; 985. [] F. Fakhar-Iza a M. Dehgha. Space me specral meho for a weakly sgular parabolc paral egro-ffereal equao o rregular omas. Compu. Mah. Appl. Vol. 67 o. 4 pp [] C. Frake a R. Schaback. Solvg paral ffereal equaos by collocao usg raal bass fucos. Appl. Mah. Compu. Vol pp [] M.E. Gur a A.C. Ppk. A geeral heory of hea couco wh fe wave spees. Arch. Rao. Mech. A. Vol. 3 o. 968 pp [3] R.L. Hary. Mulquarc equaos of opography a oher rregular surfaces. J. Geo phys. Res. Vol. 76 o pp [4] W.J. Hrusa a M. Reary. O a class of quaslear paral egro ffereal equaos wh sgular kerels. J. Dffer. Equaos Vol. 4 o. 986 pp. 95. [5] I.R.H. Jackso. Raal bass fuco mehos for mulvarable appromao. PhD hess Uversy of Cambrge 988. [8] M. Dehgha a A. Shokr. A umercal meho for soluo of

7 Apple a Compuaoal Mahemacs 5; 4(6): [6] E.J. Kasa. Mulquarcs a scaere aa appromao scheme wh applcaos o compuaoal flu-yamcs II soluos o parabolc hyperbolc a ellpc paral ffereal equaos. Compu. Mah. Appl. Vol. 9 o pp [7] C.H. Km a U.J. Cho. Specral collocao mehos for a paral egro-ffereal equao wh a weakly sgular kerel. J. Ausral. Mah. Soc. Ser. B. Appl. Mah. Vol. 39 o pp [8] E. Larsso a B. Forberg. A umercal suy of some raal bass fuco base soluo mehos for ellpc {PDEs}. Compu. Mah. Appl. Vol. 46 o pp [9] W. L a X. Da. Fe ceral fferece/fe eleme appromaos for parabolc egro-ffereal equaos. Compug Vol. 9 o. 3-4 pp. 89. [] W. Log D. Xu a X. Zeg. Quas wavele base umercal meho for a class of par al egro-ffereal equao. Appl. Mah. Compu. Vol. 8 o. 4 pp [] M. Luo D. Xu a L. L. A compac fferece scheme for a paral egro-ffereal equao wh a weakly sgular kerel. Appl. Mah. Moel. Vol. 39 o. 5 pp [] R.K. Mller. A egro ffereal equao for rg hea coucors wh memory. J. Mah. Aal. Appl. Vol. 66 o. 978 pp [3] A.S. Vasueva Murhy a J.G. Verwer. Solvg parabolc egro-ffereal equaos by a eplc egrao meho. J. Compu. Appl. Mah. Vol. 39 o. 99 pp. 3. [4] R.B. Plae a.a. Drscoll. Compug egemoes of ellpc operaors usg raal bass fucos. Compu. Mah. Appl 4. [6] R. Schaback. Appromao by raal bass fucos wh fely may ceers. Cosr. Appro. Vol. o pp [7] F. Shaker a M. Dehgha. A hgh orer fe volume eleme meho for solvg ellpc paral egro-ffereal equaos. Appl. umer. Mah. Vol pp [8]. ag. A fe fferece scheme for paral egro-ffereal equaos wh a weakly sgular kerel. Appl. umer. Mah. Vol. o pp [9] H. Wela. Error esmaes for erpolao by compacly suppore raal bass fucos of mmal egree. J. Appro. heory Vol. 93 o. 998 pp [3] E.G. Yak a G. Farweaher. Fe eleme mehos for parabolc a hyperbolc paral egro-ffereal equaos. olear Aal heor Vol. o pp [3] H. Zhag X. Ha a X. Yag Quc B-sple collocao meho for fourh orer paral egro-ffereal equaos wh a weakly sgular kerel Appl. Mah. Compu. Vol. 9 3 pp [3] X. Yag D. Xu a H. Zhag Crak colso/quas-waveles meho for solvg fourh orer paral egro-ffereal equao wh a weakly sgular kerel J. Compu. Phys. Vol pp [33] M. Q. Wag C. X. Wag Mg L a C. S. Che A umercal Scheme Base o FD-RBF o Solve Fracoal-Dffuso Iverse Hea Couco Problems umer. Hea R A-Appl. Vol 68 o 9 5 pp [34] M.K. Kaalbaoo A. Kumar a L.P. rpah A raal bass fucos base fe ffereces meho for wave equao wh a egral coo Appl. Mah. Compu. Vol pp [5] M. Reary. Mahemacal aalyss of vscoelasc flows volume 73. Sam.