Numerical Solution of Fuzzy Arbitrary Order Predator-Prey Equations

Transcription

1 Aville hp://pvm.ed/m Appl. Appl. Mh. ISSN: Vol. 8 Isse eemer pp Appliios d Applied Mhemis: A Ieriol Jorl AAM Nmeril Solio of Fzz Arirr Order Predor-Pre Eqios Smi Tpswii d S. Chrver * eprme of Mhemis Niol Isie of Teholog Rorel Odish Idi smipswii@gmil.om se_h@hoo.om *Correspodig Ahor Reeived: eemer Aeped: Sepemer 9 Asr This pper sees o ivesige he meril solio of fzz rirr order predor-pre eqios sig he Homoop Perrio Mehod HPM. Fzziess i he iiil odiios is e o me ove ormlised fzz ses viz. riglr fzz mer. Comprisos re mde ewee risp solio give ohers d fzz solio i speil ses. The resls oied re depied i plos d les o demosre he effi d powerfless of he mehodolog. Kewords: Fzz rirr order predor-pre eqios HPM fzz mer riglr fzz mer AMS-MSC No.: 4A7 4A8 95. Irodio A predor-pre model is he well-ow ssem of differeil eqios whih re sed o desrie he dmil relioship ewee predor d pre. These differeil eqios re lso ow s he Lo Volerr eqios. Lo i 95 d Volerr i 96 re he firs who sdied d pplied his model. M of heir oriios e fod i he oo Elemes of Phsil Biolog Lo 95. This model osiss of wo differeil eqios oe for he food speies pre d he oher for feedig speies predor. I his model he prmeers d iiil odiios re sll osidered s risp fied. B i l prie we m hve some vge isffiie d iomplee iformio o he vriles d 647

2 AAM: Ier. J. Vol. 8 Isse eemer 648 prmeers de o error i oservio eperime e. reslig i eriies. These eriies m e modelled hrogh proilisi iervl or fzz heor. I proilisi prie he vriles of eri re re ssmed s rdom vriles wih joi proili desi fios. Proilisi mehods reqire sffiie d o ge relile resls. I he ree dedes iervl lsis d fzz heor re eomig powerfl ools for he mhemil modellig of rel life prolems. I hese pprohes eri vriles d prmeers re represeed iervl d fzz mers. The oep of fzz heor ws firs developed Zdeh 965. As disssed ove if he prmeers d he iiil odiios re desried i impreise forms he fzz heor e pplied. Fzz iffereil Eqios FEs re iresigl sed for modellig prolems i siee d egieerig. Mos of he siee d egieerig prolems reqire he solio o FEs. I rel life ppliio i is oo omplied o oi he e solio of FEs. Chg d Zdeh 97 firs iroded he oep of fzz derivive followed ois d Prde 98 who defied d sed he eesio priiple i heir pproh. Fzz differeil eqio d fzz iiil vle prolem FIVP were sdied Klev Seil 987 d ohers. The meril mehods o solve fzz differeil eqios were iroded i Asd e l Friedm e l. 999 M e l Allhvirloo e l. 7 9 implemeed he predior orreor mehod o FEs. Shori 7 pplied he modified Eler s model for firs order fzz differeil eqios ierive mehod. Jfri e l. pplied vriiol ierio mehod for solvig he -h order fzz differeil eqios o oi lil solio. Tpswii d Chrver iroded he Eler d improved Eler pe mehods o solve fzz iiil vle prolems. Also HPM is sessfll pplied Tpswii d Chrver for he solio of he h order fzz differeil eqios. Ver reel Chrver d Tpswii lso proposed ew oep sed o dole prmeri form of fzz mers for he meril solio of he fzz em eqio. I ree ers rirr-order or friol order differeil eqios hve ee sed o model phsil d egieerig prolems. I is oo diffil o oi e solio of friol differeil eqios d so oe rs o relile d effiie meril ehiqes o solve hese eqios. A gre del of impor wor hs ee repored regrdig friol lls i he ls few dedes. Relig o his field severl oos hve lso ee wrie differe hors represeig he sope d vrios spes of friol lls sh s i Smo e l. 99 Miller d Ross 99 Oldhm e l. 974 Kirov 99 Podl 99. These oos lso provide eesive review o friol derivives d friol differeil eqios whih m help he reder dersd he si oep of friol lls. Regrdig his oep m hors hve developed vrios mehods o solve friol ordir d pril differeil eqios d iegrl eqios of phsil ssems. As he goverig differeil eqio of predor-pre model is of rirr-order reled lierre re lso reviewed d ied elow for eer dersdig of he prese ivesigio. Friol order predor-pre model or friol order Lo-Volerr eqios or friol order prsie-hos eqios is model of poplio growh of predor d pre h ws eellel eplied i Pers d Xi d Che. Nmeril solios of he predor-pre model d friol order ries model hve ee doe Ahmed e l. 7. Sheri d ehgh 7 pplied vriiol ierio mehod o solve his eqio. El-Sed e l. 9 disssed he solio of he friol order iologil poplio model

3 649 Smi Tpswii d S. Chrver he domi mehod. Reel s e l. 9 pplied HPM o solve his eqio for vrios prilr ses. Agi Li d Xi oied meril solio of olier friol pril differeil eqios risig i predor-pre ssem sig HPM. Sie oh friol d fzz eqios pl impor role i rel life phsil prolems emp hs ee mde here o omie oh for more eesive lsis. Some ree sefl oriios o he heor of fzz friol differeil eqios m e see i Agrwl e l. Arshd e l. Mohmmed e l. Wg e l. Allhvirloo e l. Slhshor e l. Slh e l.. Aordigl he oep of fzz friol differeil eqio ws iroded Agrwl e l.. Arshd d Lples proved some resls o he eisee d iqeess of solios of fzz friol differeil eqios sed o he oep of fzz differeil eqios of friol order iroded Agrwl e l.. Arshd d Lples ivesiged he friol differeil eqio wih fzz iiil odiio. Mohmmed e l. pplied he differeil rsform mehod for solvig fzz friol iiil vle prolems. Beher d Chrver hve ivesiged eri implse respose of impreisel defied friol order mehil ssem. Bodr vle prolem for fzz friol differeil eqios wih fiie del re solved Wg d Li. The eslished he eisee of solio he orio mppig priiple. Also Allhvirloo e l. sdied he eplii solio of friol differeil eqios wih eri. Slhshor e l. developed he Riem-Lioville differeiili sig Hhr differee lled Riem-Lioville H-differeiili d solved fzz friol differeil eqios Lple rsform. Slh e l. sed homoop lsis rsform mehod for oiig solio of he fzz friol he eqio. I he prese lsis he homoop perrio mehod HPM He 999 is sed o hdle he meril solio of fzz friol predor-pre ssem wih fzz iiil odiios. I he followig seios he prelimiries re firs give. Ne he si HPM is eplied. The implemeios of HPM o fzz friol predor-pre eqios wih fzz iiil odiios re disssed. Lsl meril emples d olsios re iorpored.. Prelimiries I his seio we prese some oios defiiios d prelimiries whih re sed i his pper. efiiio.. Fzz Nmer A fzz mer U is ove ormlised fzz se U of rel lie R sh h { U : R [] R} where U is lled he memership fio of he fzz se d is pieewise oios. efiiio.. Triglr Fzz Nmer A riglr fzz mer U is ove ormlized fzz se U of he rel lie R sh h:

4 AAM: Ier. J. Vol. 8 Isse eemer 65 i. here eiss el oe R wih U is lled he me vle of U where U is lled he memership fio of he fzz se d ii. is pieewise oios. U We deoe rirr riglr fzz mer s U. The memership fio U of U is he defied s follows U. The riglr fzz mer U e represeed wih ordered pir of fios hrogh he pproh viz. [ ] [ ] where []. For ll he ove pes of fzz mers he lef d righ ods of he fzz mer sisfies he followig reqiremes i. is oded lef oios o-deresig fio over [ ] ii. is oded righ oios o-iresig fio over [ ] iii.. efiiio.. Fzz Arihmei For wo rirr fzz mer [ ] [ ] d slr he fzz rihmei s re defied s follows: i. d. ii. [ r ]. iii. mi m iv. [ ]. [ ]. Lemm.. Bede 8 If z is fzz riglr mer vled fio d if is Hhr differeile he z. B sig his proper we ied o solve he fzz iiil vle prolem

5 65 Smi Tpswii d S. Chrver f where R R d : f f f f R R f. We rsle his io he followig ssem of ordir differeil eqios s elow:. f f f. Fzz Arirr Order Predor-Pre Cosider he fzz friol order predor pre eqio wih riglr fzz iiil odiios d wih d where d re he Cpo derivive of orders. I his sd we hve osidered d d s fios of ime. So he reled predor-pre ssem is ow defied s. d 4. Homoop Perrio Mehod [He 999 ] To illsre he si ide of his mehod we osider he followig olier differeil eqio of he form r r f A wih he odr odiio

6 AAM: Ier. J. Vol. 8 Isse eemer 65 B r 4 where A is geerl differeil operor B odr operor f r ow lil fio d is he odr of he domi. A e divided io wo prs whih re L d N where L is lier d N is olier. Eqio herefore e wrie s follows L N f r r. 5 B he homoop ehiqe we osr homoop U r p : R sisfies: or whih L U L v pa U f r p r H U p p 6 N U f r H U p L U L v pl v p 7 where p [] is emeddig prmeer v is iiil pproimio of Eqio. Hee i is ovios h H U L U L v 8 H U A U f r 9 d he hgig proess of p from o is js h of U r p from v r o r. I opolog his is lled deformio d L U L v A U f r re lled homoopi. Applig he perrio ehiqe He 999 de o he f h p e osidered s smll prmeer we ssme h he solio of Eqio 6 or 7 s power series i p s follows U U U pu p U p. Whe p Eqio 6 or 7 orrespods o Eqio 5. The Eqio eomes he pproime solio of Eqio 5 i.e. limu U U U U. p The overgee of he series hs ee proved i He 999. I he followig seio homoop perrio mehod is pplied o solve fzz friol order predorpre Eqios. 5. HPM Solio for Fzz Arirr Order Predor-Pre Eqios Le s osider he fzz rirr order predor-pre eqios

7 65 Smi Tpswii d S. Chrver d wih iiil odiios i he erm of riglr fzz mer viz. d. Usig - he riglr fzz iiil odiios eome ] [ ] d [. Here ] [ d ] [ re fzz fios of. Usig he Hhr differeiili fzz rirr order predorpre eqio i.e. Eqio m e reded o se of ordir differeil eqios. We redil osr homoop for Eqios s ] [ p p ] [ d p p. 4 Oe m r o oi solios of Eqios i he form p p 5 p p 6 where i i d i i re fios e o e deermied. Ssiig Eqios 5 d 6 io Eqios d 4 respeivel d eqig he erms wih ideil powers of p we hve : p : d p 7

8 AAM: Ier. J. Vol. 8 Isse eemer 654 : d p : d p d so o. The mehod is sed o pplig he operors J d J he iverse operor of Cpo derivives d respeivel o oh sides of eh Eqios 7 d he we m wrie he pproime solio ods s d. A pproimio o he solios wold e hieved ompig few erms s s lim d lim 8 lim d lim where d. 6. Prilr Cses Cse. Le s osider d d iiil odiios i erm of riglr fzz mers viz....4 d Throgh he - pproh iiil odiios eome ] [ ]..4. [. d ] [ ]..7.5 [.. Solvig Eqio 7 we hve

9 655 Smi Tpswii d S. Chrver } { } { 4 β Γ β

10 AAM: Ier. J. Vol. 8 Isse eemer 656 where d. 4 β Γ β where d. 4 Γ Γ β Γ β

11 657 Smi Tpswii d S. Chrver where d. 4 Γ Γ β Γ β. where d. I similr mer he res of he ompoes e oied. Frher we ge he pproime solio of d from Eqio 8. Cse Ne osider d wih he sme iiil odiios s osider i Cse. Agi solvig Eqio 7 we hve

12 AAM: Ier. J. Vol. 8 Isse eemer 658

13 659 Smi Tpswii d S. Chrver 4 4 where d where 4 d 4 4.

14 AAM: Ier. J. Vol. 8 Isse eemer where d where 4 d 4 4.

15 66 Smi Tpswii d S. Chrver We m oi res of he ompoes s disssed ove. Ssiig hese i Eqio 8 we m ge he pproime solio of d. I he speil se oe m see h for he fzz iiil odiios over o risp iiil odiios d he solios oied he pplied mehod el gree wih he solios of s e l.. 7. Nmeril Resls d isssios I his seio we prese meril solio of fzz rirr order predor-pre eqios sig HPM. I is gigi s o ilde here ll he resls wih respe o vrios prmeers ivolved i he orrespodig eqio. As sh few represeive resls s per he ove ses re repored. I he firs se fzz friol order predor-pre eqio for d d wih fzz iiil odiio s disssed erlier is solved HPM. The solios re give i Tles o 4 for vrios vles of d. Nmeril resls re depied i Tles o 4 respeivel vrig from o.6 d eepig os for differe pes of friol order viz. / / / d for ieger order. Crisp resls re lso ilded i hese les. Ne ig. 5 s os d vrig he vle of from o he oied fzz resls re depied i Figres i o iv for vrios order derivives viz. / / / d respeivel. Vrig oh d from o.6 d o for / / / d resls re depied respeivel i Figres o 5. Now for. 5 d he iervl solio of se wih vrios order derivives viz. / / / d re omped d hose re show i Figres 6 o 9. Similr pes of sdies hve ee doe sig he sme prmeers s Cse wih d d for Cse. Correspodig resls re give i Tles 5 o 8 d depied i Figres o 8. I is ieresig o oe h for oh he ses lower d pper ods of he fzz solios re he sme for whih pproimel mhes he risp solio of s e l.. Also from he Tles o 8 i e olded h he risp solio lies i ewee he lower d pper ods of he fzz d iervl solio. From Figres 6 o 9 for Cse oe see h he pre poplio dereses d predor poplio ireses wih ime. O he oher hd for Cse Figres5 o 8 he firs pre poplio ireses d fer h i dereses wih he irese i. I his se he predor ireses hrogho s see i Figres 5 o 8. We see h i oh he ses i es more ime for meeig predor-pre poplios s he friol ime derivive ireses d fill es he mimm ime for. I m lso e see from he figres h Cse es he les ime o mee predor-pre poplios wheres Cse es he mimm ime.

16 AAM: Ier. J. Vol. 8 Isse eemer 66 Tle. Fzz solio for / of Cse..4.6 [ ] [..4] [.865.] [ ] [ ] [ ] [.5.7] [ ] [.49.56] [ ].5 [ ] [.5.5 [ ] [.46.46] [-.99.7] [ ] [.55.65] [ ] [ ] [ ] [ ] [..] [ ] [ ] [.49.49] [ ] [.6.6] [.44.44] [.57.57] [.4.4] s e l Tle. Fzz solio for / of Cse..4.6 [ ] [..4] [.557.6] [.4.6] [ ] [ ] [.5.7] [ ] [.5.796] [.6.47].5 [ ] [.5.5] [ ] [.98.75] [.76.74] [ ] [.55.65] [ ] [ ] [ ] [ ] [..] [.88.88] [.7.7] [ ] [ ] [.6.6] [.7.7] [ ] [ ] s e l. Tle 4. Fzz solio for of Cse..4.6 [ ] [..4] [.58.4] [.87.6] [ ] [ ] [.5.7] [ ] [.664.9] [ ].5 [ ] [.5.5 [.87.9] [.9.56] [.756.8] [ ] [.55.65] [ ] [.775.8] [ ] [ ] [..] [.58.58] [.49.49] [.94.94] [ ] [.6.6] [ ] [ ] [ ] s e l Tle. Fzz solio / of Cse..4.6 [ ] [..4] [.76.] [.46.55] [ ] [ ] [.5.7] [ ] [ ] [ ] [.5 [..] [.8.8] [ ] [ ].5 [ [.8884.] [ ] [.86.9] ].5] [ ] [ ] [.8.46] [.7.449] [.55.65] [ ] s e l. [ ] [.6.6] [.99.99] [.75.75] [.7.7]

17 66 Smi Tpswii d S. Chrver Figre. Fzz solio of friol predor-pre eqios of Cse for d. 5 d [ ] i / ii / iii / iv Figre. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.6] d [ ] whe /

18 AAM: Ier. J. Vol. 8 Isse eemer 664 Figre. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.6] d [ ] whe / Figre 4. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.6] d [ ] whe / Figre 5. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.6] d [ ] whe

19 665 Smi Tpswii d S. Chrver Figre 6. Iervl solio of friol predor-pre eqios of Cse for d wih / i. 5 d ii Figre 7. Iervl solio of friol predor-pre eqios of Cse for d wih / ii. 5 d iv Figre 8. Iervl solio of friol predor-pre eqios of Cse for d wih / v. 5 d vi Figre 9. Iervl solio of friol predor-pre eqios of Cse for d wih vii. 5 d viii

20 AAM: Ier. J. Vol. 8 Isse eemer 666 Tle 5. Fzz solio for / of Cse.6..8 [ ] [..4] [ [ [ ] [ ] [.5.7 [ ] [-.565.] [ ].5 [ ] [.5.5] [ ] [-.99.5] [ ] [ ] [.55.65] [ ] [ ] [ ] [ ] [..] [.4.4] [ ] [ ] [ ] [.6.6] [ ] [.78.78] [ ] s e l. s e l Tle 6. Fzz solio for / of Cse.6..8 [ ] [..4] [.56.57] [ ] [ ] [ ] [.5.7] [ ] [.6.449] [-.6.5].5 [ ] [.5.5 [ ] [.644.7] [ ] [ ] [.55.65] [ ] [ ] [ ] [ ] [..] [.99.99] [ ] [ ] [ ] [.6.6] [ ] [ ] [ ] s e l Tle 7. Fzz solio for / of Cse.6..8 [ ] [..4] [ ] [ [ ] [ ] [.5.7] [.64.8] [.7.688] [ ].5 [ ] [.5.5] [ ] [ ] [ ] [ ] [.55.65] [ ] [.79.97] [ ] [ ] [..] [ ] [.44.44] [ ] [ ] [.6.6] [.58.58] [.7.7] [ ] s e l Tle 8. Fzz solio for of Cse.6..8 [ ] [..4] [ ] [ ] [ ] [ ] [.5.7] [ ] [.8.64] [ ].5 [ ] [.5.5] [.9.99] [.9.68] [ ] [ ] [.55.65] [ ] [.56.8] [ ] [ ] [..] [ ] [ ] [ ] [ ] [.6.6] [ ] [ ] [ ]

21 667 Smi Tpswii d S. Chrver Figre. Fzz solio of friol predor-pre eqios of Cse for d. 5 d [] i / ii / iii / iv Figre. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.8] d [] whe /

22 AAM: Ier. J. Vol. 8 Isse eemer 668 Figre. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.8] d [] whe / Figre. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.8] d [] whe / Figre 4. Fzz solio of friol order predor-pre eqios of Cse for i d ii where [.8] d [] whe

23 669 Smi Tpswii d S. Chrver Figre 5. Iervl solio of friol predor-pre eqios of Cse for d wih / i. 5 d ii Figre 6. Iervl solio of friol predor-pre eqios of Cse for d wih / iii. 5 d iv Figre 7. Iervl solio of friol predor-pre eqios of Cse for d wih / v. 5 d vi Figre 8. Iervl solio of friol predor-pre eqios of Cse for d wih vii. 5 d viii

24 AAM: Ier. J. Vol. 8 Isse eemer Colsios I his pper he HPM hs ee sessfll pplied o fid he solio of fzz friol order predor-pre eqios. The solio oied HPM is ifiie series wih pproprie iiil odiio whih m e epressed i losed form h is he e solio. The erl solio oied he fzz pproh is el eql o he risp solio oied s e l.. I is oserved h for Cse pre dereses d predor ireses wih ime d for Cse he pre firs ireses d he dereses wih irese i. Oe sees h i oh he ses i es more ime for meeig predor-pre poplios s he friol ime derivive ireses d fill es he mimm ime for. I m lso e see from he figres h Cse es he les ime o mee he predor-pre poplios wheres Cse es he mimm ime. The resl shows h he HPM is powerfl ool o fzz friol order predor-pre eqios. I seems lso o hold gre promise for solvig oher fzz olier eqios. The solios oied re show i lr d grphil form. Aowledgemes The firs hor wold lie o h he UGC Goverme of Idi for fiil sppor der Rjiv Gdhi Niol Fellowship RGNF. REFERENCES Asd S. d Allhvirloo T.. Nmeril solios of fzz differeil eqios Tlor mehod Jorl of Compiol Mehods i Applied Mhemis Vol. pp. -4. Asd S. Allhvirloo T. Lopez-Poso O. d Nieo J. J. 4. Nmeril mehods for fzz differeil ilsios Jorl of Comper d Mhemis wih Appliios Vol. 48 pp Agrwl R.P. Lshmihm V. d Nieo J. J.. O he oep of solio for friol differeil eqios wih eri Nolier Alsis Vol. 7 pp Ahmed E. El-Sed A. M. A. d El-S H. A. A. 7. Eqilirim pois sili d meril solios of friol-order predor pre d ries models Jorl of Mhemil Alsis d Appliios Vol. 5 pp Allhvirloo T. Ahmd N. d Ahmd E. 8. Errm o Nmeril solio of fzz differeil eqio predior-orreor mehod Iformio Siees Vol. 78 pp Allhvirloo T. Ahmd E. d Ahmd N. 8. Nh-order fzz lier differeil eqios Iformio Siees Vol. 78 pp Allhvirloo T. Slhshor S. d Asd S.. Eplii solios of friol differeil eqios wih eri Sof Compig Vol. 6 pp Allhvirloo T. Ahmd N. d Ahmd E. 7. Nmeril solio of fzz differeil eqios Predior-orreor mehod Iformio Siees Vol. 77 pp Allhvirloo T. Asd S. Ahmd N. d Ahmd E. 9. Improved predior-orreor mehod for solvig fzz iiil vle prolems Iformio Siee Vol. 79 pp

25 67 Smi Tpswii d S. Chrver Arshd S. d Lples V.. Friol differeil eqio wih he fzz iiil odiio Eleroi Jorl of iffereil Eqios Vol. pp. -8. Arshd S. d Lples V.. O he friol differeil eqios wih eri Nolier Alsis Vol. 74 pp Bede B. 8. Noe o Nmeril solios of fzz differeil eqios prediororreor mehod Iformio Siee Vol. 78 pp Beher ipirj d Chrver S.. Ueri implse respose of impreisel defied hlf order mehil ssem Als of Fzz mhemis d Iformis I press. Chg S. L. d Zdeh L. A. 97. O fzz mppig d orol IEEE Trs. Ssems M Cere Vol. pp. -4. Chrver S. d Tpswii S.. Nmeril Solio of Ueri Bem Eqios sig ole Prmeri Form of Fzz Nmers Applied Compiol Ielligee d Sof Compig I press. s S. Gp P. K. d Rjeev. 9. A Friol predor-pre model d is solio Ieriol Jorl of Nolier Siees d Nmeril Simlio Vol. pp s S. d Gp P. K.. A mhemil model o friol lo- volerr eqios Jorl of Theoreil Biolog Vol. 77 pp. -6. ois. d Prde H. 98. Towrds fzz differeil lls pr: iffereiio Fzz Ses d Ssems Vol. 8 pp. 5-. El-Sed A. M. A. Rid S. Z. d Arf A. A. M. 9. E solios of friolorder iologil poplio model Commiios i Theoreil Phsis Vol. 5 pp Friedm M. M M. d Kdel A Nmeril solios of fzz differeil d iegrl eqios Fzz Ses d Ssems Vol. 6 pp Hss M. 5. Applied Fzz Arihmei: A Irodio wih Egieerig Appliios Spriger-Verlg Berli. He J. H Homoop Perrio Tehiqe Comper Mehods i Applied Mehis d Egieerig Vol. 78 pp He J. H.. A oplig mehod of homoop ehiqe d perrio ehiqe for o-lier prolems Ieriol Jorl of No-Lier Mehis Vol. 5 pp Jfri H. Seid M. d Ble.. The vriiol ierio mehod for solvig h order fzz differeil eqios Cerl Erope Jorl of Phsis Vol. pp Klev O Fzz differeil eqios Fzz Ses d Ssems Vol. 4 pp Klev O. 99. The Ch prolem for fzz differeil eqios Fzz Ses d Ssems Vol. 5 pp Kirov V. S. 99. Geerlized Friol Clls d Appliios Logm Sieifi d Tehil Egld. Li Y. d Xi B.. Nmeril solios of friol predor-pre ssem Adves i ifferee Eqios Vol. pp. -. Lo A. J. 95. Elemes of phsil iolog limore willims & wlis omp. M M. Friedm M. d Kdel A Nmeril solios of fzz differeil eqios Fzz Ses d Ssems Vol. 5 pp. -8. Miller K. S. d Ross B. 99. A Irodio o he Friol Clls d Friol iffereil Eqios Joh Wile d Sos New Yor.

26 AAM: Ier. J. Vol. 8 Isse eemer 67 Mohmmed O. H. Fdhel S. F. d Fjer A. A. K.. iffereil rsform mehod for solvig fzz friol iiil vle prolems Jorl of Bsrh Reserhes Vol. 7 pp Oldhm K. B. d Spier J The friol Clls Ademi Press NY New Yor. Pers I.. Friol-Order Nolier Ssems Modelig Alsis d Simlio Higher Edio Press Spriger Heidelerg ordreh Lodo New Yor. Podl I Friol iffereil Eqios Ademi Press NY New Yor. Ross T. J. 4. Fzz Logi wih Egieerig Appliios Joh Wile & Sos New Yor. Slh A. Kh M. d Godl A. M.. A ovel solio proedre for fzz friol he eqios homoop lsis rsform mehod. Nerl Compig d Appliios Vol. OI.7/s z. Slhshor S. Allhvirloo T. d Asd S.. Solvig fzz friol differeil eqios fzz Lple rsforms Commiios i Nolier Siee d Nmeril Simlio Vol. 7 pp Smo S. G. Kils A. A. d Mrihev O. I. 99. Friol Iegrls d erivives-theor d Appliios Gordo d Breh Siee Plishers PA Lghore. Seil S O he fzz iiil vle prolem Fzz Ses d Ssems Vol. 4 pp. 9-. Sheri F. d ehgh M. 7. Nmeril solio of iologil poplio model sig He s vriiol ierio mehod Compers & Mhemis wih Appliios Vol. 54 pp Shori J. 7. Nmeril solio of fzz differeil eqios. Applied Mhemil Siees Vol. pp Tpswii S. d Chrver S.. A New Approh o Fzz Iiil Vle Prolem Improved Eler Mehod Ieriol Jorl of Fzz Iformio d Egieerig Vol. 4 pp. 9-. Tpswii S. d Chrver S.. Eler sed ew solio mehod for fzz iiil vle prolems Ieriol Jorl of Arifiil Ielligee d Sof Compig I Press Tpswii S. d Chrver S.. Nmeril solio of -h order fzz lier differeil eqios homoop perrio mehod Ieriol Jorl of Comper Appliios Vol. 646 pp. 5-. Wg H. d Li Y.. Eisee resls for friol fzz differeil eqios wih fiie del Ieriol Mhemil Form Vol. 6 pp Xio Y. d Che L.. Alsis of hree speies eo-epidemiologil model Jorl of Mhemil Alsis d Appliios Vol. 58 pp Zdeh L. A Fzz ses Iformio d Corol Vol. 8 pp Zimmerm H. J.. Fzz Se Theor d is Appliio Klwer demi plishers Lodo.