New Fuzzy Numerical Methods for Solving Cauchy Problems

Transcription

1 Article New Fuzzy Numericl Metods for Solving Cucy Problems Hussein ALKssbe, *, Irin Perfiliev, Mummd Zini Amd nd Zinor Ridzun Yy Institute of Engineering Mtemtics, Universiti Mlysi Perlis, Kmpus Tetp Pu Putr, Aru 0600, Perlis, Mlysi; M.Z.A.; Z.R.Y. Institute for Reserc nd Applictions of Fuzzy Modelling, University of Ostrv, NSC IT4Innovtions, 30. dubn, Ostrv, Czec Republic; * Correspondence: Received: 7 April 08; Accepted: 3 My 08; Publised: My 08 Abstrct: In tis pper, new fuzzy numericl metods bsed on te fuzzy trnsform F-trnsform or FT for solving te Cucy problem re introduced nd discussed. In ccordnce wit existing metods suc s trpezoidl rule, Adms Moulton metods re improved using FT. We propose tree new fuzzy metods were te tecnique of FT is combined wit one-step, two-step, nd tree-step numericl metods. Moreover, te FT wit respect to generlized uniform fuzzy prtition is ble to reduce error. Tus, new representtions formuls for generlized uniform fuzzy prtition of FT re introduced. As n ppliction, ll tese scemes re used to solve Cucy problems. Furter, te error nlysis of te new fuzzy metods is discussed. Finlly, numericl exmples re presented to illustrte tese metods nd compred wit te existing metods. It is observed tt te new fuzzy numericl metods yield more ccurte results tn te existing metods. Keywords: fuzzy prtition; fuzzy trnsform; new itertive metod; Cucy problems. Introduction In fct, most mtemticl models in engineering nd science requires te solution of ordinry differentil equtions ODEs. Generlly, it is difficult to obtin te closed form solutions for ODEs, especilly, for nonliner nd nonomogeneous cses. Mny models often led to ordinry differentil equtions wic consist of Cucy problems re n importnt brnc of modern mtemtics tt rises nturlly in different res of pplied sciences, pysics, nd engineering. Tus, mny resercers strt developing metods for solving Cucy problems re of prticulr importnce [ 3]. FT ws coined by Perfiliev s new mtemticl metod ws developed [4]. Te core ide of FT is fuzzy prtition of universe into fuzzy subsets. Te tecnique of FT s been successfully pplied into oter mtemticl problems s well including imge processing, nlysis of time series nd elsewere [5 7]. Tis ide s been pplied to Cucy problems ws first publised s well s oter numericl clssicl metods [8], by proposing generlized Euler nd Euler- Cucy metods, so tt te Mid-point FT metod ws demonstrted in [9]. Te success of tese pplictions is due in prt to te fct tt FT is cpble to ccurtely pproximte ny continuous function. Tus, we will propose new fuzzy numericl metods for Cucy problems wit elp of te FT nd new itertive metod. Te motivtion of te proposed study comes from te ppers [3,8,0]. Numeric Solution to te Cucy problem ws considered nd te utors sowed tt te error cn be reduced by using FT wit uniform fuzzy prtitions [8,9]. At te sme time, [0,], te concept of generlized fuzzy prtition ws proposed. Besides oters, necessry nd sufficient condition mking it possible to design esily Appl. Syst. Innov. 08,, 5; doi:0.3390/si0005

2 Appl. Syst. Innov. 08,, 5 of 6 te generlized fuzzy prtition ws provided []. Tis is importnt for vrious prcticl pplictions of FT. Furter [3], te utors ve proposed modifictions trpezoidl rule nd Adms-Moulton metods nd 3-step to solve ODEs bsed on te new itertive metod ws introduced []. In tis pper, we discuss te problem tt considered in [8,9]. Te tringulr nd rised cosine generting function ws replced by new representtions formuls for generlized uniform fuzzy prtition of FT suc s power of te tringulr nd rised cosine generting function. We study pproximtion properties of te FT bsed on powers of tringulr nd rised cosine generlized uniform fuzzy prtition cn be constructed in suc wy tt te FT cn reduce error. Also, we propose modifictions in te FT introduced by I. Perfiliev [4] wit respect to new representtions formuls for generlized uniform fuzzy prtition of FT nd ten te tecnique of FT is combined wit trditionl metods bsed on te new itertive metod [,3] to solve Cucy problems. It is observed tt te new metods proposed re more ccurte results tn te fuzzy pproximtion metod [8,9]. Tis pper is orgnized s follows. In Section, we introduce te bsic concepts nd results of te FT wit respect to te generlized uniform fuzzy prtition needed trougout tis pper. Te min prt of tis pper is Sections 3 nd 4, new representtions for bsic functions of FT, followed by te modified one step, -step, nd 3-step bsed on new representtions formuls for generlized uniform fuzzy prtition of FT. In Section 5, numeric exmples re discussed. Concluding remrks re presented in Section 6. Trougout te pper, we denote by N, N +, Z, R, nd R + te sets of nturl including zero, positive nturl, integer, rel, nd positive rel numbers, respectively.. Bsic Concepts In tis section, we give some definitions nd introduce te necessry nottion in [0], wic will be used trougout te pper. Trougout tis section, we del wit n intervl [, b] R of rel numbers. Definition. generlized uniform fuzzy prtition Let x i [, b], i =,..., n, be fixed nodes suc tt = x <... < x n = b, n. We sy tt te fuzzy sets A i : [, b] [0, ] constitute generlized fuzzy prtition of [, b] if for every i =,..., n tere exists > 0 suc tt x 0 = x, x n = x n+, [x i, x i + ] [, b] nd te following conditions re fulfilled:. positivity nd loclity A i x > 0 if x x i, x i+ nd A i x = 0 if x [, b] \ x i, x i+ ;. continuity A i is continuous on [x i, x i+ ]; 3. covering for x [, b], n i= A ix > 0. Fuzzy sets A,..., A n re clled bsic functions. It is importnt to remrk tt by conditions of loclity nd continuity, A ixdx > 0. A generlized of uniform fuzzy prtition of [, b] is defined for equidistnt nodes, i.e., for ll i =,..., n, x i = x i+ +, were = b / n nd two dditionl properties re stisfied, 4. A i x i x = A i x i + x for ll x [0, ], i =,..., n ; 5. A i x = A i x nd A i+ x = A i x for ll x [x i, x i+ ], i =,..., n ; ten te fuzzy prtition is clled -uniform generlized fuzzy prtition. Trougout tis pper, we will write generlized uniform fuzzy prtition insted of -uniform generlized fuzzy prtition. Definition. generting function A function K : [, ] [0, ] is clled generting function if it is ssumed to be even, continuous nd K x > 0 if x,. Te function K : [, ] R is even if for ll x [0, ], K x = K x. Te following definition recll te concept of generlized fuzzy prtition wic cn be esily extended to te intervl [, b]. We ssume tt [, b] is prtitioned by A,..., A n, ccording to Definition.

3 Appl. Syst. Innov. 08,, 5 3 of 6 Definition 3. A generlized uniform fuzzy prtition of intervl [, b], determined by te triplet K,,, cn be defined using generting function K Definition. Ten, bsic functions of generlized uniform fuzzy prtition re sifted copies of K defined by x xi A i x = K, x [x i, x i + ], for ll i =,..., n. Te prmeter is clled te bndwidt or te sift of te fuzzy prtition nd te nodes x i = + i re clled te centrl point of te fuzzy sets A,..., A n. Remrk. A fuzzy prtition is clled Ruspini if te following condition A i x + A i+ x =, i =,..., n, olds for ny x [x i, x i+ ]. Tis condition is often clled Ruspini condition. 3. New Representtions of Bsic Functions for Prticulr Cses In tis section, we propose two subsection, new representtions of bsic functions constitute generlized uniform fuzzy prtition of intervl [, b] nd ten FT tecnique bsed on new representtions of bsic functions. 3.. Power of te Tringulr nd Rised Cosine Generlized Uniform Fuzzy Prtition Two types of bsic functions, tringulr nd sinusoidl sped membersip functions, were proposed by [4,8]. Lter [3], te utors considered different spes for te bsic functions of fuzzy prtition. Furtermore, generlized fuzzy prtition ppered in connection wit te notion of iger-degree F-trnsform []. Its even weker version ws implicitly introduced to stisfy te requirements of imge compression [4]. Recently, te different conditions for generlized uniform fuzzy prtitions ws proposed by [0,]. Tble provides te definition two types of generting function, tringulr nd rised cosine generting functions [7,0,5]. Tble. Generting functions of strong uniform fuzzy prtition. Tringulr Generting Function mx { x, 0} Rised Cosine Generting Function + cos πx [,] In te following, we present new representtions for generting function. In prticulr, we present tree new representtions, bsed on te tringulr nd rised cosine generting functions: two generting function bsed on te tringulr generting functions nd one generting function bsed on te rised cosine generting function. Definition 4. nturl order tringulr generting function Let K T m i : R [0, ], i =,, be defined by. K T m x = { x m, x, 0, oterwise = min x m,,. K T m x = { x m, x, 0, oterwise = min x m,, 3 re clled power of te tringulr sped generting functions, wen m N +.

4 Appl. Syst. Innov. 08,, 5 4 of 6 Definition 5. odd nturl order rised cosine generting function Let K C m : R [0, ] be defined by K C m x = { + cos m πx, x ; 0, oterwise. 4 is clled power of te rised cosine generting function, i.e., m = k, k N +. wen m is n odd nturl number Remrk. Prticulrly, we cn ceck te vlidity of Eqution 4 using te following reltion K C m x = = { + cos m πx, x, 0, oterwise. { + sin m π x +, x, 0, oterwise. Lemm. If K T n i functions, ten x, i =,, K C m x determines power of te tringulr rised cosine generting. K T n x dx = n+,. K T n n x dx = n+, 3. K Cm x dx =, or equivlent. K t T n dt = n+,. K t T n dt = n n+, 3. K t C m dt =, were 0, n, be positive rel numbers, m is n odd nturl number nd n N +. n+ n+ Proof. Te proof cn be esily obtined by using integrtion metods witin te boundries nd ten substitution x = t/. On te bsis of Definitions 4 nd 5, Lemm, nd ccording to Definition 3, we cn lso be defined using generting function αk for α > 0 in generl, not necessrily stisfy Ruspini condition. Tus, bsic functions of generlized uniform fuzzy prtition re sifted copies of αk defined by x x0 A k x, x 0 = αk k, x [x i, x i+ ]. 5 In prticulr, let K T m, K T m, nd K C m be power of te tringulr nd rised cosine generting function defined bove. We will sy tt generlized uniform fuzzy prtition is power of tringulr or of rised cosine generlized uniform fuzzy prtition if its generting function K belongs to αk T m, αk T m, or αk C m wenever α = / Ktdt. Indeed, te equlity α immeditely follows from αk T m t dt = α = / K T. m t dt In te following, we modified te definition tringulr nd rised cosine generlized uniform fuzzy prtition by propose tt power of te tringulr nd rised cosine generlized uniform fuzzy prtitions cn be simply using te equlity α = / Ktdt. Definition 6. Let m N +. A system of fuzzy sets {A k k Z} defined by. A k x, x 0 = αk T m x x0. A k x, x 0 = αk T m x x0 k, α = m+, 6 k, α = m+ m, 7

5 Appl. Syst. Innov. 08,, 5 5 of 6 is clled power of te tringulr generlized uniform fuzzy prtition of te rel line determined by te triplet K T m i,, x 0, i =,. Furter, let m is n odd nturl number. A system of fuzzy sets {A k k Z} defined by 3. A k x, x 0 = αk C m x x0 k, α =, 8 is clled power of te rised cosine generlized uniform fuzzy prtition of te rel line determined by te triplet K C m,,, x 0. Te prmeter is bndwidt of te fuzzy prtition nd x 0 + k = x k. Definition 7. Let x <... < x n be fixed nodes witin [, b] R, suc tt x =, x n = b nd n. We consider nodes x,..., x n re equidistnt, wit distnce sift = b / n. A system of fuzzy sets B,..., B n : [, b] [0, ] be power of tringulr nd rised cosine generlized uniform fuzzy prtitions of [, b] if it is defined by B k x = { Ak x,, x [, b], 0, oterwise. or equivlent B k x = αk x xk, x [, b], 0, oterwise. were x k = + k. In te sequel, we denote K for generting function determined by te Formuls 4. Furter, α, A k x,, k =,..., n, re determined by te Formuls 6 8. Lemm. If B k x determines power of te rised cosine generlized uniform fuzzy prtition of [, b], ten B k x stisfied Ruspini condition wen m see 4 is n odd nturl number. Proof. Indeed, if x [, b], tere exists k {,..., n } suc tt x [x k, x k+ ]. By 4 nd 8, nd Remrk, we get x xk x xk+ B k x + B k+ x = A k x, + A k+ x, = αk C m + αk C m, = x + cos m xk π + x + cos m xk+ π, = + cos m π x x k + cos m π x x k+. By te properties of trigonometric functions, notice ttcos θ + π = cos θ, it is esy to see tt x cos π m xk x + cos m xk+ x π = cos m xk+ x π + π + cos m xk+ π. Tus, if m is n odd nturl number, te result is 0. In te following, if K is norml generting function i.e., K0 =, not necessrily stisfy Ruspini condition, we use generting function αk for α > 0, were αk x = α K x. Lemm 3. If bsic functions B k, k =,..., n, of generlized uniform fuzzy prtition re sifted copies of αk, α > 0, defined by te Formul 5 nd moreover, K is norml s n dditionl condition. Ten, for ec k =,..., n, B k x k = α, x k [x k, x k + ]. Proof. A generting function K is sid to be norml if K0 =. By te Formul 5 nd generting function K is norml, we get B k x k = αk xk x k = αk0 = α > 0. Corollry. Let te ssumptions of Lemm 3 be fulfilled, but fuzzy sets B k, k =,..., n, n, determined by Definition 7. Ten, for ec k =,..., n, B k x k = α, x k [x k, x k + ], were α is defined by Definition 7. 9

6 Appl. Syst. Innov. 08,, 5 6 of 6 Proof. Indeed, te proof immeditely follows from Definition 7 nd Lemm 3. Corollry. Let te ssumptions of Lemm 3 be fulfilled, but fuzzy sets B k, k =,..., n, n, determined by Definition 3. Ten, for ec k =,..., n, B k x k =, x k [x k, x k + ]. 3.. New FT Bsed Power of te Tringulr nd Rised Cosine Generlized Uniform Fuzzy Prtition In tis subsection, we present te min principles of F-trnsform detiled in [8,0,] tt re modified wit respect to power of te tringulr nd rised cosine generlized uniform fuzzy prtition. Furter, we will sow tt FT components wit respect to power of te tringulr nd rised cosine generlized uniform fuzzy prtition cn be simplified nd pproximted of n originl function, sy f. Definition 8. Let f be continuous function on [, b] nd B k t, k =,..., n, be power of te tringulr nd rised cosine generlized uniform fuzzy prtition of [, b], n. A vector of rel numbers F[ f ] = F, F,..., F n given by F k = f t B kt dt B, 0 kt dt for k =,..., n is clled te direct FT of f wit respect to power of te tringulr nd rised cosine generlized uniform fuzzy prtition B k. In te following, we ssume generting function K in te Formuls 4. We will simplify te representtion 0. Lemm 4. Let f C [, b] nd ccording to Definition 7, fuzzy sets B k, k =,..., n, n, be power of tringulr nd rised cosine generlized uniform fuzzy prtition of [, b] wit generting function K, ten representtion 0 of direct FT cn be simplified s follows for k =,..., n F k = f t + t k Kt dt Kt dt = f t + t k K t dt K t dt. Proof. In tis proof, we will write generting function K insted of 4. By Definition 7, we get t tk B k t = αk, t [t k, t k + ], for k =,..., n, t 0 = t, t n+ = t n, nd substituting u = t t k nd ten substituting t = s/. Tus, we get tk+ f t B k t dt = α f t + t k Kt dt = α f t + t k K t t k dt tk+ B k t dt = α Kt dt = α K t dt t k nd its corresponding results wit representtion 0. Indeed, te previous lemm olds for every fuzzy prtition generted by kernel. Now, we will simplify te bove given expressions for te coefficients F[ f ] = F, F,..., F n in te representtion 0 even more. Tis fct is very importnt for pplictions wic re more flexible nd consequently esier to use.

7 Appl. Syst. Innov. 08,, 5 7 of 6 Lemm 5. Let te ssumptions of Lemm 4 be fulfilled. Ten, te coefficients F[ f ] = F, F,..., F n in te expression 0 of te FT component F k of f s follows: F k = f t B k t dt = α t tk f t K dt, for k =,..., n, were intervl [, b] is prtitioned by power of te tringulr nd rised cosine generlized uniform fuzzy prtition B,..., B n nd α is defined by Definition 7. Proof. Let k {,..., n} nd consider set of fuzzy sets B k x from power of te tringulr nd rised cosine generlized uniform fuzzy prtition of [, b] in 9. We will prove te equlity t k+ t B k k t dt =. We get by virtue of Lemms nd 4, nd 6: tk+ t k B k t dt = tk+ t k A k t,, dt = tk + t k m + K T m t tk dt = m + K T m t dt =, were is te bndwidt of te fuzzy prtition nd t k = + k. Similrly, te oter Formuls 7 nd 8 will be proved nd ten its corresponding in te expression 0. Lemm 6. Let f C [, b]. Ten for ny ε > 0 tere exist n ε N nd B,..., B nε be bsic functions form power of te tringulr nd rised cosine generlized uniform fuzzy prtition of [, b]. Let F k, k =..., n, be te integrl FT components of f wit respect to B,..., B nε. Ten for ec k =..., n ε te following estimtions old: f t F i ε for ec t [, b] [t k, t k+ ] nd i = k, k +. Proof. see [4]. Corollry 3. Let te conditions of Lemm 6 be fulfilled. Ten for ec k =..., n ε te following estimtions old: F k F k+ < ε. Proof. According to [4,6], let t [, b] [t k, t k+ ]. Ten by Lemm 6, for ny k =,..., n we obtin f t F k < ε/ nd f t F k+ < ε/. Tus, F k F k+ f t F k + f t F k+ < ε + ε = ε. Te following teorem estimtes te difference between te originl function nd its direct FT wit respect to power of te tringulr nd rised cosine generlized uniform fuzzy prtition. Teorem. Let f t C [, b] nd te conditions of Lemm 5 be fulfilled. Ten for k =,..., n were α > 0 or α is defined by Definition 7. F k = α f t k + O, Proof. By loclity condition for Definition, Lemms 3 nd 5, nd ccording to te proof of Lemm 9.3 [8], using te trpezoid formul wit nodes t k, t k, t k+ to te numericl computtion of te integrl, we get for α > 0 F k = tk+ t k f t B k t dt, =. f t k B k t k + f t k B k t k + f t k+ B k t k+ + O = f t k B k t k + O = f t k A k t k, + O, = f t k αk 0 + O, = α f t k + O.,

8 Appl. Syst. Innov. 08,, 5 8 of 6 Definition 9. Let F[ f ] = F, F,..., F n be direct FT of function f C [, b] wit respect to te fuzzy prtition B k t, k =,..., n of [, b]. Ten, te function ˆf defined on [, b] is clled te inverse FT of f. ˆf t = n k= F kb k t n k= B kt, 3 Corollry 4. Let te ssumptions of Lemm nd moreover, Let ˆf t be te inverse FT of f wit respect to power of te rised cosine generting function. Ten, for ll t [, b] te following olds: ˆf t = n k= F k B k t. Proof. Tis proof immeditely follows from Defintion 9, Lemm nd ten using n k= B kt =. Te following lemm estimtes te difference between te originl function nd its inverse FT. Lemm 7. Let te ssumptions of Teorem nd let ˆf t be te inverse FT of f wit respect to te fuzzy prtition of [, b] is given by Definition 7. Ten, for ll t [, b] te following estimtion olds: ˆf t = α f t k + O. 4 Proof. Let t [, b] so tt x [t k, t k+ ] for some k =,..., n. By Teorem, ˆf t α f t k = n k= F kb k t n k= B kt = n k= F k α f t k B k t n k= B kt α f t = n k= F kb k t n k= B kt = O. n k= α f t k B k t n k= B kt Corollry 5. Let te ssumptions of Lemm 7, ten ˆf t f t < ε. Proof. Te proof esily follows from te proof of Lemm 7 nd ten using Lemm 6 s follows: ˆf t f t = n k= F k f t B k t n k= B < ε. kt Remrk 3. According to te Definitions nd, if te normlity is considered to be n dditionl condition for generting function i.e., K0 = nd generlized uniform fuzzy prtition of [, b] stisfies A k x k = α, α > 0, ten it is esy to see tt te inverse FT ˆf t k = F k for ll k =,..., n. Tis is true for Definition 7. Moreover, if ortogonlity condition Ruspini condition is replced by covering condition in Definition nd generlized uniform fuzzy prtition of [, b] stisfies A k x k = α =, ten it is esy to lso see tt te inverse FT ˆf t k = F k for ll k =,..., n. Tis is true for Formul 8 only. Importnt property of te direct FT s well s inverse FT is teir linerity, nmely, given f, g C [, b] nd α, β R, if = α f + βg, ten F [] = αf [ f ] + βf [g] nd ĥ = α ˆf + βĝ. In te next section, we present new fuzzy numericl metods bsed on te FT nd new itertive metod to numeric solution of te Cucy problem. 4. New Fuzzy Numericl Metods for Cucy Problem Consider te initil vlue problem IVP for te Cucy problem: y = f t, y, yt = y, = t t t n = b. 5

9 Appl. Syst. Innov. 08,, 5 9 of 6 were y R nd f is continuous function on [, b] R nd stisfies Lipscitz condition. In fct, te nlyticl solution of problem 5 is often difficult nd sometimes impossible to obtin. Insted, numericl nlysis is interested wit obtining pproximte solutions wit errors witin resonble bounds. Tus, usge of fuzzy numericl metods seems to be suitble. In [8,9], te utors ve presented Euler metod nd Mid-point rule, bsed on FT to numeric solution of Cucy problem 5. A new itertive metod NIM s been proposed for solving liner nonliner functionl equtions, ordinry differentil equtions nd dely differentil equtions [,3]. In tis section, we present tree new scemes to solve Cucy problem 5, tt use te FT nd NIM. Our motivtion stems from te clssicl pproc, trpezoidl rule -step nd Adms Moulton metods nd 3-step. For te rest of tis pper, suppose tt we re given te Cucy problem 5, were te function f on [, b] re sufficiently smoot nd we ssume tt ll necessry requirements for constructing te FT of te solution of Cucy problem 5 re fulfilled. Now, we present numericl Sceme I, II, nd III. Te first sceme uses -step metod, wile te second one uses -step metod, nd te tird uses 3-step metod. 4.. Numeric Sceme I: Modified Trpezoidl Rule Bsed on FT nd NIM for Cucy Problem In te present subsection, we will construct numeric sceme of te more dvnced metod known s te Trpezoidl Rule. Recll tt it is one-step metod wit second-order ccurcy, wic cn be considered s Runge Kutt metod. We propose modifiction of trpezoidl rule bsed on FT nd NIM for solving Cucy problem. Modifiction of te trpezoidl rule cn be improved by te FT to solve Cucy problem 5. We contributed to numeric metods of Cucy problem 5 by sceme provides formuls for te FT components, Y k, k =,..., n, of te unknown function yt wit respect to coose some power of te tringulr or rised cosine generlized uniform fuzzy prtition, B,..., B n, of intervl [, b] wit prmeter to pproximte solution of Cucy problem 5. Te first, coose te number n nd compute = b / n, ten construct te generlized uniform fuzzy prtition of [, b] using Definition 7. Note tt ec function B k spns over tree nodes t k, t k, t k+, k =,..., n. Neverteless, B k t k = B k t k+ = 0 nd B k t k =. Now, we pply te FT nd NIM to Cucy problem 5 nd obtin te numeric Sceme I for k =,..., n s follows see [3,8] for tecnicl detils: were Y = y, Yk+ = Y k + F k /, Yk+ = Y k+ + F k+ /, 6 Y k+ = Y k + F k + Fk+ /, F k = f t, Y kb k tdt B, Fk+ = ktdt f t, Y k+ B k+tdt B, Fk+ = k+tdt f t, Y k+ B k+tdt B k+tdt. 7 In te sequel, te pproximte solution of Cucy problem 5 cn be obtined using te inverse FT s follows: n y n t = Y k B k t. 8 k= 4.. Numeric Sceme II: Modified -Step Adms Moulton Metod Bsed on FT nd NIM for Cucy Problem Te Sceme I uses -step metod for solving Cucy problem 5. In tis subsection, we improve -step Adms Moulton metod using FT nd NIM for solving Cucy problem 5. Te -step Adms Moulton metod cn be improved to effectively pproximte te solution of 5 by te FT components, Y k, k =,..., n, of te unknown function yt wit respect to coose some power of

10 Appl. Syst. Innov. 08,, 5 0 of 6 te tringulr or rised cosine generlized uniform fuzzy prtition 9. Let Y = y nd Y = y if possible; oterwise, we cn compute FT component Y from numeric Sceme I. Anlogously to [3,8], we pply te FT nd NIM to Cucy problem 5 nd obtin te numeric Sceme II in te following form for k =,..., n : Y k+ = Y k + 8F k F k /, were F k = F k+ = Yk+ = Y k+ + 5F k+ /, Y k+ = Y k + 9 8F k F k + 5Fk+ /, f t, Y k B k tdt B, F k = k tdt f t, Y k+ B k+tdt B, nd Fk+ = k+tdt f t, Y kb k tdt B, ktdt f t, Y k+ B k+tdt B. k+tdt Ten, obtin te desired pproximtion for y by te inverse FT 8 pplied to [Y,..., Y n ] Numeric Sceme III: Modified 3-Step Adms Moulton Metod Bsed on FT nd NIM for Cucy Problem In tis subsection, we improve 3-step Adms Moulton metod using FT nd NIM for solving Cucy problem 5. Te 3-step Adms Moulton metod cn be improved to effectively pproximte te solution of 5 by te FT components, Y k, k =,..., n, of te unknown function yt wit respect to coose some power of te tringulr or rised cosine generlized uniform fuzzy prtition see Definition 7, B,..., B n, of intervl [, b] wit prmeter = b / n, n. Let Y = y, Y = y nd Y 3 = y 3 if possible; oterwise, we cn compute FT components Y nd Y 3 from numeric Sceme I. Now, we pply te FT nd NIM to Cucy problem 5 nd obtin te following numeric Sceme III for k = 3,..., n see [3,8] for tecnicl detils: were Y k+ = Y k + 9F k 5F k + F k /4, Yk+ = Y k+ + 9F k+ /4, Y k+ = Y k + 0 9F k 5F k + F k + 9Fk+ /4, F k = f t, Y k A k tdt A, F k = k tdt f t, Y k A k tdt A, F k = k tdt f t, Y ka k tdt A, ktdt F k+ = f t, Y k+ A k+tdt A, nd Fk+ = k+tdt f t, Y k+ A k+tdt A. k+tdt In te sequel, te inverse FT 8 pproximtes te solution yt of te Cucy problem Error Anlysis of Fuzzy Numeric Metod for Cucy Problem In tis subsection, we present error nlysis for numeric sceme I only, becuse te tecnique of error nlysis for rest numeric scemes Scemes II nd III cn be obtined nlogously. Consider te Formul 6. If yt k = y k nd Y k denote te exct solution nd te numericl solution nd substituting te exct solution in te Formul 6, we get y k+ = y k + F e k /, y k+ = y k+ + Fe k+ /, y k+ = y k + Fk e + Fe k+ /,

11 Appl. Syst. Innov. 08,, 5 of 6 were F e k = f t, y kb k tdt B, Fk+ e = ktdt nd te trunction error T k of te Sceme I is given by f t, y k+ B k+tdt B, Fk+ e = k+tdt f t, y k+ B k+tdt B, k+tdt Rerrnging 6, we get T k = y k+ y k 0 = Y k+ Y k F e k + F e k+. 3 Fk + F k+. 4 If we denote te error e k+ = Y k+ y k+ nd subtrcting 4 from 3, so: T k = e k+ e k F k F e k F k+ Fe k+. 5 Lemm 8. Let f is ssumed to be sufficiently smoot function of its rguments on [, b] nd stisfies te Lipscitz condition wit te constnt L wit respect to y, ten we get for k =,..., n, e k+ e k + c + T nd Fk e Fe LM were c = L + L + 3 L 3 8, T = mx T k, M is upper bound for f, nd F e k n k, Fe k+ Formul. k+ re determined by Proof. By ypotesis, f stisfies te Lipscitz condition nd using Lemm 5, Formuls 6, 7, nd, we get F k Fk e f t, Y k B k tdt f t, y k B k tdt L e k F k+ Fe k+ f t, Yk+ B k+tdt f t, y k+ B k+tdt L Y k+ y k+ F k+ Fe k+ f t, Yk+ B k+tdt f t, y k+ B k+tdt L Y k+ y k+ Y k+ y k+ Y k + F k / y k + Fk e / e k + L Y k+ y k+ Yk+ + F k+ / y k+ + Fe k+ / e k + L e k+ e k + L e k + L y e k + L e k + L e k = e k + L + L k+ Y k+ + L + 3 L T + T + T

12 Appl. Syst. Innov. 08,, 5 of 6 Furtermore, by using f t, yt M, we get F e k Fk+ e = b = b L yk y = L = L f t, yk f t +, y k+ B k tdt f t, yk f t, y k+ k+ Fk e Fe LM Tis completes te proof. k+ f t, yk + f t, y k+ B k tdt + f t, y k+ f t +, y k+ B k tdt Teorem. Consider te te numeric Sceme I 6, were f C [, b] nd stisfies te Lipscitz condition wit te constnt L wit respect to y. Ten, te solution Y k, k =,..., n, obtined by te numeric sceme I 6 for solving Cucy problem 5 stisfies e k = Y k y k M L ekc, 6 were c = L + L + 3 L 3 8, M, M re upper bound for f, f, respectively, on [, b], nd M + M L = M. Proof. By ypotesis, y exists nd bounded on [, b] wit mx t b y t = M by ssuming tt f C [, b]. Ten, using Lemm 8, 3 nd Tylor s teorem for k =,..., n, we get T k = y k+ y k were ξ k [t k, t k+ ]. Now, using Lemm 8 = y ξ k + f t k, y k F e k + Fk+ e F e k + Fk+ e = y ξ k + f t k, y k Fk e + Fe k Fe k+ = y ξ k + F e k Fk+ e T = mx T k k n y ξ k + LM M + LM = M Now, by virtue of Lemm 8 nd we ve used e = 0, + c k e kc, we get for k =,..., n e k + ck T c T L ekc M L ekc + c k L + L + L 3 8 were c = L + L + 3 L 3 8. Tus, if te step lengt 0, ten for ll k, te error, e k converges to zero. So te metod is convergent. Tis completes te proof. T

13 Appl. Syst. Innov. 08,, 5 3 of 6 5. Numericl Exmples In tis section, we present exmples of te Cucy problem 5. Exmple. Consider te following initil vlue problem wit initil conditions y 0 = nd wit smoot rigt-nd function y t = t y, t [0, ]. 7 Exmple. Consider te Cucy problem 5 wit oscillting rigt-nd function. We tke f t, y = + y cos t + sin t, t π =.95, = π nd b = 3π. Te results re listed in Tbles 4 by fuzzy numericl metods proposed in tis pper wit respect to cse K T 0 nd Tble 5 by fuzzy numericl metods proposed in tis pper wit respect to cse K T, K T 3, K T 0, K C. Te Eucliden distnce is given by Norm l defined s Y yt = k Y k yt k nd men squre error MSE defined s MSE = n Y k yt k. Tis is n esily computble quntity for prticulr smple. Concluding remrks re summrized s follows: In view of Tble, comprison between te Euler metod Euler-FT [8], te Mid-point rule Mid-FT, Sceme I nd II [9] nd tree new scemes 6, 9 nd 0 in tis pper for Exmple. We cn esily observe from Tble, te better results in comprison wit te Euler-FT metod [8] re obtined by te tree new scemes in tis pper nd te best result in comprison wit te Sceme I, II nd II is obtined by te Sceme III. Also, te better results in comprison wit te Mid-point rule Mid-FT, Sceme I nd II [9] re obtined by te Sceme II 9 nd Sceme III 0 in tis pper were ll fuzzy numericl metods used te FT components nd te best pproximtion is sown by te Sceme III 0 wit FT components. Tble. Comprison of numeric results for Exmple. Te columns contin te exct nd seven pproximte solutions of te Cucy problem 7 wit smoot rigt-nd function: te first tree pproximte solution is obtined by te tree new scems 6, 9 nd 0, te fourt pproximte solution by te Euler-FT [8] wit FT components nd te lst tree by te scemes re proposed in [9]. Te best pproximtion is sown by te Sceme III proposed bove 0 wit FT components. t i Solution yt Proposed Sceme I Proposed Sceme II Proposed Sceme III Euler-FT in [8] Mid-FT in [9] Sceme I in [9] Sceme II in [9]

14 Appl. Syst. Innov. 08,, 5 4 of 6 In Tbel 3, comprison of MSE nd comprison of Norm l for Exmples nd. We cn esily observe, te best results re obtined by te tree new scemes in tis pper nd te better results in comprison wit te oter numericl clssicl metods re obtined by ll fuzzy numericl metods used te FT components except Euler-FT [8] for tese exmples. Tble 3. Te vlues of MSE nd Norm l for Exmple. Metod Ex. Ex. Norm l MSE Norm l MSE Proposed Sceme I Proposed Sceme II Proposed Sceme III Euler-FT [8] Mid-FT [9] Sceme I [9] Sceme II [9] Trpezoidl Rule Step Adms Moulton Step Adms Moulton In view of Tble 4, comprison between te tree new scemes 6, 9 nd 0 in tis pper nd te Trpezoidl Rule, -Step Adms Moulton Metod nd 3-Step Adms Moulton Metod bsed on Euler metod for Exmple. We cn esily observe from Tble 4, te better results re obtined by te tree new scemes in tis pper nd te best result in comprison wit te Sceme I, II nd II is obtined by te Sceme III. Tble 4. Comprison of numeric results for Exmple. Te columns contin te exct nd six pproximte solutions of te Cucy problem 7 wit oscillting rigt-nd function: te first tree pproximte solution is obtined by te tree new scems 6, 9, nd 0, te lst tree pproximte solution by te Trpezoidl Rule, -Step Adms Moulton Metod nd 3-Step Adms Moulton Metod. Te best pproximtion is sown by te Sceme III proposed bove 0 wit FT components. t i Solution yt Proposed Sceme I Proposed Sceme II Proposed Sceme III Trp -Step Adms 3-Step Adms Trpezoidl Rule; -Step Adms Moulton Metod; 3 3-Step Adms Moulton Metod.

15 Appl. Syst. Innov. 08,, 5 5 of 6 In Tbel 5, comprison between computtion errors for tree scemes bsed on te FT wit respect to te power of te tringulr nd rised cosine generlized uniform fuzzy prtition determined by Formul 9, were te dvntge of te K T m for Exmples nd is evident. Tble 5. Te vlues of MSE nd Norm l for Exmples nd by te tree scemes wit respect to te power of te tringulr nd rised cosine generlized uniform fuzzy prtition re proposed in tis pper. Te best pproximtion is sown by using K T 0. Proposed Sceme I II III Cse Ex. Ex. Norm l MSE Norm l MSE K T K T K T K C K T K T K T K C K T K T K T K C Tis constitutes n importnt improvement to previous metods wic do not provide suc informtion except in te metods suc s Euler-FT proposed in [8] nd Mid-FT, Sceme I, nd Sceme II [9] for Cucy problems by te more efficient wy of computtion pproximte solutions. Tus, tis study will be of prticulr importnce. 6. Conclusions We extended pplicbility of fuzzy numeric metods to te initil vlue problem te Cucy problem. We proposed tree new numeric metods bsed on te FT nd NIM nd ten nlyzed teir suitbility. We considered in te cse of te generlized uniform fuzzy prtition is power of te tringulr rised cosine generlized uniform fuzzy prtition nd sowed tt te newly proposed scemes outperform te Euler-FT [8] nd Mid-FT, Sceme I, nd Sceme II [9] especilly on exmples were te generlized uniform fuzzy prtition is power of te tringulr generlized uniform fuzzy prtition by using generting function. Alos, te newly proposed scemes in tis pper outperform te Trpezoidl Rule, -Step Adms Moulton Metod nd 3-Step Adms Moulton Metod. Moreover, we proved tt te Sceme I determines n pproximte solution wic converges to te exct solution. Tis constitutes n importnt improvement to previous results were coined by I. Perfiliev [8]. To conclude previous sections, te proposed scemes re more ccurte nd stble. In prticulr, tese scemes cn be used for solving initil vlue problem. Autor Contributions: H. A. ALKssbe nd M. Z. Amd proposed nd designed te numericl metods. H. A. ALKssbe performed te numericl experiments. I. Perfiliev evluted te results nd supported tis work. M. Z. Amd project dministrtion. Z. R. Yy provided softwre nd dt curtion. Acknowledgments: Tis work of Irin Perfiliev s been supported by te project "LQ60 IT4Innovtions excellence in science" nd by te Grnt Agency of te Czec Republic project No S. Also, mny tnks given to Universiti Mlysi Perlis for providing ll fcilities until tis work completed successfully. Conflicts of Interest: Te utors declre no conflicts of interest.

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