The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations

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1 vlle ole t chve o ppled Scece Reech 5 6: ISSN X CODEN US SRC9 The ue o le petc ppoto uecl olvg o ole o-ooth uzz equto Mjd Hllj e Ze d l Vhd Kd Nehu Bch Ilc zd Uvet Nehu I od Bch Ilc zd Uvet od I edow Uvet Mhd I BSTRCT I th ppoch The le petc ppoto the ole ucto ppoted pecewe le ucto. The oted oluto h dele ccuc d the eo copletel cotollle. Wth eteo th ppoch we popoe ew two-tep tetve ethod o olvg ole uzz equto d ole o-ooth uzz equto. ll oe uecl eple e gve to how the ecec o the popoed ppoch to olve e equto the othe eeece. Ke wod: Tlo le epo Le Petc ppoto ole o-ooth uzz ucto. INTRODUCTION I ecet e uch tteto h ee gve to develop tetve tpe ethod o olvg ole equto lke. Becue the Ste o ulteou ole equto pl jo ole vou e uch thetc tttc egeeg d ocl cece. The cocept o uzz ue d thetc opeto wth thee ue wee t toduced d vetgted Oe o the jo pplcto o uzz ue thetc ole equto whoe pete e ll o ptll epeeted uzz ue 6. Stdd ltcl techque peeted Buckle d Qu 5. Stdd ltcl techque lke Buckle d Qu ethod 4 cot e utle o olvg the equto uch : 5 g 4 c d e Whee c d e d g e uzz ue. Moeove ou o clcl uecl ethod uch : Newto d Newto-Rpho e ule to olve the o-ooth equto uch : equto.we theeoe eed to develop the uecl ethod to d the oot o uch equto. Hee we code thee equto geel :. 49

2 Mjd Hllj et l ch. ppl. Sc. Re. 5 6:49-6 I th ppe we toduce ew ppoch to olve ppotel ole o-ooth uzz equto whch do t hve ltto upo covet d oothe o the ole uzz ucto. I th ppoch gve ole uzz ucto ppoted pecewe le ucto wth cotolled eo whch ed o geelzto o Tlo le epo o ooth ucto. lo we epeet ecet lgoth to olve o ppoted uzz pole. Oe o the dvtge o ou ppoch tht t c e eteded to pole wth ole o-ooth uzz ucto toducg ovel deto o lol Wek Deetto the ee o L-o 9. The ppe ogzed ollow: I Secto two we ecll oe udetl eult o uzz ue. I ecto thee we epl the ppoch o le petc ppoto o ole equto. We ve the outh ecto the ppoch eteded o o-ooth ole equto toducg the deto o glol wek deetto. We eteded the ppoch ecto ve o olvg uzz ole equto. I the th ecto the ppoch w eteded o olvg o o-ooth ole uzz equto. ll oe lluttve eple d cocluo e gve to how the eectvee o the popoed ppoch.. Pele Deto.. uzz ue uzz et lke u : R I whch te 968. u uppe e cotuou. u outde oe tevl c d. Thee e el ue uch tht c d d. u ootoc ceg o c. u ootoc deceg o d. u. The et o ll thee uzz ue deoted E. equvlet petc lo gve ollow. Deto.. uzz ue u petc o p u u u u whch te the ollowg equeet:. u ouded ootoc ceg let cotuou ucto. u ouded ootoc deceg let cotuou ucto. u u. o ucto popul uzz ue the tpezodl uzz ue σ β let uzze σ d ght uzze β whee the eehp ucto : u wth tevl deuzze d σ σ u β β It petc o : σ othewe. β u σ σ u β β. Let T R e the et o ll tpezodl uzz ue. The ddto d cl ultplcto o uzz ue e deed the eteo pcple d c e equvletl epeeted ollow. o t u u u v v v d k > we dee ddto u v d ultplcto cle k : 5

3 Mjd Hllj et l ch. ppl. Sc. Re. 5 6:49-6 u v u v ku ku u v u v ku ku.. The ppoch o le petc ppoto o ole equto 9 Code the ole ooth ucto. We ppote the ole ucto pecewe le ucto deed o. Let u eto the ollowg deto. Deto.. Let P e ptto o the tevl the o: P {... } Whee h d h. The o o ptto deed : P { } It e to how tht P. Deto.. The ucto deed ollow: ; K whee t pot. The ucto clled the le petc ppoto o o t the pot. I uul le epo the pot ed ut hee we ue ee pot. Now we dee g the petc le ppoto o o octed wth the ptto P ollow: g χ whee χ the chctetc ucto d deed elow: χ. The ollowg theoe e how tht g covegece uol to the ogl ole ucto whe. I the othe wod we how tht: P g uol o P The ollowg theoe e how tht g covegece uol to the ogl ole ucto whe P. the othe wod we how tht: g uol o P 5

4 Mjd Hllj et l ch. ppl. Sc. Re. 5 6:49-6 Le.. Let P e t egul ptto o. I cotuou ucto o d e t pot the: l. P Poo. The poo edte coequece o the deto. Th le how tht g pot-we o. Deto.. l o cople ucto deed o et etc pce X d to e equcotuou o o eve ε > thee et δ > uch tht wheeve d < δ. Hee d deote the etc o ee 7. Sce { g } equece < ε o le ucto t tvl tht th equece equcotuou. Theoe.. Let { } equcotuou equece o ucto o copct et d { } covege potwe o. The { } covege uol o. Poo. See 9. Theoe.. Let g pecewe le ppoto o o g uol o.. The: Poo. The poo edte coequece o Le. d Theoe. 9. Now we toduce ovel deto o glol eo o ppoted wth le petc ucto g the ee o L-o whch utle cteo to how the goode o ttg. Deto.4. Let e ole ooth ucto deed o d let g deed 4 e petc le ppoto o.let the glol eo o ppoto o the ucto wth ucto g the ee o L -o deed ollow: E g d d It e to how tht E ted to zeo uol whe. Th deto ued to ke the e ptto whch tched wth dele ccuc. 4. Eteo o le petc ppoto o olvg uzz ole equto Now ou to ot oluto o uzz ole equto. The petc o o two tep ethod ollow: P 5

5 Mjd Hllj et l ch. ppl. Sc. Re. 5 6: Theeoe ue the le petc ppoto ppoch geelzto o Tlo le epo o ooth ucto o the we dee the ucto ollow: ; ; K whee d e t pot. The ucto clled the lowe oud le petc ppoto o o t the pot d clled the uppe oud le petc ppoto o o t the pot. Now we dee the petc le ppoto o o octed wth the ptto P ollow: χ χ 4 whee d χ χ e the lowe oud d uppe oud chctetc ucto epectvel d deed elow: χ χ 5 The ollowg theoe e how tht covegece uol to the ogl ole uzz equto whe P. I the othe wod we how tht: Le 4.. Let P e t egul ptto o. I d e cotuou ucto o d e t pot the:. l P 6 Poo. The poo edte coequece o the deto. Th le how tht pot-we o. o uol P

6 Mjd Hllj et l ch. ppl. Sc. Re. 5 6: Be o deto.4 d eteo o t d e equece o le uzz ucto t tvl tht th equece equcotuou. Moeove e o Theoe. d covege uol o etc pce X X epectvel. Theoe 4.. Let { }d { } e equcotuou equece o ucto o copct et o epectvel d { } { } covege pot-we o. The { } { } covege uol o epectvel. Poo. Sce { } { } e equece o equcotuou uzz ucto o the:. ; ; L < < < < > > d d.t ε ε δ δ δ ε o ech thee et > δ uch tht U N δ U N δ.sce e copct th ope coveg o hve te u-coveg. Thu thee et te ue o pot uch : K d K uch tht U N δ U N δ.theeoe o ech d thee et epectvel o ; K uch tht: < < δ δ d d We kow e pot-we coveget equece the thee et tul ue N uch tht o ech N N we hve:. ε 7. ε 8 The ccodg to the Theoe the equece } }{ { e uol cotuou o d the poo copleted. Theoe 4.. Let d e pecewe le ppoto o epectvel o. 4. The: o uol. Poo. The poo edte coequece o Le. d Theoe..

7 Mjd Hllj et l ch. ppl. Sc. Re. 5 6: Eteo to ole o-le o-ooth uzz equto I geel t eole to ue tht the ojectve ucto o-oothe. Theeoe we dee kd o geelzed deetto o o-ooth ucto the ee o L-o. Th kd o deetto cocdg wth uul deetto o ooth ucto. Theeoe the ollowg theoe epeeted. Theoe 5.. Code the ole o-ooth ucto : R whee. The the optl oluto o the ollowg optzto pole. Mze P. K. p d Kd 9 whee K t pot d P. P. P. K P. vecto. Poo. See. Deto 5.. Let : R o-ooth ucto whee. The glol wek deetto wth epect to the ee o L-o deed the P. the optl oluto o the zto pole whch how 9. Now ed o Theoe 4. d deto 4.we popoed eteo ethod o o-le o-ooth uzz ucto ollowg: Code the ole o-ooth uzz ucto. Bed o Theoe 4.we hve o-ooth uzz pole ollow: Mze P.. p dd K Whee : p p d. p Wth ue o two tep ethod o-ooth uzz pole covet ollow: Mze K P.. p d d Mze K. p dd P. d ed we ue o VK ethod o olvg ove pole.wth uppoe.5 uzz zto pole 9 oed : Mze p.. p dd Rek: we kow ppote vlue o tegl c. o-ooth kd k c whee c pot uch : 55

8 Mjd Hllj et l ch. ppl. Sc. Re. 5 6:49-6 So ed two tep ethod VK ethod pplg ove ek d ue c edg pot utevl d ddle pot o oed : Mze p. Mze p. Whee :. p. p d. whole pole NLP pole d we ot t oluto pckge uch LgoMtl o etc. 6. Nuecl pplcto Hee we peet eple to lluttg the le petc ppoto ethod o d potve oot o ole uzz equto d ole o-ooth uzz equto. Eple d code o Buckle Qu d S. d B. d. Eple 6.. Code the uzz ole equto 45 Wthout lo o geelt ue tht potve d the the petc o o th equto ollow: 5 To ot tl gue we ue ove te o theeoe: 4 d ; 4 5 we ot the oluto o d wth the Me qued olzed eo MSE7.9e-7. o oe detl ee g.. Now uppoe egtve hece > theeoe egtve oot doe ot et. Eple 6.. Code uzz ole equto Wthout lo o geelt ue tht potve d the petc o o th equto ollow: O eqult: 8 4 We ppl popoed ethod o d how eult g. wth MSE 9.677e-7. 56

9 Mjd Hllj et l ch. ppl. Sc. Re. 5 6:49-6 Eple 6.. I th eple we code ole o-ooth uzz ucto ollow: 45 Sce ojectve ucto o-ooth uzz ucto. We d the glol wek deetto o whch the optl oluto o the ollowg optzto pole.we olve th pole ed two tep ethod d VK ethod o d wth uppoe Mze p. Mze p. p. p. Mze Mze ollow: p d 5 5 p d p 5 5 p Lt equto NLP pole d we ot t oluto Mtl otwe.the optl oluto how g.. ll we d uzz potve oot o ed popoed ethod pecewe le ppoto g.4. Tle 5. cope ppoted d ect oluto o lt eple. Copo eult how the eectvee o the popoed ppoch the peece o uzz ole o-ooth ucto Tle 5.-Nuecl Reult o Eple o Nole Noooth uzz ucto 4 l cut Me qued olzed eo Ect oluto Lowe Boud ppoted oluto Ect oluto Uppe Boud ppoted oluto e-6 57

10 Mjd Hllj et l ch. ppl. Sc. Re. 5 6: Etted Potve Root Rel Potve Root.7 L CUTS Potve Root g.. Potve oluto o popoed ethod.9 Etted Ptve Root Rel Potve Root.8.7 L CUTS Potve Root g.. Potve oluto d eo o popoed ethod 58

11 Mjd Hllj et l ch. ppl. Sc. Re. 5 6: Blck : Wek Deeto o Blue : Wek Deeto o << Red : Wek Deeto o lol Wek Deeto g-lol Wek Deetto o Nole No-Sooth ucto Etted Root Rel Root.7.6 l cut g 4. Potve oot o o-ooth uzz ucto Bed Pecewe le ppoto CONCLUSION I th ppe we hve uggeted uecl olvg ethod o o-le uzz equto ted o tdd ltcl techque whch e ot utle evewhee. lo the ppoch c e eteded o o-le o- 59

12 Mjd Hllj et l ch. ppl. Sc. Re. 5 6:49-6 ooth uzz equto ovel deto o glol wek deetto the ee o L d LP o. The dvtge o th ppoch tht we oted ppoto o the optu oluto o the uzz pole wth dele ccuc. Itll we wote ole d o-ooth uzz equto petc o d the olve t the le petc ppoto ethod. ll eple wee peeted to llutte popoed ethod. REERENCES J.J. Buckle Y. Qu uzz Set d Ste J.J. Buckle Y. Qu uzz Set d Ste J.J. Buckle Y. Qu uzz Set d Ste J.J. Buckle Y. Qu uzz Set d Ste S.S.L. Chg L.. Zdeh O uzz ppg d cotol IEEE Tcto o Ste M d Ceetc Y.J. Cho N.J. Hug S.M. Kg uzz Set d Ste 5. 7 J.E. De R.B. Schel Nuecl Method o Ucoted Optzto d Nole Equto Petce-Hll New Jee D. Duo H. Pde Joul o Ste Scece D. Duo H. Pde uzz Set d Ste: Theo d pplcto cdec Pe New Yok 98. J. g uzz Set d Ste R. oetchel W. Vo uzz Set d Ste J. M. eg uzz Set d Ste M. Mzuoto Soe popete o uzz ue : M.M. upt R.K. Rgde R.R. Yge Ed. dvce uzz Set Theo d pplcto Noth-Holld ted 979 pp M. Mzuoto K. Tk Ste Copute d Cotol S. Nh uzz Set d Ste L.. Zdeh uzz et Ioto d Cotol L.. Zdeh Ioto Scece H.J. Ze uzz Set Theo d t pplcto Kluwe cdec Pe Dodecht M. Vz.V. Kd. Jj S. Et Coputtol d ppled Mthetc Volue N. pp S. d B. d ppled Mthetc d Coputto K.P. Bdkhh.V. Kd. ze ppled Mthetc d Coputto M. Vz.V. Kd S. Et d M. chpz petc lezto ppoch o olvg ole pogg pole lgh Joul Stttc tcle pe. 6