Fuzzy Fredholm integro-differential equations with artificial neural networks

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1 Avilble online t Volume 202, Yer 202 Article ID cn-0028, 3 pges doi:0.5899/202/cn-0028 Reserch Article Fuzzy Fredholm integro-differentil equtions with rtificil neurl networks Mrym Mosleh, Mhmood Otdi () Deprtment of Mthemtics, Firoozkooh Brnch, Islmic Azd University, Firoozkooh, Irn Copyright 202 c Mrym Mosleh nd Mhmood Otdi. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this pper, we use prmetric form of fuzzy number, then feed-forwrd neurl network is presented for obtining pproximte solution for fuzzy Fredholm integro-differentil eqution of the second kind. This pper presents method bsed on neurl networks nd Newton-Cotes methods with positive coefficient. The bility of neurl networks in function pproximtion is our min objective. The proposed method is illustrted by solving some numericl exmples. Keywords: Fuzzy integro-differentil equtions; Artificil neurl networks Introduction The solutions of integrl equtions hve mjor role in the field of science nd engineering. A physicl even cn be modelled by the differentil eqution, n integrl eqution. Since few of these equtions cnnot be solved explicitly, it is often necessry to resort to numericl techniques which re pproprite combintions of numericl integrtion nd interpoltion [2, 32]. There re severl numericl methods for solving liner Volterr integrl eqution [8, 4] nd system of nonliner Volterr integrl equtions [4]. Kuthen in [28] used colloction method to solve the Volterr- Fredholm integrl eqution numericlly. Borzbdi nd Frd in [6] obtined numericl solution of nonliner Fredholm integrl equtions of the second kind. The concept of fuzzy numbers nd fuzzy rithmetic opertions were first introduced by Zdeh [43], Dubois nd Prde [20]. We refer the reder to [26] for more informtion on fuzzy numbers nd fuzzy rithmetic. The topics of fuzzy integrl equtions (FIE) Corresponding uthor. E-mil: mosleh@iufb.c.ir; Tel:

2 which growing interest for some time, in prticulr in reltion to fuzzy control, hve been rpidly developed in recent yers. The fuzzy mpping function ws introduced by Chng nd Zdeh [7]. Lter, Dubois nd Prde [2] presented n elementry fuzzy clculus bsed on the extension principle lso the concept of integrtion of fuzzy functions ws first introduced by Dubois nd Prde [2]. Bbolin et l. nd Abbsbndy et l. in [3, ] obtined numericl solution of liner Fredholm fuzzy integrl equtions of the second kind. Allhvirnloo et l. in [6] presented new method for solving fuzzy integrodifferentil eqution under generlized differentibility. Another group of reserchers tried to extend some numericl methods to solve fuzzy differentil equtins (FDEs) [, 8, 38] such s Runge-Kutt method [2], Adomin method [0], predictor-corrector method nd multi-step methods [7]. Fuzzy neurl network hve been extensively studied [5, 9] nd recently, successfully used for solving fuzzy polynomil eqution nd systems of fuzzy polynomils [4, 5], pproximte fuzzy coefficients of fuzzy regression models [33, 34, 35], pproximte solution of fuzzy liner systems nd fully fuzzy liner systems [36, 39]. Lgris et l. in [3] used multilyer perceptron to estimte the solution of differentil eqution. Their neurl network model ws trined over n intervl (over which the differentil eqution must be solved), so the inputs of the neurl network model were the trining points. The comprison of their method with the existing numericl method shows tht their method ws more ccurte nd the solution hd lso more generliztions. Recently Effti et l. in [22] nd Mosleh nd Otdi [37] used rtificil neurl networks to solve fuzzy ordinry differentil equtions. But in this pper, we extend the rtificil neurl networks to solve integro-differentil equtions. The bility of neurl networks in function pproximtion is our min objective. In this pper, we present novel nd very simple numericl method bsed upon neurl networks for solving fuzzy liner Fredholm integro-differentil equtions of the second kind 2 Preliminries X (s) = y(s) + λ b k(s, t)x(t)dt. In this section the bsic nottions used in fuzzy opertions re introduced. We strt by defining the fuzzy number. Definition 2.. [30] A fuzzy number is fuzzy set u : R I = [0, ] such tht i. u is upper semi-continuous; ii. u(x) = 0 outside some intervl [, d]; iii. There re rel numbers b nd c, b c d, for which. u(x) is monotoniclly incresing on [, b], 2. u(x) is monotoniclly decresing on [c, d], 3. u(x) =, b x c. The set of ll the fuzzy numbers (s given in Definition (2.) is denoted by E. An lterntive definition which yields the sme E is given by Klev [27]. Definition 2.2. A fuzzy number u is pir (u, u) of functions u(r) nd u(r), 0 r, which stisfy the following requirements: i. u(r) is bounded monotoniclly incresing, left continuous function on (0, ] nd right continuous t 0; 2 ISPACS GmbH

3 ii. u(r) is bounded monotoniclly decresing, left continuous function on (0, ] nd right continuous t 0; iii. u(r) u(r), 0 r. A crisp number r is simply represented by u(α) = u(α) = r, 0 α. The set of ll the fuzzy numbers is denoted by E. This fuzzy number spce s shown in [42], cn be embedded into the Bnch spce B = C[0, ] C[0, ]. Definition 2.3. [27] For rbitrry u = (u(r), u(r)), v = (v(r), v(r)), we sy tht u = v if nd only if u = v nd u = v. For rbitrry u = (u(r), u(r)), v = (v(r), v(r)) nd k R we define ddition nd multipliction by k s (u + v)(r) = (u(r) + v(r)), (u + v)(r) = (u(r) + v(r)), ku(r) = ku(r), ku(r) = ku(r), if k 0, ku(r) = ku(r), ku(r) = ku(r), if k < 0. Definition 2.4. [24] For rbitrry fuzzy numbers u, v, we use the distnce D(u, v) = sup 0 r mx{ u(r) v(r), u(r) v(r) } nd it is shown tht (E, D) is complete metric spce [40]. Definition 2.5. [23, 24] Let f : [, b] E, for ech prtition P = {t 0, t,..., t n } of [, b] nd for rbitrry ξ i [t i, t i ], i n suppose The definite integrl of f(t) over [, b] is R p = n i= f(ξ i)(t i t i ), := mx{ t i t i, i =, 2,..., n}. b f(t)dt = lim 0 R p provided tht this limit exists in the metric D. If the fuzzy function f(t) is continuous in the metric D, its definite integrl exists [24] nd lso, ( b f(t; r)dt) = b f(t; r)dt, ( b f(t; r)dt) = b f(t; r)dt. Definition 2.6. Let u, v E. If there exists w E such tht u = v +w then w is clled the H-difference of u, v nd it is denoted by u v. 3 ISPACS GmbH

4 Definition 2.7. A function f : (, b) E is clled H-differentible t ˆt (, b) if, for h > 0 sufficiently smll, there exist the H-differences f(ˆt + h) f(ˆt), f(ˆt) f(ˆt h), nd n element f (ˆt) E such tht: lim h 0 +D( f(ˆt + h) f(ˆt) h Then f (ˆt) is clled the fuzzy derivtive of f t ˆt. 3 Fuzzy integro-differentil eqution The liner Fredholm integro-differentil equtions [25] X (s) = y(s) + λ, f (ˆt)) = lim h 0 +D( f(ˆt) f(ˆt h), f (ˆt)) = 0. h b k(s, t)x(t)dt, X(s 0 ) = X 0, (3.) where λ > 0, k is n rbitrry given kernel function over the squre s, t b nd y(s) is given function of s [, b]. If X is fuzzy function, y(s) is given fuzzy function of s [, b] nd X is the fuzzy derivtive (ccording to Definition (2.7)) of X, this eqution my only possess fuzzy solution. Sufficient for the existence eqution of the second kind, re given in [3]. Let X(s) = (X(s; r), X(s; r)) is fuzzy solution of Eq.(3.), therefore by Definition (2.3), Definition (2.5) nd Definition (2.7) we hve the equivlent system X (s) = y(s) + λ b k(s, t)x(t)dt, X(s 0) = X 0, X (s) = y(s) + λ b k(s, t)x(t)dt, X(s 0) = X 0 (3.2) which possesses unique solution (X, X) B which is fuzzy function, i.e. for ech s, the pir (X(s; r), X(s; r)) is fuzzy number, therefore ech solution of Eq.(3.) is solution of system (3.2) nd conversely lso Eq.(3.) nd system (3.2) re equivlent. The prmetric form of Eqs.(3.2) is given by X (s, r) = y(s, r) + λ b k(s, t)x(t, r)dt, X(s X (s, r) = y(s, r) + λ b k(s, t)x(t, r)dt, X(s 0) = X 0 (r), 0) = X 0 (r) (3.3) for r [0, ]. suppose k(s, t) be continuous in s b nd for fix t, k(s, t) chnges its sing in finite points s s i where x i [, s ]. For exmple, let k(s, t) be nonnegtive over [, s ] nd negtive over [s, b], therefore we hve X (s, r) = y(s, r) + λ s X (s, r) = y(s, r) + λ s k(s, t)x(t, r)dt + λ b s k(s, t)x(t, r)dt, k(s, t)x(t, r)dt + λ b s k(s, t)x(t, r)dt, X(s 0 ) = X 0 (r), X(s 0 ) = X 0 (r). In most cses, however, nlyticl solution to Eq.(3.3) my not be found nd numericl pproch must be considered. 4 ISPACS GmbH

5 4 Function pproximtion The use of neurl networks provides solutions with very good generlizbility (such s differentibility). On the other hnd, n importnt feture of multi-lyer perceptrons is their utility to pproximte functions, which leds to wide pplicbility in most problems. In this pper, the function pproximtion cpbilities of feed-forwrd neurl networks is used by expressing the tril solutions for system (3.3) s the sum of two terms. The first term stisfies the intil conditions nd contins no djustble prmeters. The second term involves feed-forwrd neurl network to be trined so s to stisfy the integrodifferentil equtions. Since it is known tht multilyer perceptron with one hidden lyer cn pproximte ny function to rbitrry ccurcy, the multilyer perceptron is used s the type of the network rchitecture. If X T (s, r, p) is tril solution for the first eqution in system (3.3) nd X T (s, r, p) is tril solution for the second eqution in the system (3.3) where p nd p re djustble prmeters, (indeed X T (s, r, p) nd X T (s, r, p) re pproximtions of X(s, r) nd X(s, r) respectively) thus the problem of finding the pproximted solutions for (3.3) over some colloction point in [, b] is equivlent to clculte the functionls X T nd X T tht stisfies the following constrined optimiztion problem [29]: M min p i= {(X T (s i, r, p) y(s i, r) F (s i, r, p)) 2 + (X T (s i, r, p) y(s i, r) F (s i, r, p)) 2 }, (4.4) X T (s 0, r, p) = X 0 (r), X T (s 0, r, p) = X 0 (r) where p = (p, p) contin ll djustble prmeters (weights of input nd output lyers nd bises) nd F (s, r, p) = λ b k(s, t)x T (t, r, p)dt, F (s, r, p) = λ b k(s, t)x T (t, r, p)dt. In generl we cnnot be ble to crry out nlyticlly the integrtions, involved. In this cse we nturlly turn to numericl qudrture. We introduce qudrture rule R for the intervl [, b] with positive weights w j nd N nodes t j, i.e., Rf = N w j f(t j ) = If Ef = j= b f(t)dt Ef, where Ef is the error. If we first ignore the error of this qudrture rule then the Eq. (4.4) is replced by the pproximte eqution M min p i= {(X T (s i, r, p) y(s i, r) λ N j= w jk(s i, t j )X T (t j, r, p)) 2 + (X T (s i, r, p) y(s i, r) λ N j= w jk(s i, t j )X T (t j, r, p)) 2 }, X T (s 0, r, p) = X 0 (r), X T (s 0, r, p) = X 0 (r). (4.5) Ech tril solution X T nd X T employs one feed-forwrd neurl network for wich the corresponding networks re denoted by N nd N, with djustble prmeters p nd p, respectively. The relted tril functions will be in the form [22]: X T (s, r, p) = X(s 0, r) + (s s 0 )N(s, r, p), X T (s, r, p) = X(s 0, r) + (s s 0 )N(s, r, p), (4.6) 5 ISPACS GmbH

6 where N nd N re single-output feed-forwrd neurl networks with djustble prmeters p nd p, respectively. Here s nd r re the network inputs. This solutions by intention stisfies the initil condition in (4.6). According to (4.6) it is stright forwrd to show tht: X T (s, r, p) = N(s, r, p) + (s s 0) N s, X T (s, r, p) = N(s, r, p) + (s s 0 ) N s. (4.7) Now consider multilyer perceptron hving one hidden lyer with H sigmoid units nd liner output unit (Fig., Fig. 2). b s r w w 2 Input w 2 Input w 22 w H 2 b 2 z 2 z z H N Liner w H2 H Fig.. Three lyered perceptron with two input nd N output. b H b s r w w 2 Input w 2 Input w 22 w H 2 b 2 z 2 z H z Liner N w H2 H Fig. 2. Three lyered perceptron with two input nd N output. b H Here we hve: N = H i= v iσ(z i ), z i = w i s + w i2 r + b i, N = H i= v iσ(z i ), z i = w i s + w i2 r + b i, where σ(z) is the sigmoid trnsfer function. The following is obtined: (4.8) N s N s = H i= v iw i σ (z i ), = H i= v iw i σ (z i ), 6 ISPACS GmbH

7 where σ ( z i ) is the first derivtive of the sigmoid function. Also there re mny choices for the sigmoid function, here we choose σ(z) = /( + e z ) since it is possible to derive ll the derivtives of σ(z) in terms of the sigmoid function itself. i.e. σ (z) = σ 2 (z) + σ(z). 5 Exmple To illustrte the technique proposed in this pper, consider the following exmple. For ech fuzzy numbers, we use r = 0, 0.,...,, where we clculte the ccurcy of the method by Eq. (4.5). In the computer simultion of this section, we use the H = 0 sigmoid units in the hidden lyer. Exmple 5.. Consider the following fuzzy liner Fredholm integro-differentil eqution X (s) = ( r, 2 r)(e s s) + 0 tsx(t)dt, X(0) = ( r, 2 r); 0 r, 0 s, t. The exct solution in this cse is given by The tril functions for this problem re X = ( r, 2 r)e s. X T (s; r) = ( r) + s H i= X T (s; r) = (2 r) + s H i= v i +e w i s w i2 r b i, v i +e w i s w i2 r b i. The exct nd obtined solution of fuzzy liner Fredholm integro-differentil eqution in this exmple t s = re shown in Figure 3, lso the error by Eq. (4.5) is.2433e Exct solution Approximte solution Fig. 3. The exct nd pproximte solution for exmple ISPACS GmbH

8 Figs. 4-7 show the convergence property of the computed vlues of the weights Fig. 4. Convergence of the weights w i for exmple Fig. 5. Convergence of the weights w i2 for exmple ISPACS GmbH

9 Fig. 6. Convergence of the weights v i for exmple Fig. 7. Convergence of the weights b i for exmple Conclusion Solving fuzzy integro-differentil eqution (FIDE) by using universl pproximtors (UA), tht is, neurl network model (NNM) is presented in this pper. In this pper, the originl fuzzy integro-differentil eqution is replced by two prmetric liner Fredholm integro-differentil equtions which re then solved numericlly using UAM. The min reson for using neurl networks ws their pplicbility in function pproximtion. Our computer simultion in this pper were performed for three-lyer feedforwrd neurl networks. Since we hd good simultion result even from three-lyer 9 ISPACS GmbH

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