NUMERICAL SOLUTIONS OF NONLINEAR FUZZY FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF THE SECOND KIND

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1 Irnin Journl of Fuzzy Systems Vol. 12, No. 2, (2015) NUMERICAL SOLUTIONS OF NONLINEAR FUZZY FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF THE SECOND KIND M. MOSLEH AND M. OTADI Abstrct. In this er, we use rmetric form of fuzzy number, then n itertive roch for obtining roximte solution for clss of nonliner fuzzy Fredholm integro-differentil eqution of the second kind is roosed. This er resents method bsed on Newton-Cotes methods with ositive coefficient. Then we obtin roximte solution of the nonliner fuzzy integro-differentil equtions by n itertive roch. 1. Introduction The solutions of integrl equtions hve mjor role in the field of science nd engineering. A hysicl even cn be modelled by the differentil eqution, n integrl eqution. Since few of these equtions cnnot be solved exlicitly, it is often necessry to resort to numericl techniques which re rorite combintions of numericl integrtion nd interoltion [13, 29]. There re severl numericl methods for solving liner Volterr integrl eqution [18, 37] nd system of nonliner Volterr integrl equtions [15]. Kuthen in [26] used colloction method to solve the Volterr- Fredholm integrl eqution numericlly. Borzbdi nd Frd in [16] obtined numericl solution of nonliner Fredholm integrl equtions of the second kind. The concet of fuzzy numbers nd fuzzy rithmetic oertions were first introduced by Zdeh [40], Dubois nd Prde [19]. We refer the reder to [24] for more informtion on fuzzy numbers nd fuzzy rithmetic. The numericl solution of fuzzy nonliner eqution by Newton s method nd steeest descent method were considered [3, 5]. The toics of fuzzy integrl equtions (FIEs) nd fuzzy differentil equtions (FDEs) which growing interest for some time, in rticulr in reltion to fuzzy control, hve been ridly develoed in recent yers [1, 2, 7, 8, 6, 9, 10, 33, 34]. They used the concet of H-differentibility which ws introduced by Puri nd Rlescu [35]. The concet of fuzzy rndom vrible ws roosed by Kwkernk [28]. Then, the uthors of [30, 31, 32] considered the rndom fuzzy differentil equtions where the two kinds of uncertinties (rndomness nd fuzziness) were incororted. The fuzzy ming function ws introduced by Chng nd Zdeh [17]. Lter, Dubois nd Prde [20] resented n elementry fuzzy clculus bsed on the extension rincile lso the concet of integrtion of fuzzy functions ws first Received: November 2013; Revised: August 2014; Acceted: December 2014 Key words nd hrses: Nonliner fuzzy integro-differentil equtions, Newton-Cotes methods.

2 118 M. Mosleh nd M. Otdi introduced by Dubois nd Prde [20]. Bbolin et l. nd Abbsbndy et l. in [4, 12] obtined numericl solution of liner Fredholm fuzzy integrl equtions of the second kind. In this er, we generlize the nonliner fuzzy integrl equtions to the nonliner fuzzy integro-differentil equtions X (s) = y(s) + k(s, t, X(t))dt. In this er, we resent novel nd very simle numericl method bsed uon itertive methods for solving nonliner fuzzy Fredholm integro-differentil equtions of the second kind. 2. Preliminries In this section the bsic nottions used in fuzzy oertions re introduced. We strt by defining the fuzzy number. Definition 2.1. A fuzzy number is fuzzy set u : R 1 I = [0, 1] such tht [27]: i. u is uer semi-continuous; ii. u(x) = 0 outside some intervl [, d]; iii. There re rel numbers b nd c, b c d, for which 1. u(x) is monotoniclly incresing on [, b], 2. u(x) is monotoniclly decresing on [c, d], 3. u(x) = 1, b x c. The set of ll the fuzzy numbers (s given in definition 1) is denoted by E 1. An lterntive definition which yields the sme E 1 is given by Klev [25]. Definition 2.2. A fuzzy number u is ir (u, u) of functions u(r) nd u(r), 0 r 1, which stisfy the following requirements: i. u(r) is bounded monotoniclly incresing, left continuous function on (0, 1] nd right continuous t 0; ii. u(r) is bounded monotoniclly decresing, left continuous function on (0, 1] nd right continuous t 0; iii. u(r) u(r), 0 r 1. A cris number r is simly reresented by u(α) = u(α) = r, 0 α 1. The set of ll the fuzzy numbers is denoted by E 1. This fuzzy number sce s shown in [39], cn be embedded into the Bnch sce B = C[0, 1] C[0, 1]. For rbitrry u = (u(r), u(r)), v = (v(r), v(r)) nd k R we define ddition nd multiliction 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 2.3. For rbitrry fuzzy numbers u, v, we use the distnce [22]: D(u, v) = su 0 r 1 mx{ u(r) v(r), u(r) v(r) } nd it is shown tht (E 1, D) is comlete metric sce [35].

3 Numericl Solutions of Nonliner Fuzzy Fredholm Integro-differentil Equtions of Definition 2.4. Let f : [, b] E 1, for ech rtition P = {t 0, t 1,..., t n } of [, b] nd for rbitrry ξ i [t i 1, t i ], 1 i n suose n R = f(ξ i )(t i t i 1 ), := mx{ t i t i 1, i = 1, 2,..., n}. i=1 The definite integrl of f(t) over [, b] is f(t)dt = lim 0 R rovided tht this limit exists in the metric D [21, 22]. If the fuzzy function f(t) is continuous in the metric D, its definite integrl exists [22] nd lso, f(t; r)dt) = f(t; r)dt, ( f(t; r)dt) = f(t; r)dt. Definition 2.5. Let u, v E 1. If there exists w E 1 such tht u = v + w then w is clled the H-difference of u, v nd it is denoted by u v. Definition 2.6. A function f : (, b) E 1 is clled H-differentible t ˆx (, b) if, for h > 0 sufficiently smll, there exist the H-differences f(ˆx + h) f(ˆt), f(ˆx) f(ˆx h), nd n element f (ˆx) E 1 such tht: f(ˆx + h) f(ˆx) lim h 0 +D(, f f(ˆx) f(ˆx h) (ˆx)) = lim h h 0 +D(, f (ˆx)) = 0. h Then f (ˆx) is clled the fuzzy derivtive of f t ˆx. 3. Fuzzy Integro-differentil Eqution The nonliner Fredholm integro-differentil eqution of the second kind [23] is X (s) = y(s) + k(s, t, X(t))dt, X(s 0 ) = X 0, (1) where k is n rbitrry given kernel function 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 of X [36], this eqution my only ossess fuzzy solution. Sufficient condition for the existence eqution of the second kind, is given in [14]. For solving equton (1) we my relce eqution (1) by the equivlent system X (s) = y(s) + X (s) = y(s) + b k(s, t, X(t))dt = y(s) + k(s, t, X(t))dt = y(s) + b F (s, t, X, X))dt, X(s 0) = X 0, G(s, t, X, X))dt, X(s 0) = X 0 (2) which ossesses unique solution (X, X) B which is fuzzy function, i.e. for ech s, the ir (X(s; r), X(s; r)) is fuzzy number. The rmetric form of eqution (2) is given by X (s; r) = y(s; r) + F (s, t, X(t; r), X(t; r)))dt, X(s 0 ; r) = X 0 (r),

4 120 M. Mosleh nd M. Otdi X (s; r) = y(s; r) + G(s, t, X(t; r), X(t; r)))dt, X(s 0 ; r) = X 0 (r) (3) for r [0, 1]. In most cses, however, nlyticl solution to eqution (3) my not be found nd numericl roch must be considered. 4. The Numericl Aroch We relce the intervl [, b] by set of discrete eqully sced grid oints = s 0 < s 1 <... < s N = b t which the exct solution (X(s; r), X(s; r)) is roximted by some (x(s; r), x(s; r)). The exct nd roximte solutions t s i, 0 i N re denoted by X i (r) = (X i (r), X i (r)) nd x i (r) = (x i (r), x i (r)), resectively. The grid oints t which the solution is clculted re s i = s 0 + ih, h = (b )/N; 1 i N. The first-order roximtion of X (s; r) nd X (s; r) is given by Z Z(s + h; r) Z(s; r) (s; r) h (4) where Z(s; r) is X(s; r) nd X(s; r) lterntively. By virtue of eqution (4) we obtin X i+1 (r) = X i (r) + h[y i (r) + F (s i, t, X(t; r), X(t; r))dt] + h2 2 X (ζ i ), X(s 0 ; r) = X 0 (r), X i+1 (r) = X i (r) + h[y i (r) + G(s i, t, X(t; r), X(t; r))dt] + h2 2 X (ζ i ), X(s 0 ; r) = X 0 (r), i = 0, 1,..., N, (5) where s i < ζ i, ζ i < s i+1. The Newton-Cotes method [11] is given by Z(t)dt = w j Z(t j ) + O(h ν ) where Z is F nd G lterntively nd ν deends uon the used method of Newton- Cotes with ositive coefficient for estimting of the integrl in eqution (6). By virtue of eqution (6) we obtin X i+1 (r) = X i (r) + h[y i (r) + w j F (s i, t j, X j (r), X j (r)))] + h2 2 X (ζ i ) + O(h ν+1 ), X(s 0 ; r) = X 0 (r), X i+1 (r) = X i (r) + h[y i (r) + w j G(s i, t j, X j (r), X j (r)))] + h2 2 X (ζ i ) + O(h ν+1 ), X(s 0 ; r) = X 0 (r), i = 0, 1,..., N. (7) (6)

5 Numericl Solutions of Nonliner Fuzzy Fredholm Integro-differentil Equtions of Following eqution (7) we define The olygon curves x i+1 (r) = x i (r) + h[y i (r) + x i+1 (r) = x i (r) + h[y i (r) + w j F (s i, t j, x j (r), x j (r)))], x(s 0 ; r) = x 0 (r), w j G(s i, t j, x j (r), x j (r)))], x(s 0 ; r) = x 0 (r), i = 0, 1,..., N. (8) x(s; h; r) {[s 0, x 0 (r)], [s 1, x 1 (r)], },..., [s N, x N (r)]}, x(s; h; r) {[s 0, x 0 (r)], [s 1, x 1 (r)], },..., [s N, x N (r)]} (9) re the roximtes to X(s; r) nd X(s; r), resectively, over the intervl s 0 s s N. Let F (s, t, u, v) nd G(s, t, u, v) be the functions F nd G of eqution (2) where u nd v re constnts nd u v. In other words F (s, t, u, v) nd G(s, t, u, v) re obtined by substituting X = (u, v) in eqution (2). The domin where F nd G re defined is therefore B = {(s, t, u, v) s, t b, < v < +, < u v}. Theorem 4.1. Let F (s, t, u, v) nd G(s, t, u, v) belong to C 1 (B), let the rtil derivtives of F, G be bounded over B nd D(X, x ) = mx 0 i N {D(X i, x i )}. Then, for rbitrry fixed r : 0 r 1, Proof. Let nd we hve: lim h 0 x (r) = X (r), X (r) = X 1 (r) + h[y 1 (r) + lim h 0 x (r) = X (r). w j F (s 1, t j, X j (r), X j (r)) + h2 2 X (ζ 1 ) + O(h ν+1 ), X (r) = X 1 (r) + h[y 1 (r) + x (r) = x 1 (r) + h[y 1 (r) + x (r) = x 1 (r) + h[y 1 (r) + w j G(s 1, t j, X j (r), X j (r)) + h2 2 X (ζ 1 ) + O(h ν+1 ) (10) w j F (s 1, t j, x j (r), x j (r))], w j G(s 1, t j, x j (r), x j (r))]. (11)

6 122 M. Mosleh nd M. Otdi Consequently X (r) x (r) = X 1 (r) x 1 (r) + h[ w j (F (s 1, t j, X j (r), X j (r)) F (s 1, t j, x j (r), x j (r)))] + h2 2 X (ζ 1 ) + O(h ν+1 ), X (r) x (r) = X 1 (r) x 1 (r) + h[ w j (G(s 1, t j, X j (r), X j (r)) G(s 1, t j, x j (r), x j (r)))] + h2 2 X (ζ 1 ) + O(h ν+1 ). Denote W = X (r) x (r), V = X (r) x (r). Then W W 1 + 2Lh(b )D(X, x ) + h2 2 M + O(hν+1 ), V V 1 + 2Lh(b )D(X, x ) + h2 2 M + O(hν+1 ), M = mx s0 s s N X (s; r), M = mx s0 s s N X (s; r) nd L > 0 is bound for the rtil derivtives of F, G. Thus, we hve W W 0 + 2Lh(b )D(X, x ) + h2 2 M + O(hν+1 ), V V 0 + 2Lh(b )D(X, x ) + h2 2 M + O(hν+1 ). Since W 0 = V 0 = 0 we obtin W 2Lh(b )D(X, x ) + h2 2 M + O(hν+1 ), V 2Lh(b )D(X, x ) + h2 2 M + O(hν+1 ) nd if h 0 we get W 0, V 0 which concludes the roof. So fr, we cme to the nonliner eqution system (8) with secil form tht let us offer numericl roch for obtining the roximte solution. Itertive methods re widely used for finding roximte solution of nonliner equtions systems [38]. The nonliner equtions system (8) lso hs structure tht ermits to roximte its solution by n itertive method. For this urose, we ly successive substitution, similr to Jcobi method of solving liner equtions systems nd therefore define n itertive rocess leding to the sequence of vectors x (k) nd x (k), where the comonents of the vectors stisfy the itertion formuls, x (k+1) i+1 (r) = x (k) i (r) + h[y i (r) + x(s 0 ; r) = x 0 (r), w j F (s i, t j, x (k) j (r), x (k) j (r))],

7 Numericl Solutions of Nonliner Fuzzy Fredholm Integro-differentil Equtions of x (k+1) i+1 (r) = x (k) i (r) + h[y i (r) + w j G(s i, t j, x (k) j (r), x (k) j (r))], x(s 0 ; r) = x 0 (r), i = 0, 1,..., N, k = 0, 1,..., K. (12) However, we should first study the conditions tht gurntee the convergence of the roximte solution. Theorem 4.2. Considering ssumtions of Theorem 1 nd D(x (k), x ) = mx 0 i N {D(x (k) i, x i ), } the roduced sequence x (k) from the itertion rocess (12) tends to the exct solution of (8), sy x, for ny rbitrry fuzzy initil vector x (0) with x (k) (s 0 ; r) = x 0 (r) for ll k. Proof. By (8) nd (12) we hve, h h n n x (k+1) (r) x (r) x(k) 1 (r) x 1 (r) + w j F (s 1, t j, x (k) j (r), x (k) j (r)) F (s 1, t j, x j (r), x j (r)), x (k+1) (r) x (r) x (k) 1 (r) x 1(r) + w j G(s 1, t j, x (k) j nd ccording to the conditions of theorem 4.1, (r), x (k) j (r)) G(s 1, t j, x j (r), x j (r)) x (k+1) (r) x (r) x(k) 1 (r) x 1 (r) + 2Lh(b )D(x(k), x ), x (k+1) (r) x (r) x (k) 1 (r) x 1(r) + 2Lh(b )D(x (k), x ). Denote W (k+1) Thus, we hve = x (k+1) (r) x (r), V (k+1) = x (k+1) (r) x (r). Then W (k+1) W (k) 1 + 2Lh(b )D(x(k), x ), V (k+1) V (k) + 2Lh(b )D(x(k), x ). 1 W (k+1) W (k) 0 + 2Lh(b )D(x (k), x ), V (k+1) V (k) 0 + 2Lh(b )D(x (k), x ). Since W (k) 0 = V (k) 0 = 0 for ll k we obtin W (k+1) V (k+1) nd if h 0 we get W (k+1) roof. 2Lh(b )D(x (k), x ), 2Lh(b )D(x (k), x ). 0, V (k+1) 0 for ll k which concludes the

8 124 M. Mosleh nd M. Otdi 5. Numericl Exmles To illustrte the technique roosed in this er, consider the following exmles. In this exmles we tke mx 0 i n {D(x (k+1) i, x k i )} < Exmle 5.1. Consider the following nonliner fuzzy Fredholm integro-differentil eqution X (s) = (r r2 12 4r 1 r2 t, ) X2 (t)dt, The rmetric equtions re X(0) = 0; 0 s, t 1, 0 r 1. X (s; r) = (r r2 8 ) + X (s; r) = ( t min((t; r))dt, r 1 r2 t ) + mx((t; r))dt, X(0; r) = 0, X(0; r) = 0, 0 s, t 1, 0 r 1, Figure 1. Comres the Exct Solution nd Obtined Solution t s = 0.5 where (t; r) = {X(t; r)x(t; r), X(t; r)x(t; r), X(t; r)x(t; r)}. The exct solution in this cse is given by X(s; r) = rs, X(s; r) = (2 r)s. The exct nd obtined solution of nonliner fuzzy Fredholm integro-differentil eqution in this exmle t s = 0.5 is shown in Figure 1. Exmle 5.2. Consider the following nonliner fuzzy Fredholm integro-differentil eqution 1 X (s) = y(s) + 2ste X(t) dt, X(0) = 0, 0 s, t 1, 0 s y(s; r) = 2s( r) r (e r 1), y(s; r) = 2s( r) + The exct solution in this cse is given by s r (e r 1), 0 r 1. X(s; r) = ( r)s 2,

9 Numericl Solutions of Nonliner Fuzzy Fredholm Integro-differentil Equtions of Figure 2. Comres the Exct Solution nd Obtined Solution X(s; r) = ( r)s 2, 0 r 1. The rmetric equtions re X (s; r)(s) = y(s; r) + X (s; r)(s) = y(s; r) ste X(t;r) dt, X(0; r) = 0, 2ste X(t;r) dt, X(0; r) = 0, 0 s, t 1. The exct nd obtined solution of nonliner fuzzy Fredholm integro-differentil eqution in this exmle t s = 0.5 is shown in Figure Conclusions We roose generl numericl rocedure for treting nonliner fuzzy Fredholm integro-differentil equtions of the second kind. The originl nonliner fuzzy Fredholm integro-differentil eqution is relced by two rmetric nonliner Fredholm integro-differentil equtions which re then solved numericlly using clssicl lgorithm. In this er the stndrd Newton-Cotes method is designed for roximting integrl. Also we cn execute this method in comuter simly. Acknowledgements. We would like to resent our sincere thnks to the Prof. M. Mshinchi, Prof. R. A. Borzooei, Prof. S. Abbsbndy nd referees for their vluble suggestions. References [1] S. Abbsbndy nd T. Allhvirnloo,Numericl solution of fuzzy differentil eqution by Runge-Kutt method, Nonliner studies, 11(1) (2004), [2] S. Abbsbndy, T. Allvirnloo, O. Loez-Pouso nd J. J. Nieto, Numericl methods for fuzzy differentil inclusions, Comuters & mthemtics with lictions, 48(10-11) (2004), [3] S. Abbsbndy nd B. Asdy, Newtons method for solving fuzzy nonliner equtions, Alied Mthemtics nd Comuttion, 159(2) (2004), [4] S. Abbsbndy, E. Bbolin nd M. Alvi, Numericl method for solving liner Fredholm fuzzy integrl equtions of the second kind, Chos Solitons & Frctls, 31(1) (2007), [5] S. Abbsbndy nd A. Jfrin, Steeest descent method for solving fuzzy nonliner equtions, Alied Mthemtics nd Comuttion, 175(1) (2006), [6] S. Abbsbndy, J. J. Nieto nd M. Alvi, Tuning of rechble set in one dimensionl fuzzy differentil inclusions, Chos, Solitons & Frctls, 26(5) (2005),

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