To Possibilities of Solution of Differential Equation of Logistic Function

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1 Arnold Dávd, Frnše Peller, Rená Vooroosová To Possbles of Soluon of Dfferenl Equon of Logsc Funcon Arcle Info: Receved 6 My Acceped June UDC 7 Recommended con: Dávd, A., Peller, F., Vooroosová, R. (). The Possbles of Soluon of Dfferenl Equon of Logsc Funcon. Inernonl Scenfc Journl of Mngemen Informon Sysems, 9 (), -. Summry The conens of he pper reles o rcles of uhors usng he mhemcs n economcs wh pplcon n Ecel. I focuses upon he nformon flow usng he logsc funcon, whch descrbes he spred of nformon n sysem wh lmed cpcy nd he soluon by Bernoull s dfferenl equon. The m of he rcle s o nlyse he pplcon of he logsc funcon o descrbe he spred of nformon or number of cusomers n hypermre dependng on me. I s epressed by Bernoull s dfferenl equon wh consn coeffcens. Theorecl srng pons re beng llusred on prccl emples usng he pprome mehod for solvng he negrl by sysemc negron. I presens he process, nlyss nd specfcon of he relve mse. There s n emple showng he progrmmng of he soluon of prmve funcon n Ecel, s process, dscusson on resuls. Fnlly here s nlyss of he soluon of economc clculons n Ecel, he procedure of s rnsformon nd he comprson of obned resuls wh he rely. A gven pplcon my be used s n effecve ool for modellng ss on sysem suron. Keywords logsc funcon, Bernoull s dfferenl equon, sysemc negron, Ecel. Inroducon The m of logsc s o cheve he opml soluon of cern opc n s pplcon feld. One of possble mehods for solvng s he usge of logsc funcon. Ths funcon ws nroduced no he logsc by Perre Frncos Verhuls n he 9h cenury nd found s pplcon n mny felds. Auhors focused on he srem of nformon employng he logsc funcon whch descrbes he spred of nformon n sysem wh lmed cpcy. They gve soluon by Bernoull s dfferenl equon. In ddon o hs uhors presen possbles for s soluon usng mehod of sysemc negron nd s pplcon n Ecel or smlr spredshees.. Dfferenl equon of logsc funcon Logsc funcon () descrbng he spred of nformon or he number of cusomers n hypermre n dependence on he me, n sysem wh lmed cpcy N hs he followng form () c. b.e where, b, c re he prmeers, s he me vrble. By dervon of () wh respec o me we ge. b. e. c /. c. b. e c (). b e c. b e c... Le us suppose, h he velocy of spredng he quny () of economc nformon mong N prcpns s proporonl o he produc of enes sendng nd recevng he nformon hus fs he followng equon () =.[N ()].() =.N.(). () () By comprng () nd (), we ge fer rrngemen = N, c =.N, b = [N ()]/(), where () s number of enes sendng he nformon n he me. Epresson () s Bernoull s dfferenl equon wh consn coeffcens.n nd. Is soluon s he funcon

2 To Possbless of Soluon of Dfferenl Equon of Logsc Funcon N N. N..e To evlue he coeffcen s necessry o e he nformon from he neror of he nervl (, ) ). Emple. Spredng of he fme []. Le us suppose h here re sudens n he college re. 8 from hose sudens sr o spred cern fme. Furherr on we suppose h 8 hours ler here were 8 sudens fmlr wh hs fme. How long does e o spred he fme mong more hn sudens? Soluon. To solve hs problem we use he funcon () wh prmeers N =, () = 8. The prmeer s evlued from he condon (8) = 8 = 8.e 8 where =,67 The nswer s on fgure. I es hours o spred he fme mong 6 sudens...8, () () The equon () s Bernoull s dfferenl equon wh consn coeffcens (wh funcons f() =.N, g() = ) nd wh prmeer m =. If he funcons f nd g re no consn, (f for emple, he cpcy of N depends on me ) s necessry o compue he negrls n (7). Generlly, s mpossble o epress he negrl n squre brces n (7) n eplc form usng elemenry funcons. Therefore s necessry o choose he pprome mehod for s compuon. The sysemc negron represens one of possble mehods.. Sysemc negron Inervl, b of he vrble wll be dvded on n equdsn prs wh he sep h ( b ) / n,, b n. (8) Equdsn nodl pons re:. h,,,... n (9) To deduce he formuls for sysemc negron, we sr wh Lgrnge s formul for he pons,,... : L l.... f, l.. () where n numeror nd n denomnor he -h prenhess s omed [. e. n he numeror he prenhess ( ) nd n he denomnor he prenhess ( )] The polynoml l() of degrees s clled he Lgrnge nerpolon polynoml wh he propery Fgure Suron of sysem wh consn cpcy A generl form of Bernoull s dfferenl equon s he followng, f f g m. g, re. m, m, connous funcons Is soluon for m = s gven by formul (6) l(j) = for j, l(j) = for = j,,, j =,,... < n () The frs hree Lgrnge nerpolon polynomls for he pons =, =, h =, = re presened on fgure. E f s E s s d s C, where E s e g u du (7) Inernonl Scenfc Journl of Mngemen Informon Sysems

3 Arnold Dávd, Frnše Peller, Rená Vooroosová Fgure Lgrnge nerpolon polynomls of he h degree Lgrnge nerpolonn polynoml depends only on nerpolon pons. Lgrnge s formul s lner combnon of Lgrnge nerpolon polynomls. The coeffcens of hs lner combnon re he vlues of nerpoled funcon n pons (9). These pons re clled nerpolon nodes or qudrure nodes. The Lgrnge s formul s heorecl bss for deducng he negron nd compound qudrure formuls. We wll choose equdsn nerpolon nodes (9) nd compued qudrure formuls. These wll hve he followng form I H. f ( ), where H We shll consder h, he negrl I s compued for vrous vlues of he upper bound b. These vlues re he nerpolon nodes (9). For = we ge (fer longer compuon) he followng formuls for sysemc negron: h f h f h f h f h f n d d d d d h 7f 7f 798f h 8 f 9f f 9 h 7 f 7f 8f 6 h 7 f 9 h 9f 88 f f Becuse s vld f ( ) d h 7f f f 7f f ( ) d h h we ge for b = +r.h (where r s nurl number) rule (lgorhm) o compue he vlues of prmve funcon o he funcon f() n f ( ) d b f l ( d ) f 6f 8f 7f 8f h h 7f 7f f f f f ( ) d.. 9f. n () 7f n h () f ( ) d () nerpolng nodes (9). In prccl pplcons we choose r = unl nd we ge n mjory of cses he resuls wh relve error of % nd less. Emple. Progrmmng compuon of vlues of prmve funcon F C f d for, + h () n Ecel we proceed followng wy: Ino he bloc C6:C6 we sep by sep nser he vlues f(), f(+h),, f(+h),... f( (+h). Ino he cell F we nser he sep h.. Ino he cell F6 we nser negron consn C. Ino he cells n bloc F7:F6 we grdully nser he nsrucons (6) =F6+(7*C6+ +7*C7 798*C8+8*C9 7*C+7*C)*$F$/ =F6+(8*C6+ 9*C7+*C8+ +*C9 6*C+ +C)*$F$/9 =F6+(7*C6+ 7*C7+8*C8+ 8*C9 7*C+C)*$F$*/6 =F6+(7*C6+*C7+*C8+*C9+7*C+* *C)*$F$*/ 9 =F6+(9*C6+ 7*C7+*C8+ *C9+7*C+ +9*C)*$F$ */88 =F+(7*C+7*C 798*C+8*C 7*C+7*C6)*$F$/ =F+(8*C+9*C+* *C+*C 6*C+C6)*$F$/9 =F+(7*C+7*C+8*C+8*C 7*C+C6)*$F$*/6 =F+(7*C+ +*C+*C+*C+7*C+*C6)*$F $*/9 =F+(9*C+7*C+*C+*C+7*C+9*C6) *$F$*/88 The vlues F() = C, F(+ +h), F(+h),... F(+h) pper n bloc F6:F6. Noe. The vlues of funcon f() cn be genered by formul or, hey cn be observed vlues. If he number of nodes n r +, he formuls for mssng nodes re dded by less ec qudrure formuls, e. g. by he rpezodl formul =F6+(C6 + C7)*$F$ / for he s followng node or by he Smpson s formul =F6+(C6 +*C7+C8)*$F$/ for he nd followng node or by hree-eghs rule =F6+(C6+*C7+*C8+C9)*$F$ $*/8 for he rd followng node or by Boole s rule =F6+(7*C6+*C7+ *C8+*C9+7*C) *$F$*/9 for he h followng node. Inernonl Scenfc Journl of Mngemen Informon Sysems

4 To Possbles of Soluon of Dfferenl Equon of Logsc Funcon For he h followng node we cn use he ls formuls n () for he cells shfed down rows.. Applcon n Ecel Ecel s used mnly n economc compuons. We wll consruc n pplcon performng he formul (7) n Ecel. Is consrucon n oher spredshee progrm s smlr. The pplcon s bsed upon he sysemc negron, whch does no gve he nmes of negrls n formul (7), bu, for prcce, gves suffcenly ccure vlues n nodes (). In ddon o, provdes wh grph of he soluon () n dependence on he me. On screen ppers s on fgure. Choosng n = nd suffcenly shor nervl [, n] he resuls re ccure les on decml plces. Fgure presens he compuon of he emple. grph () of he process of suron s depced on fgure. LOGISTIKA ( ) = ( ). [N ( ) - ( )]. ( ), ( ) = [ n ] s room, () = 6,78 = ( ) N ( ) =,6,68E- 8 Ep[-F()] -*Ep[-F()] negrál,68 8,68,,,68 8,6687 -,796,789,86,68,8,68 9,8 -,9,67,7,67,7,68, -,788,99,, Fgure Applcon LOGISTIKA [ n ] s room, () = 6,868 = ( ) N ( ) = *SQRT()+,6, 9,96 7,, 6,,,,666,69 -,68,,, 7,897,96 -,97,,6, 9,96,678 -,968,,8,,7,68 -,678, N,,799 -,797,,,,,8 -,9,,,,989, -,7677,6,,789,677 -,6,8, 6,9969,9 -,7,,,,,, [ n ] s room, () = 9,69 = ( ) N ( ) = *(-)^+,6,,,,,,, 98,,96 -,,, 86,8 7,669 -,96,,N, 7,8,989 -,8778,,, 6,9 9,9 -,66,,, 6,6 -,,,6,,,7 -,66,,7,,88,68 -,8897,8,,,,68, 6, 6,76 8,, -,68,,9,,7 7,778 -,6 Fgure Suron of sysems wh vrble cpcy If he funcon n() s no decresng, () < n() for ll. see emple on fgure. If he funcon n() s decresng, he () > n() for some. See he emple on fgure. In h cse s convenen o plce he number of hose nformed by he funcon Y(T) =MIN(X(T), N(T)), (7) becuse he number of nformed enes cnno eceed he number of ll prcpns n he sysem. Ths mnmum s ndced by bold lne on fgure.. Concluson Economc problem s solved n such wy, h s rnsformed no correspondng mhemcl model. For dynmc ss s mosly he common or prl dfferenl equon. If he soluon s nown n eplc form, hen he nlyss s performed. The rbues of soluon re compred wh rely. Accordng o hs, one cn consder how suble s o replce he rel s by mhemcl model. I s shown h, he dfferenl equon () descrbes he suron of sysem well f he cpcy of he N sysem s no decresng. If he cpcy of he sysem s decresng hn he correcon s ppropre (7). The gven pplcon cn serve s n effecve ool for modellng he s on sysem suron. Fgure Suron of sysems wh ncresng cpcy Inernonl Scenfc Journl of Mngemen Informon Sysems

5 Arnold Dávd, Frnše Peller, Rená Vooroosová References Brezn, I. (). Kvnívne meódy v logse. Brslv: EKONÓM. Brezn, I., Ččová, Z., Reff, M. (9). Kvnívne meódy n podporu logscých procesov. Brslv: EKONÓM. [Dávd, A., Peller, F. (7). Zobrzovne funčných závslosí v Ecel. Brslv: Iur Edon. Feceno, J., Pnd, Ľ. (6). Mem. Brslv: ELITA/IURA EDITION. Kme, E. (99). Dfferenlglechungen. Lepzg: Ademsche Verlgsgesellschf Gees & Porg. Šolés, V., Hudec, O., Réveszová, L. (). Mem (). Košce: Techncá unverz v Košcch. Šolés, E., Šolésová, T. (). The regresson credbly model nd s pplcon. In J. Pocech, D nlyss mehods for modellng nd forecsng economc processes (p. 9 ). Krów: Crcow Unversy of Economc Press Šolés, E. (8). Regresná orelčná nlýz s plácm. Brslv: Iur Edon. Arnold Dávd Unversy of Economcs n Brslv Fculy of Economc Informcs Dolnozemsá ces /b 8 Brslv Slov Eml: dvd@deeub.s Frnše Peller Unversy of Economcs n Brslv Fculy of Economc Informcs Dolnozemsá ces /b 8 Brslv Slov Eml: peller@deeub.s Rená Vooroosová Techncl Unversy of Košce Fculy of Economcs Nemcovej Košce Slov Eml: ren.vooroosov@ue.s Inernonl Scenfc Journl of Mngemen Informon Sysems

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