FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS

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1 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS Muhmmd Zini Ahmd Nor Hsyimh Abdul Rz Wn Suhn Wn Dud Elyn Sib nd Norzrizl Aswd Abdul Rhmn Institute of Engineering Mthemtics Puh Putr Min Cmpus Universiti Mlysi Perlis Aru Perlis Mlysi E-Mil: mzini@unimpedumy ABSTRACT The im of this pper is to propose n lterntive method to pproimte the solution of fuzzy nonliner equtions The proposed method will be referred to s Fuzzy Homotopy Continution Method (FHCM) In FHCM fuzzy nonliner eqution is embedded in one of the prmeter fmily of the problems ie where is considered s n pproimte vlue to the originl problem nd is corresponding to problem with nown solution A continution method is then ttempted in order to determine the sequence of problems corresponding to m A numericl emple on obtining the solution of fuzzy nonliner eqution will be demonstrted to illustrte the cpbility of FHCM Keywords: fuzzy nonliner eqution homotopy continution method fuzzy homotopy method runge-utt method INTRODUCTION In the literture lot of ttention hs been given into developing new method for solving nonliner equtions This is due to the significnt role of nonliner equtions where it is used to model mny rel life problems We cn see the pplictions of nonliner equtions in mny res such s mthemtics medicines engineering nd socil sciences Zdeh [ ] Mizumoto [] nd Nhmis [] re the erliest reserchers who introduced nd investigted the concept of fuzzy numbers Zdeh [] suggested tht there is sitution when the estimtion of the system coefficients is imprecise nd only some vgue nowledge bout the ctul vlues of the prmeters is vilble For instnce the initil vlue of model in rel life problem my possess fuzziness or uncertinties Thus it my be esier to represent them with fuzzy numbers insted of using crisp number One of the mjor pplictions of fuzzy rithmetic is in nonliner systems whose prmeters ll or prt represented by fuzzy numbers [5 6 7] Mny methods hve been proposed nd studied by reserchers to solve systems of nonliner eqution In this pper fuzzy Homotopy Continution method will be introduced nd used to solve the problem of fuzzy nonliner eqution This method hs been chosen becuse it provides n lterntive to void divergence problems tht often occurred in trditionl numericl methods Plus the solutions cn be globlly convergent rther thn loclly convergent This method is lso cpble in locting multiple solutions rther thn just one Nevertheless this method cn be modified to locte rel solutions only An insight to properties of the solutions obtined cn lso be obtined using this method For this reson integrting the Homotopy Continution Method into fuzzy setting will be beneficil Runge-Kutt method is powerful method when deling with nonliner equtions There re severl dvntges of Runge-Kutt method tht hve been widely discussed For instnce it is esy to be implemented nd the method is very stble It might require some significnt more computing time but the fct tht the reltive simplicity of this method outweighs it disdvntge For comprison we my compre the method with Newton method It is cler tht Runge-Kutt method is better since for deling with Newton method functionl representtion of the function s derivtive must eists which is not lwys possible if we use only specific dt given In Runge-Kutt method this problem is not concern since function s derivtive is not needed The present wor hs been orgnized in the coming section The net section describes some preliminries on the topic Lter method for solving fuzzy nonliner eqution using FHCM will be discussed nd Runge-Kutt method of order four will be integrted in this section Then numericl illustrtion will be provided to illustrte FHCM used In the finl section conclusions will be mde PRELIMINARIES Some bsic definitions of fuzzy set theory nd fuzzy rithmetic re dopted in this section Note tht the Rel numbers re denoted by R nd the multipliction symbol denotes the multipliction between mtrices Definition [ 8 9] A fuzzy number is fuzzy set U : R I [] which stisfies the following conditions i U is upper semi-continuous ii There re rel numbers bcd c b d nd U( ) is monotoniclly incresing on [ c ] such tht 976

2 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom b U( ) is monotoniclly decresing on [ bd ] iii U( ) if lies outside the intervl [ cd ] Definition [ 8 9] A fuzzy numbers U in prmetric form is pir [ uu ] of function ur () ur () r which stisfies the following conditions i ur () is bounded monotonic incresing left continuous function over [] ii ur () is bounded monotonic decresing left continuous function over [] iii ur () ur () r Definition [8 ] Given fuzzy set A in X nd ny rel number r then the r -cut or r -level or cut worthy set of A denoted by r A is the crisp set r A X : r A For emple let A be fuzzy set whose membership function is given s: b b A c b c c b To find the r -cut of r to both left nd right reference functions of A Then epress in terms of b r nd cc b r which gives the r -cut of A s follows r A b r c c b r A set Definition [] If y b b yb b nd yb b Therefore the ddition of A nd B denoted by A B is defined s follows then A B b b b b While the subtrction of A nd B denoted by A defined s follows A B b b b b B is Definition 5 [] The multipliction of two closed A B b b in R denoted by intervls nd A B is defined s follows B b b b b b b b b b b A min m Definition 6 [] Let A be closed intervl in R nd R Identifying the sclr s the closed intervl the sclr multipliction A is defined s follows A Also for A in R if then A denoted by A A for nd Therefore the inverse of is defined s Definition 7 [] The division of two closed intervls A B b b A / B is nd in R denoted by defined s the multipliction of nd b b Therefore / / b b A B b b min m b b b b b b b b for b b Definition 8 [] A tringulr fuzzy number is represented with three points s follows A A membership functions is interpreted s 977

3 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom A( ) Figure- shows grphicl representtion of tringulr fuzzy numbers A f f r r By using FHCM Eqution () is considered s fmily of problems described using prmeter tht ssumes vlues in [ ] A problem with nown solution X corresponds to the sitution when * nd the problem with the unnown solution X X corresponds to For emple suppose X is n * initil pproimtion to the solution of F( X ) then define G: n n FR ( ) FR () or in prmetric form for ll r [] Figure- Tringulr fuzzy number A Let crisp intervl by r -cut opertion intervl r A shll be obtined s follows From ( r ) Hence r ( r) ( ) r ( r ) nd ( ) r ( r ) Thus A r ; r [] [( ) r ( ) r ] ( r) ( r) ( r) [ ] A GENERAL FUZZY HOMOTOPY CONTINUATION METHOD In this section the etended Homotopy Continution method or referred to s FHCM for solving fuzzy nonliner eqution is proposed Consider the following fuzzy nonliner eqution F X () First rewrite Eqution () using prmetric representtion of fuzzy numbers s r g f r f r f r g r f r f r f r It cn be esily simplified to f r f r g r g r f r f r where the r -cut representtion of X X ( ) is the initil pproimtion of the Eqution () For vrious vlues of solution to g r g r (5) when is Eqution () rewritten s g r f r f r g r f r f r () () 978

4 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom So the solution is X However when Eqution () becomes g r f r g r f r By differentition Eqution (5) becomes ( ) ( ) ( ) ( ) g r g r g ( ) ( ) r g ( ) ( ) r) g ( ) ( ) r g ( ) ( ) r Simplify Eqution (6) in mtri form then g g ( ) ( ) r ( ) ( ) r g g ( ) ( ) r ( ) ( ) r g ( ) ( ) r g ( ) ( ) r where g g g ( ) ( ) r ( ) ( ) r g ( ) ( ) r ( ) ( ) r is the Jcobin mtri nd g ( ) ( ) r f r g ( ) ( ) r f r Then the following system of differentil equtions is obtined (6) r r g g r r g g r r f r f r In this pper r [] nd fied [] Runge-Kutt of order four is used to solve this system The following steps re the procedures to solve Eqution (8) Step : Integer N is chose nd let (8) ( ) h The N intervl is prtitioned into N subintervls with the mesh points jh for ech j N j Step : To denote n pproimtion to X i( j) the nottion w ij for ech i n nd j N is used For initil conditions set w () w () w () n Step : Ting Eqution (8) gives X() X( ) w nd for ech j N gives g g ( ) ( ) r ( ) ( ) r h g g ( ) ( ) r ( ) ( ) r f r f r g g ( ) ( ) r ( ) ( ) r h g g ( ) ( ) r ( ) ( ) r f r f r n 979

5 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom g g () () r () () r h g g () () r () () r f r f r nd g g ( ) ( ) r ( ) ( ) r h g g ( ) ( ) r ( ) ( ) r f r f r nd j j 6 j j w j w j 6 Finlly n * n is our pproimtion to * nd NUMERICAL ILLUSTRATION In this section numericl emple is provided to illustrte the Fuzzy Homotopy Continution method for solving fuzzy nonliner eqution The right hnd side of this eqution is considered s fuzzy numbers Runge- Kutt method of order four is used to solve this fuzzy nonliner eqution Emple Consider the following fuzzy nonliner eqution r r The nlyticl solutions of this problem is X (688) nd its r -cut representtion is X 68 r8 8 r Let consider X = bout 5 s initil guess tht is X 5 The r -cut of this initil guess is X r r 5 5 By using Homotopy method using prmeter tht ssumed vlues in given problem is embedded in one prmeter fmily of problems When this nonliner eqution is ssumed to possess the form of g r r r r r r g r r By simplifiction it becomes = g r r r g r r r When this nonliner eqution is ssumed to possess the form of X X Without ny loss of generlity then the prmetric form of this eqution is r r 975

6 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom g r r r r method we obtin the solution of w nd re plced in Tble- The solution of w cn be further visulized in Figure- to Figure-5 g r r r r By simplifiction it becomes = g r r g r r The Jcobin mtri of this eqution is J g g ( ) ( ) r ( ) ( ) r r g g ( ) ( ) r ( ) ( ) r r r Figure- Grphicl representtion of the solutions of w w re considered s initil pproimtion for this eqution when r where w 5 r w r 5 nd f r = f r Figure- Grphicl representtion of the solutions of w To use Runge-Kutt method of order four for solving this eqution first we choose n integer N h / N so h 5 Computing the nd let solutions by the proposed Fuzzy Homotopy Continution 975

7 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom Tble- The solution of w using proposed method Grphicl representtion of w cn be clerly presented in Figure-6 nd Figure-7 Figure- Grphicl representtion of the solutions of w Figure-6 Grphicl representtion of w using Euler Method Figure-5 Grphicl representtion of the finl solutions of w Computtions hve lso been crried out using Euler method s comprison with the Runge-Kutt method of order four in solving fuzzy nonliner eqution Figure-7 Grphicl representtion of w using Runge-Kutt of order four CONCLUSIONS This pper described procedure in finding the solution of fuzzy nonliner equtions by Fuzzy Homotopy 975

8 VOL NO 6 AUGUST 6 ISSN Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom Continution method Firstly fuzzy nonliner eqution is rewritten in prmetric form nd then it is solved using the proposed method Runge-Kutt method of order four is used to obtin the solution of this problem since by using this method the highest possible order will be obtined From the grphicl representtion shown bove it cn be emined tht by using Runge-Kutt method of order four the pproimtion solution is ner to the ect solution The Euler method is first order Runge-Kutt procedure A second order of Runge-Kutt method is n improvement of Euler method The error hs been eliminted s the order used is higher [] Bector C R nd Chndr S (5) Fuzzy mthemticl progrmming nd fuzzy mtri gmes Springer Berlin Heidelberg REFERENCES [] Zdeh LA (965) Fuzzy sets Informtion Control 8() pp8 5 [] Zdeh LA (975) The concept of linguistic vrible nd its ppliction to pproimte resoning Informtion Sciences 8() pp99 9 [] Mizumoto M nd Tn K (979) Some properties of fuzzy numbers Advnces in Fuzzy Sets Theory nd Applictions North-Hollnd Amsterdm () pp56 6 [] Nhmis S (978) Fuzzy vribles Fuzzy Sets nd Systems () pp97 [5] Cho YJ Hung NJ nd Kng SM () Nonliner equtions for fuzzy mpping in probbilistic normed spces Fuzzy Sets nd Systems () pp5 [6] Fng J () On nonliner equtions for fuzzy mppings in probbilistic normed spces Fuzzy Sets nd Systems () pp57-6 [7] M J nd Feng G () An pproch to H control of fuzzy dynmic systems Fuzzy Sets nd Systems 7() pp67-86 [8] Dubois D nd Prde H (98) Fuzzy Sets nd Systems: Theory nd Appliction Acdemic Press New Yor 99 [9] Zimmermnn HJ (99) Fuzzy Sets Theory nd its Appliction Kluwer Acdemic Press Dordrecht [] Kufmnn A nd Gupt MM (99) Introduction to Fuzzy Arithmetic: Theory nd Applictions Vn Nostrnd Reinhold New Yor NY USA 975