Fuzzy transform to approximate solution of boundary value problems via optimal coefficients

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1 217 Interntionl Conference on Hig Performnce Computing & Simultion Fuzzy trnsform to pproximte solution of boundry vlue problems vi optiml coefficients Zr Alijni Mtemtics nd Sttistics Dept University of Trtu Trtu, Estoni Alirez Kstn Mtemtics Dept IASBS Znjn, Irn Snjy K Kttri Fculty of Tecnology HVL Stord, Norwy snjykttri@vlno Stefni Tomsiello CORISA nd DIEM University of Slerno Fiscino (SA), Itly stomsiello@unisit Abstrct In tis study, we propose fuzzy-bsed pproc iming t finding numericl solutions to some clssicl problems We use te tecnique of fuzzy trnsform to solve second-order ordinry differentil eqution wit boundry conditions To determine te optiml coefficients in expnsion of function, we minimize te integrl squred error in 2-norm Some exmples re presented to illustrte te pplicbility of te metod Index Terms Fuzzy trnsform, Differentil eqution, Boundry vlue problem, Optiml coefficients I INTRODUCTION Fuzzy trnsform ws proposed by Perfiliev in [1] nd it is n pproximtion metod bsed on fuzzy sets Te core ide of F-trnsforms is fuzzy prtition of universe into fuzzy subsets Two types of bsic functions, tringulr nd sinusoidl sped membersip functions were proposed by Perfiliev in [1] Lter, in [2], te utors considered different spes for te bsic functions of fuzzy prtition Tey defined F- trnsforms bsed on severl clssicl pproximtion opertors, suc s B-splines, Seprd kernels, Bernstein bsis polynomils nd Fvrd-Szász-Mirkjn opertors Oter considertions on te properties of F-trnsforms nd teir improvements cn be found in [3] [5] In tis contribution, we continue te study of F-trnsform bsed numericl metods for ordinry differentil equtions (ODEs) We discuss new metod for pproximtion of functions nd its ppliction in numericl solution of boundry vlue problems (BVPs) From certin perspective, suc metod s its counterprt in dt compression by te lestsqures pproc described in [6], [7], [8] Te nlysis of existence nd uniqueness of solution nd solution metods for BVPs re more sopisticted Moreover, BVP requires solution of corresponding liner or non-liner system of equtions wic my cuse dditionl tecnicl efforts [9] Te proposed pproc stems from [1] were te Euler-like metod for te IVPs ws discussed Te sme pproc s been successfully used in [1] for second order IVPs For te sme problem, some numericl scemes bsed on te iger degree F-trnsform ve been introduced in [11] Te nlysis of te F-trnsform-bsed pproc to BVPs s been initited in [12] were one prticulr cse of tis problem ws nlyzed nd efficiently solved [13] In tis contribution, we continue tis nlysis nd pply te F- trnsform metod to liner nd nonliner second order BVPs using te optiml coefficients II PRELIMINARIES In tis section, we give some definitions nd introduce te necessry nottion wic will be used trougout te pper, see for exmple [1] Definition 21: Let x 1 < < x n be fixed nodes witin [, b], suc tt x 1 =, x n = b nd n 3 We sy tt fuzzy sets A 1,, A n, identified wit teir membersip functions A 1 (x),, A n (x) defined on [, b], form fuzzy prtition of [, b] if tey fulfill te following conditions for k = 1,, n, 1) A k : [, b] [, 1], A k (x k ) = 1; 2) A k (x) = if x (, x k+1 ), were for te uniformity of te nottion, we put x = nd x n+1 = b; 3) A k (x) is continuous; 4) A k (x), k = 2,, n, strictly increses on [, x k ] nd A k (x), k = 1,, n 1, strictly decreses on [x k, x k+1 ]; 5) for ll x [, b], n k=1 A k(x) = 1 Te membersip functions A 1,, A n re clled bsic functions Let us remrk tt bsic functions re specified by set of nodes x 1 < < x n nd te properties bove Definition 22: Te fuzzy sets A 1,, A n identified by membersip functions 1 x x 1, x 1 x x 2, A 1 (x) = 1, oterwise, (x ), x x k, k 1 A k (x) = 1 (x x k), x k x x k+1, k, oterwise, (x x n 1 ), x n 1 x x n, A n (x) = n 1, oterwise, /17 $ IEEE DOI 1119/HPCS

2 Figure 1 An exmple of tringulr bsic function over [, 5] Figure 2 An exmple of sinusoidl bsic function over [, 5] is clled tringulr bsic functions (TFs), were k = 2, 3,, n 1, k = x k+1 x k, see Figure 1 Remrk 23: We sy tt fuzzy prtition is uniform, if nodes x 1,, x n re equidistnt, ie, x k = + (k 1), k = 1,, n, were = b, nd two dditionl properties re n 1 met: A k (x k x) = A k (x k + x), for ll x,k = 2,, n 1 A k+1 (x) = A k (x ), for ll x,k = 2,, n 2 In te cse of uniform fuzzy prtition, is te lengt of te support of A 1 (x) or A n (x) wile 2 is te lengt of te oter bsic functions A k, k = 2, 3,, n 1 It is esy to see tt TFs form uniform fuzzy prtition on [, b] Definition 24: Te fuzzy sets A 1,, A n identified by membersip functions 5(cos( π (x x 1) + 1), x 1 x x 2, A 1 (x) =, oterwise, 5(cos( π (x x k) + 1), x x k+1, A k (x) =, oterwise, 5(cos( π (x x n) + 1), x n 1 x x n, A n (x) =, oterwise, is clled sinusoidl bsic functions (SFs), were k = 2, 3,, n 1, see Figure 2 Lemm 25: [1] Let te uniform prtition of [, b] be given by bsic functions A 1 (x),, A n (x) Ten we ve x2 x 1 A 1 (x)dx = nd for k = 2,, n 1, xn x n 1 A n (x)dx = 2, A k (x)dx = Definition 26: Let A 1,, A n be bsic functions wic form fuzzy prtition of [, b] nd f be ny continuous function on [, b] We sy tt te n-tuple of rel numbers [ F 1,, F n ] given by F k = f(x)a k(x)dx A, k = 1,, n, (1) k(x)dx is te F-trnsform of f wit respect to A 1,, A n Lemm 27: [1] Let f be ny twice continuously differentible function on (, b) nd A 1,, A n be bsic functions wic form n uniform fuzzy prtition of [, b] Let F n [f] = [F 1,, F n ] be te F-trnsform of f wit respect to A 1,, A n Ten for ec k = 1,, n, we ve F k = f(x k ) + O( 2 ), k = 1, 2,, n Definition 28: (Inverse F-trnsform) Let A 1,, A n be bsic functions wic form fuzzy prtition of [, b] nd f be continuous function on [, b] Let F n [f] = [F 1,, F n ] be te F-trnsform of f wit respect to A 1,, A n Ten te function f F,n (x) = F k A k (x), (2) k=1 is clled te inverse F-trnsform of f Te following teorem gurntees te convergence of sequence on inverse F-trnsforms wic re bsed on rbitrry bsic functions Tis mens tt te convergence olds irrespective of spes of bsic functions However, te speed of convergence my be influenced by concrete spe of bsic functions [2] Teorem 29: [1] For sequence A (n) 1,, A(n) n of uniform prtitions of [, b] te respective sequence f F,n (x) of inverse F-trnsforms of f C[, b] uniformly converges to f 467

3 III BEST APPROXIMATION IN 2-NORM USING FUZZY BASIC FUNCTIONS In tis section, we propose metod to obtin te pproximtion of continuous function in te spce of fuzzy bsic functions We determine te expnsion coefficients (optiml coefficients) using te minimiztion of 2-norm Here we use tringulr bsic functions It is esy to extend te results for oter bsic functions [2] Let L 2 [, b] = { f : [, b] R (f(x)) 2 is integrble on [, b] } Tis is n inner product spce wit te inner product < f, g >= f(x)g(x)dx ( ) 1 b Te induced norm is f = f(x) 2 2 dx Tis norm is clled te L 2 norm, or briefly, 2-norm Let f be continuous function on [, b] nd A 1,, A n be tringulr bsic functions wic form fuzzy prtition of [, b] Let { } P n = q : [, b] R q = α i A i (x), for some α j R Our im is to find te pproximtion p n = n α ia i (x) P n suc tt f p n = inf q P n f q, were p n is clled te best pproximtion to te function f in te 2-norm on [, b], wit respect to fuzzy bsic functions Let M = f q 2 = (f(x) α i A i (x)) 2 dx To find te optiml coefficients, we ve M =, k = α k 1, 2,, n So we obtin f(x)a k (x)dx = α i A k (x)a i (x)dx, or equivlently F k A k (x)dx = α i A k (x)a i (x)dx, k = 1, 2, n Terefore by dopting uniform prtition, we obtin 2 F 1 = α 1 x2 x 1 (A 1 (x)) 2 dx + α 2 x2 x 1 A 1 (x)a 2 (x)dx, (3) F k = α k 1 xk A k (x)a k 1 (x)dx + α k (A k (x)) 2 dx +α k+1 x k A k (x)a k+1 (x)dx, k = 2, 3,, n 1, (4) nd 2 F n = α n 1 xn x n 1 A n (x)a n 1 (x)dx + α n xn x n 1 (A n (x)) 2 dx (5) Clculting te integrls for tringulr bsic functions, we obtin xk A k (x)a k 1 (x)dx = A k (x)a k+1 (x)dx = x k 6, nd xk+1 (A k (x)) 2 dx = 2 x k (A k (x)) 2 dx = 2 3 So by re-writing te equtions (3)-(5) s tridigonl system, we ve F 1 F 2 F n = α 1 α 2 α n Te liner system of equtions (6) is nonsingulr tridigonl system nd it s unique solution Ten we my use ny efficient metod for solving (6) to obtin optiml coefficients α i, i = 1, 2,, n Remrk 31: In similr wy, we cn obtin te optiml coefficients of expnsion wit respect to sinusoidl bsic functions It is esy to ceck tt in tis cse, in te forwrd stge we ve xk A k (x)a k 1 (x)dx = A k (x)a k+1 (x)dx = x k 8, nd xk+1 (A k (x)) 2 dx = 2 x k (A k (x)) 2 dx = 3 4 Terefore we deduce te liner system of equtions 6 2 F F 2 = F n α 1 α 2 α n (6) Here our im is to obtin α k for tringulr bsic functions For simplifying, let us denote k s elements of lower digonl, c k s elements of upper digonl nd b k s elements of min digonl Terefore, te solution of tis system using te Toms metod in [14] is s following; for k = 2 until n, we ve p = k, b k 1 b k = b k pc k 1, (8) F k = F k pf k 1 (7) 468

4 By bckwrd substitution, we ve α n = F n, b n α k = F k c k α k+1, k = n 1,, 1 b k Exmple 32: Let us consider n = 3, = 1 nd functions f(x) = exp( x) nd g(x) = x 2 Using te proposed metod, we obtin optiml coefficients α i s nd α i for functions f(x) nd g(x), respectively s tbulted in Tble 1 So using te optiml coefficients, we cn find n pproximtion to function f(x) on [, 2] by inverse fuzzy trnsform s f(x) 3 α ia i (x) i α i α i Tble 1: Optiml coefficients for Exmple 32 Remrk 33: Similrly to Lemm 27, it is esy to sow tt for ny twice continuously differentible function f on (x 1, x n ), we ve α k = f(x k ) + O( 2 ), k = 1, 2,, n We wis to point out tt metods suc s te weigted residuls (eg te Glerkin metod), im to minimize (in integrl form) te residul of te differentil eqution multiplied by weigt, by ssuming n pproximtion similr to te inverse F-trnsform It is cler tt te pproc erein proposed is different nd it is independent of te differentil eqution IV APPLICATION TO BVPS In te following, we modify te generl sceme of finite differences metod by optiml components In [12], te utors presented new pproc to numericl solution of BVPs Here we reformulte te problem using te optiml coefficients Let us consider te boundry vlue problem { y + my = f(x, y), x [x 1, x n ], (1) y(x 1 ) = y 1, y(x n ) = y n, were f is continuous, y(x) nd f(x, y(x)) re sufficiently smoot on [x 1, x n ] Let x k = x 1 + (k 1), k = 1, 2,, n, were = x n x 1 Now, similr to results in [12], to solve n 1 (1) using te optiml coefficients (9), we ve te following 2 nd c = 2 scemes were = 1 2 m 2, b = m 2 Sceme 1 (m ) c Y k+1 = c 2 b Y 2 = 1 c (α 1 Y 1 cy 3 ), Y 1 = y 1, Y n = y n (9) (α k c α k 1 by k ), k = n 2,, 2, (11) Sceme 2 (m = ) Y k+1 = 4 ( α k + 1 3c 2 α k 1 + c ) 2 Y k, k = n 2,, 2 Y 2 = 1 ( α 1 + c c 2 Y 1 + c ) 2 Y 3, Y 1 = y 1, Y n = y n (12) Finlly, te pproximte solution of (1) cn be obtined using te inverse fuzzy trnsform s y n (x) = Y k A k (x), (13) k=1 were A k (x), k = 1, 2,, n re te given bsic functions Ten te solution lgoritm cn be summrized s follows: I compute α k from equtions (8)-(9) II compute Y k, by replcing te computed vlues of α k in sceme 1 of eqution (11) or sceme 2 of eqution (12), ccording to te vlue of p III compute te pproximte solution by Eq (13) Exmple 41: Let us consider y = 12x 2, y() =, y(1) = (14) Te exct solution is y = x 4 x Te numericl results nd comprisons wit finite difference metod (FDM) [9], new metod nd fuzzy trnsform (FT) [12] for = 1 nd = 1 re sown in Tble 2 nd 3, respectively We see tt for smller vlue of, we obtin better results In Figure 3, comprison between FDM nd new metod is sown for = 1 In Figures 4-5, we depict te errors of pproximte solutions for =1 nd =1, respectively x k Solution FDM FT New metod Tble 2 A comprison for Exmple 41, = 1 x k Solution FDM FT New metod Tble 3 A comprison for Exmple 41, = 1 Te ccurcy of te obtined pproximte solutions in te form of te mximl error vlue for = 1 is 5e 7 were for FT metod is e 5 469

5 Figure 3 (-) Exct solution, (o) pproximte solution by FDM nd (*) pproximte solution by new metod Using clssicl metods like Sooting metod [9], we trnsform (15) into te Cucy problem of te form { y = f(x, y, y ), x [x 1, x n ], y(x 1 ) = α, y (x 1 ) = γ (16) were γ is n unknown number Te solution of te problem (16) depends on te coice of te unknown prmeter γ Terefore we use y(x, γ) for te solution of tis problem Our im is to define γ suc tt y(x n, γ) = β Let G : R R be function defined s G(γ) = y(x n, γ) β Hence, we serc for γ defined s G(γ ) = Te root cn be found vi suitble numericl metods We use Newton metod for finding te root Assume tt G is differentible nd G Ten γ n+1 = γ n G(γ n) G (γ) n, n =, 1, Let y = y(x, γ) Ten from (15) we ve { y (x, γ) = f(x, y(x, γ), y (x, γ) ), x (x 1, x n ), y(x 1, γ) = α, y (x 1, γ) = γ (17) Figure 4 Error of pproximte solution by FDM (blue), FT (red) nd new metod (green), = Let us define te derivtive of (17) wit respect to te prmeter γ ( y f x, y(x, γ), y (x, γ) ) (x, γ) = γ y(x, γ) f ( x, y(x, γ), y (x, γ) ) y (x, γ) y, (x, γ) γ y(, γ) =, y (, γ) = 1 γ γ y(x, γ) + γ (18) Te problem (18) is Cucy problem, wic cn be solved by te metod proposed in te previous section Exmple 42: Let us consider y = y + 2(y ) 2, y( 1) = y(1) = (e + e 1 ) 1 y Te exct solution is y = (e x +e x ) 1 Comprisons between sooting metod (SM) [9] nd te new metod for = 1 nd = 1 re sown in Tble 4 nd 5, respectively In Figures 6-7, we depict te errors of pproximte solutions for =1 nd =1, respectively Figure 5 Error of pproximte solution by FDM (blue), FT (red) nd new metod (green), =1 A Te cse of nonliner BVPs Consider te two point boundry vlue problem { y = f(x, y, y ), x [x 1, x n ], y(x 1 ) = α, y(x n ) = β (15) x k Solution SM New metod Tble 4 A comprison for Exmple 42, = 1 47

6 x k Solution SM New metod Tble 5 A comprison for Exmple 42, = Figure 6 Error of pproximte solution by SM (blue), nd new metod (red), =1 [5] V Loi, S Tomsiello, nd L Troino, Improving pproximtion properties of fuzzy trnsform troug non-uniform prtitions in: Petrosino A, Loi V, Pedrycz W (eds) Fuzzy Logic nd Soft Computing Applictions WILF 216 Lecture Notes in Computer Science, vol 1147 Springer, Cm, 216, pp [6] G Ptne, Fuzzy trnsform nd lest-squres pproximtion: nlogies, differences,nd generliztions, Fuzzy Sets nd Systems, vol 18, p 41 54, 211 [7] M Get, V Loi, nd S Tomsiello, Cubic b-spline fuzzy trnsforms for n efficient nd secure compression in wireless sensor networks, Informtion Sciences, vol 339, pp 19-3, 216 [8] V Loi, S Tomsiello, nd A Vccro, Fuzzy trnsform bsed compression of electric signl wveforms for smrt grids, IEEE Trnsctions on Systems, Mn, nd Cybernetics: Systems, vol 47, pp , 217 [9] K E Atkinson, An introduction to numericl nlysis Jon Wiley & Sons, 28 [1] W Cen nd Y Sen, Approximte solution for clss of secondorder ordinry differentil equtions by te fuzzy trnsform, Journl of Intelligent & Fuzzy Systems, vol 27, no 1, pp 73 82, 214 [11] A Kstn, I Perfiliev, nd Z Alijni, A new fuzzy pproximtion metod to cucy problems by fuzzy trnsform, Fuzzy Sets nd Systems, vol 288, pp 75 95, 216 [12] A Kstn, Z Alijni, nd I Perfiliev, Fuzzy trnsform to pproximte solution of two-point boundry vlue problems, Mtemticl Metods in te Applied Sciences, 216 [13] I Perfiliev, P Śtevuliková, nd R Vlásek, F-trnsform for numericl solution of two-point boundry vlue problem, Irnin Journl of Fuzzy Systems, vol, no 1, p In Press, 217 [14] L Toms, Elliptic problems in liner differentil equtions over network: Wtson scientific computing lbortory, Columbi Univ, NY, Figure 7 Error of pproximte solution by SM (blue), nd new metod (red), =1 REFERENCES [1] I Perfiliev, Fuzzy trnsform: ppliction to reef growt problem, Fuzzy Logic in Geology, pp 275 3, 21 [2] B Bede nd I J Ruds, Approximtion properties of fuzzy trnsforms, Fuzzy Sets nd Systems, vol 18, no 1, pp 2 4, 211 [3] L Troino nd P Kriplni, A men-reverting strtegy bsed on fuzzy trnsform residuls 212 IEEE Conference on Computtionl Intelligence for Finncil Engineering nd Economics, CIFEr, 212, pp [4] L Troino, Fuzzy co-trnsform nd its ppliction to time series Proceedings of te 21 Interntionl Conference of Soft Computing nd Pttern Recognition, SoCPR, 21, pp