A parameter robust numerical method for a singularly perturbed Volterra equation in security technologies

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1 Procdng of 5 WSEAS In. onfrnc on Informon Scry nd Prvcy Vnc Ily ovmbr A prmr rob nmrcl mod for nglrly prrbd Volrr qon n cry cnolog E ZHOGDI In of Mmc Zng Wnl Unvry ngbo 5 P. R. n XI IFEG ompr Scnc nd Informon Tcnology ollg Zng Wnl Unvry ngbo 5 P. R. n Abrc: In ppr w dy convrgnc propr of fn dffrnc cm on Sn m ppld o nglrly prrbd Volrr ngro-dffrnl qon n cry cnolog. W drv pror rror m rob w rpc o prrbon prmr nd prov fn dffrnc cm lmo cond ordr ccr. mrcl rl ppor orcl rl. y-wor: Snglrly prrbd Volrr ngro-dffrnl qon fn dffrnc Sn m nform convrgnc Inrodcon Snglr prrbon problm r n vrl brnc of ngnrng nd ppld mmc wc ncld fld dynmc cry cnolog cmcl rcor ory g poro lcro ory c. To olv yp of problm vro mo r propod n lrr mor dl cn b fond n boo of Frrll l. nd Roo l.. In ppr w condr followng nglrly prrbd Volrr ngro-dffrnl qon n cry cnolog: f I : A wr < << prrbon prmr > f I nd I I r ffcnly moo fncon nd A gvn conn. On png n qon w obn rdcd qon f wc Volrr ngrl qon of cond nd. T olon of bondry lyr.g. 5. T ympoc rcr of olon o qon w mnd by Angll nd Olmd 67. T nmrcl crzon of nglrly prrbd Volrr ngro-dffrnl qon nd Volrr ngrl qon by non p collocon mo n crn non p pc r condrd n Rf.. An ponnlly fd dffrnc cm on nform m cd n Rf. 9. T prn dy dvod o fn dffrnc mod for Volrr ngro-dffrnl qon on Sn m. W fr prn bon for nd drvv. T bon nbl o conrc pcl pcw nform m on wc w cn prov fn dffrnc cm lmo cond-ordr ccr nformly n. Or nly bd on cr compron prncpl rncon rror nly nd ppropr brrr fncon. An o of ppr follow: n con w om propr of c olon. Bd on rl w nrodc Sn m nd fn dffrnc cm n con. In con w nly convrgnc propr of cm. Fnlly nmrcl rl r prnd n con 5. oon. Trogo ppr wll dno gnrc pov conn pobly bcrpd ndpndn of nd of m. o no ncrly m c occrnc. Propr of c olon To conrc lyr-dpd m corrcly crcl o v prc nowldg of ympoc bvor of c olon. mm. T olon of problm f followng bond I.

2 Procdng of 5 WSEAS In. onfrnc on Informon Scry nd Prvcy Vnc Ily ovmbr - 6 Proof. S 9 for proof w nd rgmn wor lo for. nd mlrly for f nd. T Sn m nd fn dffrnc cm In con w crb pcw-nform Sn m nd fn dffrnc cm bd on rpzodl ngron. T conrcon of Sn m bd on bon of c olon nd drvv. λ dno m rnon prmr dfnd by λ mn }. Ampon. W now m mld mpon λ orw ponnlly mll comprd w. W ll lo m rogo ppr gnrlly c n prcc. W dvd c of bnrvl λ nd λ no qn bnrvl. Tn or m λ λ. W dno by nd H m w nd nd od bondry lyr.. H λ. 5 T fn dffrnc cm w prn con of mdpon dffrnc opror nd rpzodl ngron n ppromng Volrr ngrl. Bd on Sn m w propo fn dffrnc cm for problm : 6 f wr A 7 > Anly of cm T nly bd on cr compron prncpl nd brrr fncon cnq nrodcd n. mm. Am H >. 9 Tn opror l dfnd by y y l y y f cr compron prncpl.. f v } nd w } r m fncon fy v w nd l v l w for n v w for ll. Proof. I y o vrfy mr ocd w l n M-mr n proof of mm.. An mmd conqnc of cr compron prncpl followng bly rl. mm. Undr condon 9 olon of dffrnc nl vl problm l y F y B f followng m y B F wr F nondcrng. Proof. Applyng mm o brrr fncon W B F ± y w cn ly g rd rl. For Sn m w v followng rl wll b d lr. mm. Tr conn c d for. Proof. For w v d.

3 For w v. d ombn wo nql o compl proof. T n lmm gv fl forml for rncon rror of rpzodl ngron n ppromng Volrr ngrl. mm 5. For r conn c τ wr gvn by. Proof. T rncon rror of rpzodl ngron n ppromng Volrr ngrl f d τ m d d d d wr w v d Tylor pnon mm nd mpon rnl nd drvv r bondd. To bondd w ll followng nqly b b d g d g } wc ol r for ny pov monoonclly dcrng fncon g on b nd for rbrry. By nqly w v. } d d From nqly nd mm w g rd rl. ow w cn g or mn rl for dffrnc cm. Torm. Undr condon 9 for dffrnc problm 67 w v. Proof. For from 6 nd w v d d f l l } Procdng of 5 WSEAS In. onfrnc on Informon Scry nd Prvcy Vnc Ily ovmbr - 6 9

4 Procdng of 5 WSEAS In. onfrnc on Informon Scry nd Prvcy Vnc Ily ovmbr d d} wr w v lo d Tylor pnon mm 5 nd mpon rnl nd drvv r bondd. Ung cr compron prncpl w v w wr w olon of problm < l w w. From r by vr of mm follow w nd conqnly. Applcon of rcrrnc nqly gv for. T proof of orm compld. 5 mrcl prmn In con w prn wo mpl o llr mod crbd n ppr. Empl. ondr problm f for wr f con c. Empl. ondr problm f for wr f con c n. For or w wc ffcnly mll coc o brng o nglrly prrbd nr of problm. W mr ccrcy n cr mmm norm r of convrgnc r log nd conn n rror m. Tbl mrcl rl for mpl Error R onn Tbl mrcl rl for mpl Error R onn T Tbl nd corrpond o bov mpl rpcvly. T nmrcl rl r clr llron of convrgnc m of Torm. Ty ndc orcl rl r frly rp. Acnowldgmn T wor w ppord by onl rl Scnc Fondon Grn o of n.

5 Procdng of 5 WSEAS In. onfrnc on Informon Scry nd Prvcy Vnc Ily ovmbr Rfrnc: P. A. Frrll A. F. Hgry J. J. H. Mllr E. ORordn nd G. I. Sn Rob omponl Tcnq for Bondry yr pmn & HllR Pr. H.-G. Roo M. Syn nd. Tob mrcl Mo for Snglrly Prrbd Dffrnl Eqon Sprngr Br996. A. S. odg J. B. Mcod J. A. ol A nonr nglrly prrbd Volrr ngrodffrnl qon occrrng n polymr rology Proc. Roy. Soc. Ednbrg Sc. A 97 pp G. S. Jordn A nonr nglrly prrbd Volrr ngrodffrnl qon of nonconvolon yp Proc. Roy. Soc. Ednbrg Sc. A 97 pp G. S. Jordn Som nonr nglrly prrbd Volrr ngro-dffrnl qon n: Volrr Eqon Proc. Hln Sympo. Ingrl Eqon Onm 97 cr o n Mmc Vol. 77 Sprngr Br 979 pp J. S. Angll W. E. Olmd Snglrly prrbd Volrr ngrl qon SIAM J. Appl. M. Vol. 7 o. 97 pp - 7 J. S. Angll W. E. Olmd Snglrly prrbd Volrr ngrl qon II SIAM J. Appl. M. Vol. 7 o pp 5-6. V. Horv M. Rogn Tnon p collocon mo for nglrly prrbd Volrr ngro-dffrnl nd Volrr ngrl qon J. omp. Appl. M. Vol. o. - pp -. 9 G. M. Amrlyv S. Svgn Unform dffrnc mod for nglrly prrbd Volrr ngro-dffrnl qon Appl. M. omp. Vol. 79 o. 6 pp 7-7. M. Syn H. -G. Roo T mdpon pwnd cm Appl. mr. M. Vol. o. 997 pp 6-7. R. B. llogg nd A. Tn Anly of om dffrnc ppromon for rnng pon M. omp. Vol. o. 97 pp d Boor Good ppromon by p w vrbl no n : A. Mr A. Srm E. Sp Fncon nd Appromon Tory Procdng of Sympom ld Unvry of Albr Edmonon My 9-Jn 97 Brär Bl 97.

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