Appl Math Inf Sci 11, No 4, 1059-1074 (017 1059 Applied Mathematics & Infomation Sciences An Intenational Jounal http://dxdoiog/1018576/amis/11041 A Novel Iteative Numeical Algoithm fo the Solutions of Systems of Fuzzy Initial Value Poblems Oma Abu Aqub 1, Shahe Momani,3,, Saleh Al-Mezel 3 and Mawan Kutbi 3 1 Depatment of Mathematics, Faculty of Science, Al Balqa Applied Univesity, Salt 19117, Jodan Depatment of Mathematics, Faculty of Science, The Univesity of Jodan, Amman 1194, Jodan 3 Nonlinea Analysis and Applied Mathematics (NAAM Reseach Goup, Faculty of Science, King Abdulaziz Univesity (KAU, Jeddah 1589, Saudi Aabia Received: 3 May 017, Revised: 9 Jun 017, Accepted: 13 Jun 017 Published online: 1 Jul 017 Abstact: Behavios of many dynamic systems with uncetainty can be modelled effectively by systems of fuzzy diffeential equations In this pape, we develop new numeical iteative method fo solving systems of fuzzy initial value poblems based on the epoducing kenel theoy unde the assumption of Hukuhaa diffeentiability The exact and appoximate solutions ae given with seies fom in tems of thei paametic fom, whee two smooth epoducing kenel functions ae used thoughout the evolution of the algoithm to obtain the equied nodal values Futhemoe, eo estimation is poved in ode to captue the behavio of fuzzy solutions Applicability, potentiality, and efficiency of the poposed algoithm fo the fuzzy solutions of diffeent fuzzy systems ae investigated using compute tables and gaphical epesentation Keywods: Fuzzy diffeential systems; Repoducing kenel theoy; Hukuhaa deivative 1 Intoduction Theoy of systems of diffeential equations plays a vital ole to model physical, engineeing, and economical poblems, such as in solid and fluid mechanics, dynamic supply and demand, mathematical biology, plasma physics, contol theoy, and othe aeas of science [1,, 3, 4, 5, 6, 7, 8, 9, 10] But in actual case, the paametes, vaiables, o initial conditions involved in the diffeential systems may be uncetain, o a vague estimation of those ae found in geneal by some obsevation, expeiment, expeience, data collection, o maintenance induced eo So, to ovecome the uncetainty and vagueness, one may use fuzzy envionment in paametes, vaiables, and initial conditions in place of cisp ones So, with these uncetainties the geneal diffeential systems tun into fuzzy diffeential systems Numeical techniques ae widely used by scientists and enginees to solve thei poblems A majo advantage fo numeical techniques is that a numeical answe can be obtained even when a poblem has no analytical solution Anyhow, in most eal-life applications, it is too complicated to obtain the exact solutions to systems of fuzzy initial value poblems (FIVPs in tems of elementay functions in a simple manne, so an efficient, eliable numeical algoithm fo the solutions of such systems is equied; it is little wonde that with the development of fast, efficient digital computes, the ole of numeical methods in mathematics and engineeing poblem solving has inceased damatically in ecent yeas In this pape, we intoduce a novel iteative technique based on the use of epoducing kenel Hilbet space (RKHS method fo numeically appoximating solutions of systems of FIVPs in the space η ν=1 W [a,b] unde the Hukuhaa diffeentiability The new method has the following chaacteistics; fist, it is of global natue in tems of the solutions obtained as well as its ability to solve othe mathematical and engineeing poblems; second, it is accuate, need less effot to achieve the esults, and is developed especially fo the nonlinea case; thid, in the poposed method, it is possible to pick any point in the inteval of integation and as well the appoximate solutions and thei fist Hukuhaa deivatives will be applicable; fouth, the method does not equie discetization of the vaiables, and it is not effected by Coesponding autho e-mail: smomani@juedujo c 017 NSP
1060 O Abu Aqub et al: A novel iteative numeical algoithm fo computation ound off eos and one is not faced with necessity of lage compute memoy and time; fifth, the poposed appoach does not esot to moe advanced mathematical tools; that is, the algoithm is simple to undestand, implement, and should be thus easily accepted in the mathematical and engineeing application s fields Moe pecisely, we povide numeical appoximate solutions on the inteval [a, b] fo systems of FIVPs of the fom x 1 (t= f 1(t,x 1 (t,x (t,,x η (t, x (t= f (t,x 1 (t,x (t,,x η (t, x η (t= f η(t,x 1 (t,x (t,,x η (t, subject to the fuzzy initial conditions (1 x 1 (a=α 1,x (a=α,,x η (a=α η, ( whee f υ : [a,b] R η F R F ae continuous η-tuples fuzzy-valued functions, x υ : [a,b] R F, α υ R F, a,b R, and υ = 1,,,η Thoughout this paperthe set of eal numbes and R F denote the set of fuzzy eal numbes on R Repoducing kenel theoy has impotant applications in numeical analysis, diffeential equations, integal equations, intego-diffeential equations, pobability and statistics, and so fouth [11, 1, 13] Recently, a lot of eseach wok has been devoted to the applications of RKHS method fo wide classes of stochastic and deteministic poblems involving opeato equations, diffeential equations, integal equations, and intego-diffeential equations The RKHS method was successfully used by many authos to investigate seveal scientific applications side by side with thei theoies The eade is kindly equested to go though [14, 15, 16, 17, 18, 19,0,1,,3,4,5,6,7,8,9,30,31,3,33,34,35, 36, 37] in ode to know moe details about RKHS method, including its histoy, its modification fo use, its scientific applications, its kenel functions, and its chaacteistics The numeical solvability fo systems of FIVPs have been pusued by seveal authos To mention a few, in [38] the authos have discussed the geometic appoach to solve linea systems of FIVPs Futhemoe, the vaiational iteation method is caied out in [39] fo linea fuzzy diffeential system The homotopy analysis method (HAM has been applied to solve the linea fuzzy system as descibed in [40] Recently, the fuzzy neual netwok appoach fo solving linea system of FIVPs is poposed in [41] On the othe aspect as well, the numeical solvability of othe vesion of FIVPs can be found in [4, 43, 44, 45] and efeences theein As a esult, none of pevious studies popose a methodical way to solve systems of FIVPs in geneal Moeove, pevious studies equie moe effot to achieve the esults, they ae not accuate and usually they ae suited fo linea fom This pape is compised of 6 sections including the intoduction In the next section, oveview of fuzzy calculus theoy is collected In Section, η dimensional inne poduct spaces ae constucted in ode to apply the method In Section 3, seies epesentation of exact and appoximate solutions and theoetical basis of the method ae intoduced In Section 4, an iteative algoithm fo numeically appoximating the solutions is descibed and the n-tuncation appoximate solutions ae poved to convege to the exact solutions Softwae libaies and numeical expeiment ae pesented in Section 5 This aticle ends in Section 6 with some concluding emaks Oveview of fuzzy calculus theoy The contents of this section is basic in some sense, fo the eade s convenience, we pesent some necessay definitions fom fuzzy calculus theoy and peliminay esults Afte that, a numeical algoithm fo the solutions of systems of FIVPs based on thei -cut epesentation fom is intoduced Let S be a nonempty set A fuzzy set u in S is chaacteized by its membeship function u : S [0,1] Thus, u(s is intepeted as the degee of membeship of an element s in the fuzzy set u fo each s S A fuzzy set u on R is called convex if fo each s,t R and λ [0,1], u(λs+(1 λt min{u(s,u(t}; is called uppe semicontinuous if the set {s R u(s } is closed fo each [0,1]; and is called nomal if thee is s R such that u(s = 1 The suppot of a fuzzy set u is defined as {s R:u(s>0} Definition 1 [46] A fuzzy numbe u is a fuzzy subset ofr with nomal, convex, and uppe semicontinuous membeship function of bounded suppot The concept of a fuzzy eal numbe aises fom the fact that many quantifiable phenomena do not lend themselves to being chaacteized in tems of absolutely pecise numbes In fact, a fuzzy numbe is one which is descibed in tems of a numbe wod and a linguistic modifie, such as appoximately, nealy, o aound Fo each (0,1], set [u] = {s R:u(s } and [u] 0 ={s R:u(s>0}, whee{ } denote the closue of { } Then, it easily to establish that u is a fuzzy numbe if and only if [u] is a compact convex subset of R fo each [0,1] and [u] 1 φ [47] Thus, if u is a fuzzy numbe, then[u] =[u 1 (,u (], whee u 1 (=min{s : s [u] } u (=max{s : s [u] }, fo each [0,1] The symbol [u] is called the -cut epesentation o paametic fom of a fuzzy numbe u The question aises hee is, if we have an inteval-valued function [z 1 (,z (] defined on [0,1], then is thee a fuzzy numbe u such that c 017 NSP
Appl Math Inf Sci 11, No 4, 1059-1074 (017 / wwwnatualspublishingcom/jounalsasp 1061 [u] = [z 1 (,z (] The next theoem is chaacteizes fuzzy numbes though thei -cut epesentations Theoem 1 [47] Suppose that u 1,u : [0,1] R satisfy the following conditions; fist, u 1 is a bounded inceasing function and u is a bounded deceasing function with u 1 (1 u (1; second, fo each k (0,1], u 1 and u ae left-hand continuous functions at = k; thid, u 1 and u ae ight-hand continuous functions at = 0 Then defined by u :R [0,1], u(s=sup{ : u 1 ( s u (}, is a fuzzy numbe with paameteization [u 1 (,u (] Futhemoe, if u : R [0,1] is a fuzzy numbe with paameteization [u 1 (,u (], then the functions u 1 and u satisfy the afoementioned conditions In geneal, we can epesent an abitay fuzzy numbe u by an ode pai of functions (u 1,u which satisfy the equiements of Theoem 1 Fequently, we will wite simply u 1 and u instead of u 1 ( and u (, espectively Definition [48,49] The complete metic stuctue onr F is given by the Hausdoff distance mapping such that D :R F R F R + {0}, D(u,v= sup max{ u 1 v 1, u v }, 0 1 fo abitay fuzzy numbes u and v Let u,v R F If thee exists an element w R F such that u=v+w, then w is called the Hukuhaa diffeence of u and v, denoted by u v Hee, the sign stands always fo Hukuhaa diffeence and let us mention that u v u+( 1v Usually, we denote u+( 1v by u v, while u v stands fo the Hukuhaa diffeence Definition 3 [50] Let x : [a,b] R F and t 0 [a,b] We say that x is Hukuhaa diffeentiable at t 0, if thee exists an element x (t 0 R F such that fo each h > 0 sufficiently close to 0, the Hukuhaa diffeences x(t 0 + h x(t 0, x(t 0 x(t 0 h exist and x (t 0 =lim h 0 + x(t 0+h x(t 0 h = lim h 0 + x(t 0 x(t 0 h h Hee, the limit is taken in the metic space (R F,D and at the endpoints of [a,b], we conside only one-sided deivatives Next theoem shows us a way to tanslate a diffeential system fom fuzzy setting into odinay setting Theoem [51,5] Let x : [a,b] R F be Hukuhaa diffeentiable function and [x(t] =[x 1 (t,x (t] Then the endpoints functions x 1 and x ae diffeentiable on [a,b] and fo each [0,1] [ x (t ] = d dt [x(t] = [ x 1(t,x (t ], In some applications, the behavio of an object is detemined by physics laws and is cisp Howeve, if the initial values ae obtained fom measuements, fo example, this value can be uncetain and often thee ae moe suitable to model them using fuzzy numbes Next, we conside and study systems involving fuzzy equations and/o fuzzy initial conditions In othe wod, if the initial values ae fuzzy numbes, the solutions ae fuzzy functions, and consequently the deivatives must be consideed as fuzzy deivatives Let us conside the following system of fist-ode equations descibed the cisp odinay diffeential equations (ODEs on the inteval[a, b]: x 1 (t= f 1(t,x 1 (t,x (t,,x η (t, x (t= f (t,x 1 (t,x (t,,x η (t, x η (t= f η(t,x 1 (t,x (t,,x η (t, subject to the cisp initial conditions (3 x 1 (a=α 1,x (a=α,,x η (a=α η, (4 whee f υ : [a,b] R η R ae continuous η-tuples ealvalued functions, x υ : [a,b] R, α υ,a,b R, and υ = 1,,,η Assume that the initial conditions α υ in Eq (4 ae uncetain and modeled by fuzzy numbes Also, assume that the function f υ in system of ODE (3 contain uncetain paametes modeled by fuzzy numbes Then, we obtain system of FIVP (1 and ( Anyhow, in ode to solve this new system, we ewite the fuzzy functions x υ (t as [x υ (t] = [ x (υ 1 (t,x (υ (t ] and [x υ (a] = [ α (υ 1,α (υ ] Indeed, accoding to Nguyen theoem [53, 54] it follows that: [ f υ (t,x 1 (t,x (t,,x η (t] whee υ = 1,,,η = f υ (t,[x 1 (t],[x (t],,[x η (t] = { f υ (t,y 1,y,,y η : y υ [x υ (t] } = [ f (υ 1 (t,x (t, f (υ (t,x (t ], Definition 4 Let x υ :[a,b] R F such that x υ exists If x υ and x υ satisfy system of FIVP (1 and (, we say that x υ ae system fuzzy solutions, whee υ = 1,,, η Befoe using RKHS method as an efficient solve fo fuzzy diffeential systems, we shall now intoduce and implement a pocedue to tansfom system of FIVP (1 c 017 NSP
106 O Abu Aqub et al: A novel iteative numeical algoithm fo and ( into paametic fom in ode to find system fuzzy solutions Algoithm 1 To find fuzzy solutions of system of FIVP (1 and (, thee ae fou main steps: Input: The inteval [a,b], the unit inteval [0,1], and the endpoints functions f (υ 1 (t,x (t, f (υ (t,x (t of [ f υ (t,x 1 (t,x (t,,x η (t] Output: Exact fuzzy solutions x υ (t fo each t [a,b] Step 1: Fo ν = 1,,η, do the following: Set [x υ (t] = [ x (υ 1 (t,x (υ (t ] ; ] Set [x υ (t] = [x (υ 1 (t,x (υ (t ; Set [x υ (0] = [ α (υ 1,α (υ ] ; Set [ f υ (t,x (t] = [ f (υ 1 (t,x (t, f (υ (t,x (t ] ; Step : Solve the following system of ODEs fo x (t: subject to x 1 (t= f 1(t,x (t, x (t= f (t,x (t, x 3 (t= f 3(t,x (t, x 4 (t= f 4(t,x (t, x (η 1 (t= f (η 1(t,x (t, x (η (t= f (η(t,x (t, x 1 (t 0 =α 1, x (t 0 =α, x 3 (t 0 =α 3, x 4 (t 0 =α 4, x (η 1 (t 0 =α (η 1, x (η (t 0 =α (η Step 3: Fo υ = 1,,η and each t [a,b] and [0,1], do the following: Ensue that the solutions [ x (υ 1 (t,x (υ (t ] ae valid level sets; ] Ensue that the deivatives [x (υ 1 (t,x (υ (t ae valid level sets; Constuct the fuzzy solutions x υ (t such that [x υ (t] = [ x (υ 1 (t,x (υ (t ] Step 4: Stop 3 Multidimensional inne poduct spaces In functional analysis, RKHS is a Hilbet space of functions in which pointwise evaluation is a continuous linea functional Equivalently, they ae spaces that can be defined by epoducing kenels In this section, we fistly (5 (6 fomulate seveal epoducing kenel functions in ode to geneate and constuct an othogonal nomal basis on the spaces W [a,b] and W 1 [a,b] Afte that, new spaces η ν=1 W [a,b] and η ν=1 W 1 [a,b] ae building in ode to fomulate and utilize the solutions of system of FIVP (1 and ( using RKHS method An abstact set is supposed to have elements, each of which has no stuctue, and is itself supposed to have no intenal stuctue, except that the elements can be distinguished as equal o unequal, and to have no extenal stuctue except fo the numbe of elements Definition 5 [14] Let E be a nonempty abstact set A function K : E E C is a epoducing kenel of the Hilbet space H if 1 t E; K(,t H, t E and ϕ H; ϕ(,k(,t =ϕ(t Remak 1 The condition( in Definition 5 is called the epoducing popety which means that the value of a function ϕ at a point t is epoducing by the inne poduct of ϕ( with K(,t A Hilbet space which possesses a epoducing kenel is called a RKHS An impotant subset of the RKHSs ae the RKHSs associated to a continuous kenel These spaces have wide applications, including complex analysis, hamonic analysis, quantum mechanics, statistics and machine leaning Next, in ode to apply the RKHS method, we shall define and constuct a epoducing kenel space W [a,b] in which evey function satisfies the initial conditions z(a=0 Definition 6 [15] The inne poduct space W [a,b] is defined as W [a,b] = {z(t : z,z ae absolutely continuous eal-valued functions on [a,b], z L [a,b], and z(a = 0} The inne poduct and the nom in W [a,b] ae given by z 1 (t,z (t W = z 1 (az (a + z 1 (az (a+ b a z 1 (tz (tdt, (7 and z 1 W = z 1 (t,z 1 (t W, espectively, whee z 1,z W [a,b] Definition 7 [14] The Hilbet space W [a,b] is called a epoducing kenel if fo each fixed t [a,b] and any z(s W [a,b], thee exist G(t,s W [a,b] (simply G t (s and s [a,b] such that z(s,g t (s W = z(t It is vey impotant to obtain the epesentation fom of the epoducing kenel function G t (s, because it is the basis of ou algoithm In the following theoem, we will give the epesentation fom of the epoducing kenel function G t (s in the space W [a,b] Afte that, we constuct the space W 1 [a,b] in ode to define the linea bounded opeatos as shown late in the next section c 017 NSP
Appl Math Inf Sci 11, No 4, 1059-1074 (017 / wwwnatualspublishingcom/jounalsasp 1063 Theoem 3 [15] The Hilbet space W [a,b] is a complete epoducing kenel and its epoducing kenel function G t (s is given by { Λ(s,t,s t, G t (s= Λ(t,s,s> t whee Λ(s,t = 1 6 (s a (a s + 3t(+s a(6+3t+ s Definition 8 [16] The inne poduct space W 1 [a,b] is defined as W 1 [a,b] = {z(t : z is absolutely continuous eal-valued function on [a,b] and z L [a,b]} The inne poduct and the nom in W 1 [a,b] ae defined as z 1 (t,z (t W 1 = b a (z 1 (tz (t+z 1(tz (tdt and z 1 W 1 = z 1 (t,z 1 (t W 1, espectively, whee z 1,z W 1 [a,b] Theoem 4 [16] The Hilbet space W 1 [a,b] is a complete epoducing kenel and its epoducing kenel function H t (s is given by { (s,t,s t, H t (s= (t,s,s> t whee (s,t = 1 csch(b a (cosh(t+ s b a+ cosh(t s b+a The spaces W [a,b] and W 1 [a,b] ae complete Hilbet with some special popeties So, all the popeties of the Hilbet space will be hold Futhe, theses spaces possesses some special and bette popeties which could make some poblems be solved easie Fo instance, many poblems studied in L [a,b] space, which is a complete Hilbet space, equies lage amount of integal computations and such computations may be vey difficult in some cases Thus, the numeical integals have to be calculated in the cost of losing some accuacy Howeve, the popeties of W [a,b] and W 1 [a,b] equie no moe integal computation fo some functions, instead of computing some values of a function at some nodes In fact, this simplification of integal computation not only impoves the computational speed, but also impoves the computational accuacy Hencefoth and not to conflict unless stated othewise, we denote W[a,b]= η ν=1 W [a,b] H[a,b]= η W 1 [a,b] Definition 9 The inne poduct space W[a, b] can be constucted as W[a,b]={(z 1 (t,z (t,,z η (t T }, whee z j W [a,b] and j = 1,,η The inne poduct and the nom in W[a,b] ae building as and z W = z,w W[a,b] z(t,w(t W = η η z j W z j (t,w j (t W, espectively, whee Clealy, W[a, b] is a Hilbet space On the othe aspect as well, the inne poduct space H[a,b] can be defined in a simila manne with simila inne poduct and nom, and it is also a Hilbet space 4 Seies epesentation of solutions In this section, fomulation of diffeential linea opeato and implementation method ae pesented in the spaces W[a, b] and H[a, b] Meanwhile, we constuct an othogonal function system of the space W[a, b] based on Gam-Schmidt othogonalization pocess in ode to obtain the exact and appoximate solutions of system of FIVP (1 and ( Though emainde sections, the lowecase lette wheneve used means fo each [0,1] Now, to apply the RKHS method, we will define the diffeential linea opeato L j : W [a,b] W 1 [a,b] such that L j x j (t = x j (t, j = 1,,,η Put f = ( T f 1, f,, f (η, x = ( T x 1,x,,x (η, α = ( α 1,α,,α (η T, and L = diag ( L 1,L,,L (η, whee L : W[a,b] H[a,b] Based on this, the system of ODEs (5 and (6 can be conveted into the equivalent fom as follows: subject to L x (t= f (t,x (t = f ( t,x1 (t,x (t,,x (η (t, in which x W[a,b] and f H[a,b] (8 x (a=α, (9 Lemma 1 The opeatos L j : W [a,b] W 1[a,b], j = 1,,,η ae bounded and linea Poof The lineaity pat is obvious, fo boundedness pat, we need to pove that L j x j W 1 M j x j W, whee M j > 0 Fom the definition of the inne poduct and the nom of W 1 [a,b], we have L j x j W 1 = b a { [(L j x j (t ] +[(L j x j (t] } dt c 017 NSP
1064 O Abu Aqub et al: A novel iteative numeical algoithm fo By epoducing popety of the kenel function G t (s, we have x j (t= x j (s,g t (s W (L j x j (t= x j (s,(l j G t (s W (L j x j (t= x j (s,(l j G t (s W Again, by Schwaz inequality, we get (L j x j (t = x j (t,(l j G t (t W (L j G t (t W x j (t W = M 1 j x j (t W, (L j x j (t = x j (t,(l j G t (t W whee M 1 j,m j > 0 Thus, (L j G t (t W x j (t W = M j x j (t W, L j x j W 1 = b { [(L j x j (t ] +[(L j x j (t] } dt a ( M 1 j + M j (b a x j (t W o L j x W j 1 M j x j (t W, ( whee M j = M 1 j + M j (b a Theoem 5 The opeato L : W[a,b] H[a,b] is bounded and linea Poof Clealy, L is a linea opeato A boundedness is shown as follows: fo each x W[a,b], one can wite L x H = η L j x j W 1 η L j x j W ( η ( L j η x j W = η L j x W The boundedness of L j implies that L is bounded So, the poof of the theoem is complete Next, we constuct an othogonal function system of W[a,b] as follows: put ϕ i j (t = H ti (te j and ψ i j (t = L ϕ i j (t, whee e j = ( 0,0,1 jth,0,,0 T, L (L = diag 1,L,,L (η is the adjoint opeato of L, H t (s is the epoducing kenel function of W 1[a,b], and {t i } i=1 is dense on [a,b] The othonomal function { } (,η system ψ i j (t of W[a,b] can be deived fom (i, j=(1,1 Gam-Schmidt othogonalization pocess of { } (,η ψ i j (t as follows: set (i, j=(1,1 ψ i j (t= i j l=1 k=1 lk ψ lk (t, (10 whee i = 1,,3,, j = 1,,,η and lk ae othogonalization coefficients The subscipt s by the opeato L, denoted by L s, indicates that the opeato L applies to the function of s Indeed, it is easy to see that, ψ i j (t = L ϕ i j (t = L ϕ i j (s,g t (s = W ϕ i j (s,l s G t (s = L sg t (s s=ti W[a,b] Thus, H ψ i j (t can be expessed in the fom ψ i j (t= L s G t (s s=ti Theoem 6 Fo Eqs (8 and (9, if {t i } i=1 is dense on { } (,η [a,b], then ψ i j (t is the complete function (i, j=(1,1 system of the space W[a,b] Poof x (t W[a,b], let gives Whilst x (t,ψ i j (t = 0, which W x (t,ψ i j (t W = x (t,l ϕ i j (t = L x (t,ϕ i j (t = L x (t i =0 x (t= η x j (te j = η x (,G t ( e j W e j Hence, L x (t = η L x (t,ϕ i j (t e j = 0 But W since {t i } i=1 is dense on [a,b], we must have L x (t=0 It follows that x (t = 0 fom the existence of L 1 So, the poof of the theoem is complete The intenal stuctue of the following theoem is to utilize the epesentation fom of the exact and appoximate solutions of system of FIVP (1 and ( in the space W[a,b] W H c 017 NSP
Appl Math Inf Sci 11, No 4, 1059-1074 (017 / wwwnatualspublishingcom/jounalsasp 1065 Theoem 7 If {t i } i=1 is dense on [a,b] and the solution of Eqs (8 and (9 is unique, then the exact solution of Eqs (8 and(9 satisfies the expansion fom x (t= η i j i=1 l=1 k=1 ik f k(t l,x (t l ψ i j (t (11 Poof Applying Theoem 6, it is easy to see that { } (,η ψ i j (t is the complete othonomal basis of (i, j=(1,1 W[a,b] Thus, using Eq (10, we have x (t η i=1 = η i=1 = = x (t, ψ i j (t x (t, η i j i=1 l=1 k=1 η i j i=1 l=1 k=1 = η i j i=1 l=1 k=1 = η i j i=1 l=1 k=1 = i l=1 j k=1 W ψ i j (t lk ψ lk (t W ψ i j (t lk x (t,l ϕ lk (t W ψ i j (t lk L x (t,ϕ lk (t H ψ i j (t lk f k(t,x (t,ϕ lk (t H ψ i j (t lk f k(t l,x (t l ψ i j (t Theefoe, the fom of Eq (11 is the exact solution of Eqs(8 and(9 The poof is complete Remak We mention hee that, the appoximate solution x n (t of x (t fo Eqs(8 and(9 can be obtained diectly by taking finitely many tems in the seies epesentation fom of x (t fo Eq(11 and is given as x n (t= n i=1 η i l=1 k=1 j lk f k(t l,x (t l ψ i j (t (1 5 Implementation of iteative algoithm In this section we develop an iteative algoithm to find the solutions of system of FIVP (1 and ( in the space W[a,b] fo linea and nonlinea case Also, the solutions of same system, obtained by using poposed method with existing fuzzy numbes ae poved to convege to the exact solutions with deceasing absolute diffeence between the exact values and the values obtained using RKHS method The basis of ou RKHS solutions method fo solving Eqs (8 and (9 is summaized below fo the exact and appoximate solutions Fistly, we shall make use of the following facts about linea and nonlinea case depending on the intenal stuctue of the function f Case 1 If Eq(8 is linea, then the exact and appoximate solutions can be obtained diectly fom Eqs(11 and(1, espectively Case If Eq (8 is nonlinea, then the exact and appoximate solutions can be obtained by using the following iteative pocess Accoding to Eq (11, the epesentation fom of the solution of Eqs (8 and (9 will be whee L i j = i x (t= j l=1 k=1 i=1 η L i j ψ i j (t, ik f k(t l,x (t l Put t 1 = a, it follows that x (t 1 is known fom the initial conditions of Eq (9; so f (t 1,x (t 1 is known Fo numeical computations, we put initial function x 0 (t 1 = x (t 1 and define the n-tem appoximations to x (t by x n (t= n η i=1 B i j ψ i j (t, (13 whee the coefficients B i j and the successive appoximations x i (t, i=1,,,n ae given as follows: B 1 j = 1 j l=1 k=1 β 1 j 1k f k( t1,x 0 (t 1 ; x 1 (t= η B 1 j ψ 1 j (t, B j = j l=1 k=1 x (t= B n j = n η i=1 j l=1 k=1 x n (t= n η l=1 β j lk f ( k tl,x l 1 (t l ; B i j ψ i j (t, β n j lk f ( k tl,x l 1 (t l ; B i j ψ i j (t (14 In the iteative pocess of Eq (14, we can guaantee that the appoximation x n (t satisfies the initial condition of Eq (9 Now, we will poof that x n (t in the iteative fomula(14 is convege to the exact solution x (t of Eq (8 In fact, this esult is a fundamental ule in the RKHS theoy and its applications Lemma If z(t W [a,b], then ( z(t 1+b a+ (b a 3 z W z (t ( 1+ b a z W Poof Fo the fist pat, noting that z (t z (a = t a z (pd p, whee z (t is absolute continuous on [a,b] If this is integated again fom a to t, the esult is z(t itself as; t ( y z(t z(a z (a(t a= z (pd p dy a a c 017 NSP
1066 O Abu Aqub et al: A novel iteative numeical algoithm fo So, z(t z(a + z (a b (b a+(b a z (p d p By using Holde s inequality and Eq (7, we can note the following elation: z(a z W, z (a z W, and b a z (p d p (b a z W Thus, ( z(t 1+b a+ (b a 3 z W Fo the second pat, since z (t=z (a+ t a z (pd p, this means that z (t z (a + b ( a z (p d p In othe wod, one can find z (t 1+ (b a z W Theoem 8 If x n (t x (t W 0, t n s as n, x n W is bounded, and f (t,x (t is continuous, then f (t n,x n 1 (t n f (s,x (s as n Poof Fistly, we will pove that x n 1 (t n x (s Since, we can note that x n 1 (t n x (s x n 1 = x n 1 x n 1 (t n x n 1 (t n x n 1 (s+x n 1 (s x (s (s + x n 1(s x (s ( x n 1 (ξ tn s + x n 1 (s x (s, whee ξ lies between t n and s Fom Lemma, it follows that (s x (s (1+b a+ (b a 3 x n 1 (s x (s W, which is gives x n 1 (s x (s 0 as n, while on the othe hand, we have ( x n 1 (ξ ( 1+ x n 1 (b a (ξ W In tems of the boundedness of x n 1 (t W, one obtains that x n 1 (t n x (s 0 as n Thus, by means of the continuation of f (t,x (t, it is implies that f (t n,x n 1 (t n f (s,x (s as n So, the poof of the theoem is complete Theoem 9 Suppose that x n W is bounded in Eq (13, and Eqs (8 and (9 has a unique solution If {t i } i=1 is dense on [a, b], then the n-tem appoximate solution x n (t in the iteative fomula of Eq (13 conveges to the exact solution x (t of Eqs (8 and (9, and x (t= η i=1 B i j ψ i j (t Poof Simila to the poof of Theoem 4 in [] a 6 Softwae libaies and numeical expeiment In ode to solve system of FIVP (1 and ( appoximately on a compute, the system is appoximated by a discete one Continuous functions ae appoximated by finite aays of values Algoithms ae then sought which appoximately solve the mathematical poblem efficiently, accuately and eliably While scientific computing focuses on the design and the implementation of such algoithms, numeical analysis may be viewed as the theoy behind them To show behavio, popeties, efficiency, and applicability of the pesent RKHS method, two linea and one nonlinea fuzzy diffeential systems will be solved numeically in this section An algoithm is a finite sequence of ules fo pefoming computations on a compute such that at each instant the ules detemine exactly what the compute has to do next Next algoithm is utilizes to implement a pocedue to solve FIVP (1 and ( in numeic fom in tems of thei gid nodes based on the use of RKHS method Algoithm To appoximate the solution x n (t of x (t fo Eqs(8 and(9, we do the following steps: Input The inteval[a,b], the unit inteval[0,1] the integes n, the integes m, the kenel functions G t (s and H t (s, the diffeential opeato L, and the function f Output Appoximate solution x n (t of x (t Step 1 Fixed t in [a,b] and set s [a,b]; If s t, set G t (s= Λ(s,t; Else set G t (s= Λ(t,s; Fo i=1,,,n, h=1,,,m, and j = 1,,,η, do the following: Set t i = n 1 i 1 ; Set h = h 1 m 1 ; Set ψ i, j (t=l h s[g t (s] s=ti ; Output: the othogonal function system ψ i, j (t Step Fo l =,3,n 1 and k = 1,,l 1, do the following: Set ψ i j (t= i j l=1 k=1 lk ψ lk (t; Output: the othonomal function system ψ i j (t Step 3 Set x 0 h (t 1 =x h (t 1 =0; Set B i j = i l=1 k=1 Set x i h (t= i i=1 lk f k h ( tl,x l 1 h (t l ; B i j ψ i j (t; Output: the appoximate solution x n (t of x h (t Step 4 Stop Remak 3 Thoughout this pape, we will ty to give the esults of the thee examples; howeve, in some cases we c 017 NSP
Appl Math Inf Sci 11, No 4, 1059-1074 (017 / wwwnatualspublishingcom/jounalsasp 1067 will switch between the esults obtained fo the examples in ode not to incease the length of the pape without the loss of geneality fo the emaining examples and esults In the pocess of computation, all the symbolic and numeical computations ae pefomed by using MAPLE 13 softwae package Next, we show by example that the system of cisp initial value poblems can be modeled in a natual way as system of FIVPs To illustate this, conside the dynamic supply and demand system The system of ODE coesponding to this poblem is p (t = θ k 1 (s s 0 and s (t = k (p p 0, whee p is the pice, s is the supply, p 0 is the equilibium pice, s 0 is equilibium supply, θ is the ate of inflation, and k 1,k ae positive constant coesponding to the dynamic natue of the system Hee, we ae consideing an item such that inceasing its pice p esults in an incease in supply s but that inceasing its supply s will ultimately decease its pice p Futhemoe, we will assume thee ae two factos that influence pice; inflation and supply The facto s s 0 means that; fistly, if s>s 0, the supply is too lage and pice is to decease; secondly, if s < s 0, supply is too low and pice tends to incease, while on the othe hand, the facto p p 0 means that; fistly, if p> p 0, pice is high and supply inceasing; secondly, if p< p 0, pice is low and supply deceases Uncetainty in detemining the initial values, inaccuacy in element modeling, and othe paametes cause uncetainty in the afoementioned system Consideing them instead as system of FIVPs yields moe ealistic esults Example 1 [41] Conside the following dynamic supply and demand diffeential system of fuzzy equations on [0,1]: p (t=θ k 1 (s s 0, s (t=k (p p 0, (15 fom ae [p(t] =[ ( 45 ( 45 45 45 e t ( 55 55 e, t e t + ( 55 55 e ] t + 6001 10 sin( t + 5, [s(t] =[ ( 45 45 e t ( 55 55 e, t ( 45 45 e t + ( 55 55 e ] t 6001 10 cos( t + 1001 10 Using RKHS method, taking t i = n 1 i 1, i = 1,,,n, n = 51 and j = j 1 m 1, j = 1,,,m, m = 5 with the epoducing kenel functions G t (s and H t (s on [0,1] in which Algoithms 1 and ae used thoughout the computations; some gaphical esults and tabulate data ae pesented and discussed quantitatively to illustate the fuzzy appoximate solutions and the appoximate Hukuhaa deivatives As we mentioned ealie, it is possible to pick any point in the inteval of integation [0,1] and as well the fuzzy appoximate solutions and thei fist Hukuhaa deivatives will be applicable Next, numeical esults of appoximating the sets [p(t] and [p (t] of system of FIVP (15 and (16 at t = 1/ and vaious ae given in Tables 1 and, espectively, while in Tables 3 and 4 the appoximate solutions fo [s(t] and [s (t] have been tabulated Example [41] Conside the following linea diffeential system of fuzzy equations on[0,1]: x 1 (t=x 1(t+x (t, x (t= x 1(t+ x (t, subject to the fuzzy initial conditions (17 x 1 (0=α 1,x (0=α, (18 subject to the fuzzy initial conditions whee x 1 (0=α 1,x (0=α, (16 whee and { s 1, 1 s, α 1 (s= 3 s, s 3, { s, 0 s 1, α (s= s, 1 s, [α 1 ] =[0+5,30 5] [α ] =[550+50,650 50] Fo numeical esults and compaisons, the following values, fo paametes, ae consideed [41]: θ = 005, s 0 = 100, p 0 = 5, and k 1 = k = 05 The exact fuzzy solutions of system of FIVP (15 and (16 in paametic The exact fuzzy solutions of system of FIVP (17 and (18 in fuzzy setting ae whee x 1 (t=α 3 (se t + e t sin(x+e t cos(t x (t=α 3 (se t + e t cos(t e t sin(t, { s+1, 1 s 0, α 3 (s= 1 s, 0 s 1, c 017 NSP
1068 O Abu Aqub et al: A novel iteative numeical algoithm fo Table 1: The fuzzy exact and appoximate solutions of [p(t] fo system of FIVP (15 and (16 at t = 1/ [ ( p 1/ ] [ p 51( 1/ ] 0 [09410747871800, 561388113835788] [094107457778047, 56138809696631] 05 [15517558107898, 50978034446090] [1551753759875, 50978016481808] 05 [1097637497568, 444567955056391] [1097617407695, 444567936666986] 075 [6933771688795, 38615787566669] [69337697519, 38615785685160] 1 [3774779676994, 3774779676994] [37747777037349, 37747777037349] Table : The Hukuhaa deivative of fuzzy exact and appoximate solutions of [p (t] fo system of FIVP (15 and (16 at t = 1/ [ p ( 1/ ] [ (p 51( 1/ ] 0 [5401015077088, 308976650864563] [54010146919974, 3089766190539] 05 [60880465060399, 30103507970087] [6088046195008, 30103475454491] 05 [677507793497783, 9530365075638] [677507756700765, 9530331703741] 075 [746109363936, 883617181178] [74610900451513, 88361718795993] 1 [81491407986687, 81491407986687] [81491404408, 81491404408] Table 3: The fuzzy exact and appoximate solutions of [s(t] fo system of FIVP (15 and (16 at t = 1/ [ ( s 1/ ] [ s 51( 1/ ] 0 [58154669870840, 6907969845833] [581546694374060, 690796981604634] 05 [595895984059777, 6783390698793396] [5958959807788, 678339069570081] 05 [609635969848714, 6645984413004459] [609635966181703, 6645984409796985] 075 [633765555637650, 650857817155] [6337655508559, 650857813893166] 1 [637117184146586, 637117184146586] [6371171837989350, 6371171837989350] Hee, α 1 (s, α (s, and α 3 (s ae vanished outside the intevals [1, 3], [0, ], and [ 1, 1], espectively In fact this system is a genealization of the system of ODE x 1 (t = x 1(t+x (t and x (t = x 1(t+ x (t subject to initial conditions x 1 (0 and x (0 1 Anyhow, if one put = s 1, then s = +1, again if = 3 s, then s = 3 ; hence, [α 1 ] = [+1,3 ]; similaly, [α ] = [, ] and [α 3 ] = [ 1,1 ] In ode to apply the RKHS method, we fist apply Algoithm 1 as follows; put [x 1 (t] = [x 1 (t,x (t] and [x (t] = [x 3 (t,x 4 (t] Then we have the following system of ODE: x 1 (t=x 1(t+x 3 (t, x (t=x (t+x 4 (t, x 3 (t= x (t+ x 3 (t, x 4 (t= x 1(t+ x 4 (t, subject to the initial conditions x 1 (0=+1, x (0=3, x 3 (0=, x 4 (0= (19 (0 Using RKHS method, taking t i = i 1 n 1, i = 1,,,n, n = 51 and j = j 1 m 1, j = 1,,,m, m = 5 with the epoducing kenel functions G t (s and H t (s on [0,1] in which Algoithms 1 and ae used thoughout the computations; some gaphical esults, compaison feedback, and tabulate data ae pesented and discussed quantitatively to illustate the fuzzy appoximate solutions Result fom numeical analysis is an appoximation, in geneal, which can be made as accuate as desied Because a compute has a finite wod length, only a fixed numbe of digits ae stoed and used duing computations Next, the absolute diffeence between the exact values and the values obtained using RKHS method (absolute eo of numeically appoximating x (t by x 51 (t fo system of ODE (19 and (0 have been calculated fo vaious t and as shown in Tables 5, 6, 7, and 8 Fom the tables, it can be seen that with the few tens of iteations, the RKHS appoximate solutions with high accuacy ae achievable Numeical compaisons fo system of FIVP (17 and (18 ae studied next The numeical methods that ae used fo compaison with RKHS method include the vaiational iteation method [39], the HAM [40], and the fuzzy neual netwok method [41] Anyhow, Table 9 shows a compaison between the absolute eos of ou method togethe with othe afoementioned methods in appoximating x 1 (t and x (t of [x 1 (t] at t = 0 and vaious, while Table 10 shows a compaison in appoximating x 3 (t and x 4 (t of [x (t] at t = 0 and vaious It is clea fom the tables that the absolute eos of the RKHS method ae the lowest one among all othe numeical and analytical ones c 017 NSP
Appl Math Inf Sci 11, No 4, 1059-1074 (017 / wwwnatualspublishingcom/jounalsasp 1069 Table 4: The Hukuhaa deivative of fuzzy exact and appoximate solutions of [s (t] fo system of FIVP (15 and (16 at t = 1/ [ s ( 1/ ] [ (s 51( 1/ ] 0 [90537393591667, 1155694056917909] [9053749563444, 115569404546889] 05 [95158779053987, 1164890173145] [951587650151613, 116489005368070] 05 [980463818748887, 1097839775846] [9804638050739791, 10978396530944] 075 [1009668858443705, 1068078937833370] [100966884513806, 1068078955047] 1 [1038873898138538, 1038873898138538] [1038873885191619, 1038873885191619] Table 5: The absolute eo of appoximating x 1 (t fo system of ODE (19 and (0 t = 0 = 05 =05 =075 = 1 01 699146 10 8 50816 10 8 317177 10 8 1619 10 8 6479 10 9 0 19834 10 7 93631 10 8 56691 10 8 0111 10 8 164500 10 8 03 17864 10 7 16074 10 7 738839 10 8 16936 10 8 304967 10 8 04 1319 10 7 147598 10 7 819769 10 8 163557 10 8 49655 10 8 05 318 10 7 155730 10 7 79331 10 8 93410 10 9 734639 10 8 06 317 10 7 14787 10 7 63931 10 8 199643 10 8 10386 10 7 07 07895 10 7 10601 10 7 333079 10 8 539855 10 8 14179 10 7 08 15559 10 7 699966 10 8 155357 10 8 101068 10 7 186600 10 7 09 68846 10 8 897415 10 9 8639 10 8 16349 10 7 40750 10 7 1 616605 10 8 1419 10 7 183178 10 7 43936 10 7 304695 10 7 Table 6: The absolute eo of appoximating x (t fo system of ODE (19 and (0 t = 0 = 05 =05 =075 = 1 01 88731 10 8 637746 10 8 446761 10 8 55777 10 8 6479 10 9 0 16734 10 7 16163 10 7 89590 10 8 53010 10 8 164500 10 8 03 3958 10 7 187068 10 7 134877 10 7 86870 10 8 304967 10 8 04 311750 10 7 4619 10 7 180508 10 7 114887 10 7 49655 10 8 05 379056 10 7 30658 10 7 660 10 7 14986 10 7 734639 10 8 06 43944 10 7 355546 10 7 71651 10 7 187755 10 7 103860 10 7 07 490453 10 7 403159 10 7 315866 10 7 857 10 7 14179 10 7 08 58730 10 7 443197 10 7 357665 10 7 7133 10 7 186600 10 7 09 549786 10 7 4757 10 7 39568 10 7 318009 10 7 40750 10 7 1 54779 10 7 486971 10 7 461 10 7 365454 10 7 304695 10 7 Nonlinea phenomena s ae of fundamental impotance in vaious fields of science and engineeing, and othe disciplines, since most phenomena in ou wold ae essentially nonlinea and ae descibed by nonlinea equations Anyhow, in most eal-life situations, the diffeential systems that models the uncetainty systems ae too complicated to solve analytically, and thee is a pactical need to appoximate the solutions In the next example, the fuzzy Hukuhaa diffeentiable exact solutions cannot be found analytically in tems of closed fom solutions Example 3 Conside the following nonlinea diffeential system of fuzzy equations on[0,1]: subject to the fuzzy initial conditions x 1 (t=ex (t + α, x (t=x3 1 (t, (1 x 1 (0=0,x (0=β, ( whee α(s=max s R (0,1 (4s 3 β(s=max s R (0,1 (5s Fo the conduct of poceedings in the solution and depending on Algoithm 1, it is clea that [ x 3 1 (t ] =[x 3 1 (t,x 3 (t] [ e x (t] =[e x 3 (t,e x 4(t ] This is due to the fact that s 3 and e s ae stictly inceasing continuous functions on(, On the othe hand, if one set = 1 (4s 3, then s= 4 1(1 3 o s= 1 4 (1 3 ; hence, [ ] [α] = 1 4 (1 3, 1 4 (1 3 [β] = [ 1 5 1, 1 5 1 ] c 017 NSP
1070 O Abu Aqub et al: A novel iteative numeical algoithm fo Table 7: The absolute eo of appoximating x 3 (t fo system of ODE (19 and (0 t =0 =05 =05 = 075 =1 01 118488 10 7 993898 10 8 80914 10 8 61199 10 8 40945 10 8 0 9597 10 7 19306 10 7 156455 10 7 119884 10 7 83316 10 8 03 333077 10 7 80887 10 7 8696 10 7 176506 10 7 14316 10 7 04 4836 10 7 36705 10 7 97084 10 7 31463 10 7 165841 10 7 05 51431 10 7 437914 10 7 361516 10 7 85118 10 7 0870 10 7 06 589478 10 7 505583 10 7 41687 10 7 33779 10 7 53896 10 7 07 651618 10 7 56435 10 7 477031 10 7 389738 10 7 30444 10 7 08 69773 10 7 61190 10 7 56658 10 7 44116 10 7 355593 10 7 09 73783 10 7 64655 10 7 56966 10 7 49007 10 7 414748 10 7 1 74547 10 7 663789 10 7 603030 10 7 5471 10 7 481513 10 7 Table 8: The absolute eo of appoximating x 4 (t fo system of ODE (19 and (0 t =0 =05 =05 = 075 =1 01 34994 10 8 15009 10 8 389753 10 9 9960 10 8 40945 10 8 0 69715 10 8 64005 10 8 101706 10 8 467416 10 8 83316 10 8 03 844453 10 8 3551 10 8 19935 10 8 7155 10 8 14316 10 7 04 966433 10 8 3101 10 8 345991 10 8 1000 10 7 165841 10 7 05 968714 10 8 04735 10 8 55945 10 8 133 10 7 0870 10 7 06 816858 10 8 0970 10 9 86105 10 8 170001 10 7 53896 10 7 07 46797 10 8 405638 10 8 17857 10 7 15151 10 7 30444 10 7 08 134640 10 8 989963 10 8 18459 10 7 70061 10 7 355593 10 7 09 105713 10 7 1897 10 7 6031 10 7 337489 10 7 414748 10 7 1 38478 10 7 9937 10 7 359996 10 7 40754 10 7 481513 10 7 Table 9: Numeical compaison of appoximate solution [x 1 (t] fo system of FIVP (17 and (18 at t = 0 method of [7] method of [6] method of [5] RKHS method x 1 (t x (t x 1 (t x (t x 1 (t x (t x 1 (t x (t 0 1 10 5 14 10 5 67 10 6 53 10 6 1 10 4 59 10 5 13 10 7 16 10 7 0 60 10 6 14 10 5 55 10 6 41 10 6 11 10 4 40 10 5 10 10 7 13 10 7 04 13 10 5 16 10 5 43 10 6 9 10 6 88 10 5 10 5 71 10 8 10 10 7 06 16 10 5 19 10 5 31 10 6 17 10 6 69 10 5 39 10 6 4 10 8 75 10 8 08 80 10 6 13 10 5 19 10 6 5 10 7 51 10 5 14 10 5 13 10 8 46 10 8 1 13 10 5 13 10 5 69 10 6 69 10 7 33 10 5 33 10 5 83 10 8 83 10 8 Fo finding fuzzy appoximate solutions of system of FIVP (1 and (, which ae coesponding to thei paametic fom, we have the following system of ODE: x 1 (t=ex3(t 1 4 (1 3, x (t=ex4(t + 1 4 (1 3, (3 x 3 (t=x3 1 (t, x 4 (t=x3 (t, subject to the initial conditions x 1 (0=0, x (0=0, x 3 (0= 1 5 1, x4 (0= 5 1 (4 1 Ou next goal is to pesent the HAM appoximate solutions fo system of ODE (3 and (4 in ode to measue the extent of ageement with unknowns closed fom solutions which ae inapplicable in geneal fo such nonlinea systems, in ode to employ again the obtained expansions to measue the accuacy of the RKHS method in finding and pedicting the fuzzy appoximate solutions To do so, we epot the seies fomulas fo the HAM solutions in which the obtained esults ae geneated fom the 10-tuncated seies solutions fo each x j (t, j = 1,, 3, 4 Hencefoth, fo simplicity and not to conflict, we will let x HAM j (t, j = 1,,3,4 to denote the HAM seies solutions of x j (t, as follows: x HAM 1 (t= ( e β ( 1 + α 1 t+ 1 0 e β 3 1 + α 1 e β 1 t 5 +( 1 ( 88 eβ 1 e β 6 1 + α 1 ( e β 1 5t + α 9 1 + 1 480 eβ 1 c 017 NSP
Appl Math Inf Sci 11, No 4, 1059-1074 (017 / wwwnatualspublishingcom/jounalsasp 1071 Table 10: Numeical compaison of appoximate solution [x (t] fo system of FIVP (17 and (18 at t = 0 method of [7] method of [6] method of [5] RKHS method x 1 (t x (t x 1 (t x (t x 1 (t x (t x 1 (t x (t 0 4 10 5 14 10 5 45 10 6 75 10 6 79 10 5 10 10 4 3 10 7 63 10 8 0 13 10 5 40 10 6 33 10 6 63 10 6 61 10 5 85 10 5 0 10 7 34 10 8 04 0 10 5 13 10 5 1 10 6 51 10 6 43 10 5 67 10 5 17 10 7 45 10 9 06 13 10 5 08 10 5 93 10 7 39 10 6 4 10 5 49 10 5 14 10 7 5 10 8 08 14 10 5 90 10 6 8 10 7 7 10 6 61 10 6 30 10 5 11 10 7 54 10 8 1 10 10 5 10 10 5 15 10 6 15 10 6 1 10 5 1 10 5 16 10 8 16 10 8 Table 11: The values of absolute esidual eo functions fo system of ODE ( 3 and (4 at t = 05 Res HAM 1 (t Res HAM (t Res HAM 3 (t Res HAM 4 (t 0 11656990040 10 8 344814053 10 3 133559190 10 7 83801483 10 5 05 5844108311 10 8 136393444 10 3 50779808894 10 7 4408930308 10 5 05 1773461074 10 7 7710469 10 4 1493603705 10 6 4555858754 10 5 075 49090555176 10 7 3867144886 10 4 37071763640 10 6 158680654 10 5 1 179439533 10 6 179439533 10 6 11855064168 10 5 11855064168 10 5 Table 1: Numeical compaison of appoximate solution of [x 1 (t] fo system of FIVP (1 and ( at t = 05 HAM solution RKHS solution 0 [084601058889784, 07417813098616158] [084601060460641, 07417813504005433] 05 [0339704479611809, 068033313735007] [033970450717690, 068033349733797] 05 [039051809340397, 0635854675651437] [039051807739875, 063585490513175] 075 [0437740698550377, 0570749345416490] [0437740699449165, 0570749515434681] 1 [05015733506944444, 05015733506944444] [0501573361371539, 0501573361371539] (t= ( e β ( + α t+ 1 0 e β 3 + α e β t 5 ( e β 6 1 + α 1 x HAM +( 1 88 eβ 1 + 1 1( 480 eβ e β 5 1 + α 1 t 9, x HAM 3 (t=β 1 + 1 ( 4 e β 3 1 + α 1 t 4 + 3 1( 160 eβ e β 5 1 + α 1 t 8, x HAM 4 (t=β + 1 ( 4 e β 3 + α t 4 ( e β 5 + α t 8 + 3 160 eβ While one cannot know the absolute eo without knowing the exact solution, in most cases the esidual eo, denoted by Res(t, can be used as a eliable indicatos in the iteation pogesses In Table 11, the value of the following esidual eo functions: Res HAM 1 Res HAM Res HAM (t = dt d xham 1 (t = 3 (t = 4 (t = Res HAM dt d xham dt d xham 3 dt d xham 4 ( (t e xham 3 (t 1 4 (1 3, ( (t e xham 4 (t + 1 4 (1 3, (t ( x HAM 1 (t 3, (t ( x HAM (t 3, (5 fo the 10-tuncated seies HAM appoximate solutions x HAM j (t, j = 1,,3,4 have been calculated at t = 05 and vaious fo system of ODE (3 and (4 Fom the table, it can be seen that the HAM povides us with the accuate appoximate solutions with attention to that, moe accuate solution can be found at the beginning values of in compaison with lage Now, we will etun to ou RKHS method in ode to display new numeical and compaison esults Anyhow, using RKHS method, taking t i = n 1 i 1 n = 51 and j = j 1 m 1, j = 1,,,m, m = 5 with the epoducing kenel functions G t (s and H t (s on [0,1] in which Algoithms 1 and ae used thoughout the computations; some gaphical esults, compaison feedback, and tabulate data ae pesented and discussed quantitatively to illustate the fuzzy appoximate solutions Numeical compaisons ae caied out to veify the mathematical esults and the theoetical statement of the solutions Next, some tabulated data ae pesented to show the extent between the HAM solutions and the RKHS method solutions Howeve, Table 1 shows a compaison of appoximate solution fo[x 1 (t] at t = 05 and vaious fo system of FIVP (1 and (, while Tables 13 shows a compaison of appoximate solution fo [x (t] at t = 05 and vaious As it is evident fom the compaison esults, it was found that ou method in c 017 NSP
107 O Abu Aqub et al: A novel iteative numeical algoithm fo Table 13: Numeical compaison of appoximate solution of [x (t] fo system of FIVP (1 and ( at t = 05 HAM solution RKHS solution 0 [ 0197107850593, 050393858174381] [ 019710669419136, 050395964361660] 05 [ 0168313816545731, 011685336748] [ 01683137968506898, 0116869488047] 05 [ 0133995706908855, 017148149464561] [ 01339956769374649, 01714816163367717] 075 [ 0089549366447843, 0130851973330545] [ 0089549341507733, 01308591940308] 1 [001569841875000, 001569841875000] [00156983060546904, 00156983060546904] compaison with the mentioned method is simila with a view to accuacy and utilization The afoementioned computational esults povide a numeical estimate fo the RKHS solutions Also, it is clea that the accuacy obtained using pesent method is in advanced by using only few tens of iteation, whee highe accuacy can be achieved by inceasing the numbe n in Algoithms 7 Concluding emaks In vaious subjects of science and engineeing, nonlinea systems of fuzzy diffeential equations subject to given fuzzy initial conditions, as well as thei exact and numeical solutions, ae essentially impotant, theefoe systems of FIVPs should be solved In the pesent pape, we have studied exact and numeical solutions fo system of FIVPs (1 and ( based on the epoducing kenel theoy Some esults on the behavio of fuzzy solutions, convegence theoem, and eos estimation have also been studied In tems of numeical computations, seveal impovements have been made; fist, the dependency poblem has been highlighted in constucting numeical methods fo the solutions of systems of FIVPs Second, an efficient computational algoithm has been poposed in ode to guaantee the validity of fuzzy solutions on the given inteval, especially fo nonlinea cases, whee this issue had been lagely neglected in the liteatue on numeically solving systems of FIVPs The solving pocedue eveals that the RKHS method is a staightfowad, succinct, and pomising tool fo solving linea and nonlinea systems of FIVPs of odinay types Acknowledgments This poject was funded by the deanship of scientific eseach of KAU unde gant numbe (8-130-35-HiCi The authos, theefoe, acknowledge technical and financial suppot of KAU Refeences [1] II Vabie, Diffeential Equations: An Intoduction to Basic Concepts, Results and Applications, Wold Scientific Pub Co Inc, 004 [] PG Dazin, RS Jonson, Soliton: An Intoduction, Cambidge, New Yok, 1993 [3] GB Whitham, Linea and Nonlinea Waves, Wiley, New Yok, 1974 [4] L Debnath, Nonlinea Wate Waves, Academic Pess, Boston, 1994 [5] L Collatz, Diffeential Equations: An Intoduction with Applications, John Wiley & Sons Ltd, 1986 [6] MW Hisch, S Smale, Diffeential Equations, Dynamical Systems, and Linea Algeba, Academic Pess, 1974 [7] MR Spiegel, Applied Diffeential Equations, Pentice Hall, Englewood Cliffs, NJ, 1981 [8] O Abu Aqub, A El-Ajou, S Momani, Constucting and pedicting solitay patten solutions fo nonlinea timefactional dispesive patial diffeential equations, Jounal of Computational Physics 93 (015 385-399 [9] A El-Ajou, O Abu Aqub, S Momani, D Baleanu, A Alsaedi, A novel expansion iteative method fo solving linea patial diffeential equations of factional ode, Applied Mathematics and Computation 57 (015 119-133 [10] A El-Ajou, O Abu Aqub, S Momani, Appoximate analytical solution of the nonlinea factional KdV- Buges equation: A new iteative algoithm, Jounal of Computational Physics 93 (015 81-95 [11] A Belinet, CT Agnan, Repoducing Kenel Hilbet Space in Pobability and Statistics, Kluwe Academic Publishes, 004 [1] M Cui, Y Lin, Nonlinea Numecial Analysis in the Repoducing Kenel Space, Nova Science Publishe, New Yok, 008 [13] A Daniel, Repoducing Kenel Spaces and Applications, Spinge, 003 [14] F Geng, Solving singula second ode thee-point bounday value poblems using epoducing kenel Hilbet space method, Applied Mathematics and Computation 15 (009 095-10 [15] O Abu Aqub, M Al-Smadi, S Momani, Application of epoducing kenel method fo solving nonlinea Fedholm- Voltea intego-diffeential equations, Abstact and Applied Analysis, vol 01, Aticle ID 839836, 16 pages, 01 doi:101155/01/839836 [16] C Li, M Cui, The exact solution fo solving a class nonlinea opeato equations in the epoducing kenel space, Applied Mathematics and Computation 143 (003 393-399 [17] O Abu Aqub, M Al-Smadi, N Shawagfeh, Solving Fedholm intego-diffeential equations using epoducing kenel Hilbet space method, Applied Mathematics and Computation 19 (013 8938-8948 c 017 NSP
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