A Bernstein polynomial approach for solution of nonlinear integral equations

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1 Avilble online t wwwisr-publictionscom/jns J Nonliner Sci Appl, 10 (2017), Reserch Article Journl Homepge: wwwtjnscom - wwwisr-publictionscom/jns A Bernstein polynomil pproch for solution of nonliner integrl equtions Neşe İşler Acr, Ayşegül Dşcıoğlu b, Deprtment of Mthemtics, Fculty of Arts nd Sciences, Mehmet Akif Ersoy University, Burdur, Turkey b Deprtment of Mthemtics, Fculty of Arts nd Sciences, Pmukkle University, Denizli, Turkey Communicted by A Atngn Abstrct In this study, colloction method bsed on the generlized Bernstein polynomils is derivted for solving nonliner Fredholm-Volterr integrl equtions (FVIEs) in the most generl form vi the qusilineriztion technique Moreover, qudrtic convergence nd error estimte of the proposed method is nlyzed Some exmples re lso presented to show the ccurcy nd pplicbility of the method c 2017 All rights reserved Keywords: Bernstein polynomil pproch, nonliner integrl equtions, qusilineriztion technique, colloction method 2010 MSC: 45A05, 45G10, 65L60 1 Introduction Fredholm nd Volterr integrl equtions re well-known tht liner nd nonliner integrl equtions rise in mny scientific fields such s the popultion dynmics, spred of epidemics nd semi-conductor devices The principl investigtors of the theory of these equtions re Vit Volterr ( ) nd Ivr Fredholm ( ) Qusilineriztion pioneered by Bellmn nd Klb [6] is n effective technique tht solves the nonliner equtions itertively by sequence of liner equtions Since this method is generliztion of the Newton-Rphson method, it cn be introduced by using the Tylor series expnsion The min dvntge of this method is tht it converges qudrticlly to the solution of the originl eqution Some systemtic studies of this property hve been given in [1, 5 7, 11, 13 15, 18] This method is lso powerful tool to obtin the pproximte solution of nonliner problems included such s differentil equtions [1, 3, 5, 9, 14, 17], functionl differentil equtions [2, 7], integrl equtions [13, 15] nd integro-differentil equtions [4, 18] The Bernstein polynomils nd their bsis forms defined on the intervl [0, 1] cn be generlized to the intervl [, b] by considering trnsformtion t = x b s follows Corresponding uthor Emil ddresses: nisler@mehmetkifedutr (Neşe İşler Acr), kyuz@puedutr (Ayşegül Dşcıoğlu) doi: /jns Received

2 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), Definition 11 The generlized Bernstein bsis polynomils cn be defined on [, b] by ( ) 1 n p i,n (x) = (b ) n (x ) i (b x) n i, i = 0, 1,, n i Definition 12 Let y be continuous function on the intervl [, b] The generlized Bernstein polynomils of n-th-degree tht liner combintion of the generlized Bernstein bsis polynomils p i,n (x) re defined by B n (y; x) = n i=0 ( y + ) (b ) i p i,n (x) n Besides, the generlized Bernstein polynomils nd their bsis forms hve some useful properties such s the positivity, continuity, recursion s reltion, symmetry, unity prtition of the bsis set over the intervl [, b], uniform pproximtion, differentibility nd integrbility These properties cn be shown esily by following the some studies given with the properties of the Bernstein polynomils nd their bsis forms [8, 10, 12] Theorem 13 If y (x) is continuous function on the intervl [, b], then lim n B n (y; x) = y (x) converges uniformly Proof The bove theorem is proved with the similr wys given for the proof of theorem presented by Phillips [16] Definition 14 Nonliner Fredholm-Volterr integrl eqution is defined s follows: g (x, y(x)) = λ 1 f(x, t, y(t))dt + λ 2 v (x, t, y(t)) dt, x, t [, b], (11) where λ 1 nd λ 2 re constnts, y(x) is n unknown function, g : [, b] R, the kernels f : [, b] [, b] R nd v : [, b] [, b] R re continuous functions stisfying Lipschitz condition with respect to the lst vribles: such tht L g, L f, L v 0 for x, t [, b] nd w, z R g (x, z) g(x, w) L g z w, f (x, t, z) f(x, t, w) L f z w, v (x, t, z) v(x, t, w) L v z w, The reminder of this pper follows: In Section 2, colloction method is developed itertively to get the numericl solution of the nonliner integrl equtions by mens of the generlized Bernstein polynomils nd qusilineriztion technique In Section 3, the uniqueness of the nonliner Fredholm- Volterr integrl equtions is nlyzed, nd the error estimtes of the proposed method re given In Section 4, some nonliner exmples re considered for showing the pplicbility nd efficiency of the method Numericl results re lso compred with different methods Finlly, some inferences of the study re mentioned in the lst section 2 Method of solution In this pper, the purpose is to pproximte the solution of nonliner FVIE (11) by using the qusilineriztion technique itertively with the generlized Bernstein polynomils: y (x) = B n (y; x) = n i=0 ( y + ) (b )i p i,n (x) (21) n

3 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), Theorem 21 Let x s [, b] be colloction point, y C [, b], g, f nd v hve Tylor series expnsion with respect to y Then, nonliner FVIE (11) hs the following itertion mtrix form: [G r P λ 1 F r λ 2 V r ] Y r+1 = H r, r = 0, 1, (22) Here mtrices G r = dig ( [g (x s, y r (x )] s ))], P = [p i,n (x s )], F r = [F r,s,i ] nd V r = [V r,s,i ] re (n + 1) (n + 1) mtrices, Y r+1 = [y r+1 + (b )i n nd H r = [h r (x s )] re (n + 1) 1 mtrices for i, s = 0, 1,, n Besides, elements of these mtrices re defined s h r (x) = g y (x, y r (x)) y r (x) g (x, y r (x)) + λ 1 [f(x, t, y r (t)) f y (x, t, y r (t)) y r (t)] dt + λ 2 [v (x, t, y r (t)) v y (x, t, y r (t)) y r (t)] dt, F r,s,i = f (x, t, y r (t)) p i,n (t)dt, V r,s,i = v (x, t, y r (t)) p i,n (t)dt Proof Since functions g, f nd v re ble to be expnded by Tylor series with respect to y, nonliner FVIE cn be expressed s sequence of liner integrl equtions by using qusilineriztion technique for r = 0, 1, s follows g y (x, y r (x)) y r+1 (x) = g y (x, y r (x)) y r (x) g (x, y r (x)) + λ 1 [f (x, t, y r (t)) + f y (x, t, y r (t)) (y r+1 (t) y r (t))] dt + λ 2 [v (x, t, y r (t)) + v y (x, t, y r (t)) (y r+1 (t) y r (t))] dt Here g y, f y nd v y re prtil derivtives of g, f nd v with respect to function y, y 0 (x) is n initil pproximtion function, y r (x) is lwys known function nd y r+1 (x) is obtined by the former one By considering function h r (x) given in the expression of Theorem 21, the eqution (23) cn be rerrnged shortly g y (x, y r (x)) y r+1 (x) = h r (x) + λ 1 f y (x, t, y r (t)) y r+1 (t) dt + λ 2 v y (x, t, y r (t)) y r+1 (t) dt (24) Since (11) hs the generlized Bernstein polynomil solution nd y(x s ) = B n (y; x s ) (s = 0, 1,, n) from the colloction method, the expression (21) cn be written s (23) y r+1 (x s ) = P(x s )Y r+1, r = 0, 1, (25) Substituting the colloction points nd reltion (25) into (24), we obtin the liner lgebric system g y (x s, y r (x s )) P(x s )Y r+1 λ 1 F r (x s )Y r+1 λ 2 V r (x s ) Y r+1 = h r (x s ) (26) Here F r (x s ) nd V r (x s ) re denoted by F r (x s ) = f y (x, t, y r (t)) P (t) dt = [ ] F r,s,0 F r,s,1 F r,s,n,

4 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), V r (x s ) = v y (x, t, y r (t)) P (t) dt = [ ] V r,s,0 V r,s,1 V r,s,n F r = Considering the mtrices g y (x 0, y r (x 0 )) g y (x 1, y r (x 1 )) 0 G r =, 0 0 g y (x n, y r (x n )) F r (x 0 ) V r (x 0 ) h r (x 0 ) F r (x 1 ) V r (x 1 ) h r (x 1 ) F r (x n ), V r= V r (x n ), H r = h r (x n ) for s = 0, 1,, n, the eqution (26) cn be written s mtrix form (22) This completes the proof Let the following steps be given to solve the nonliner FVIE (11) Step 1 The eqution (22) cn lso be written in the compct form W r Y r+1 = H r or [W r ; H r ], r = 0, 1,, so tht W r = G r P λ 1 F r λ 2 V r This eqution corresponds to liner lgebric eqution system with y r+1 unknown coefficients for itertions r Step 2 An initil pproximtion function y 0 (x) should be selected for computing the W r nd H r This function cn be determined s source or constnt function roughly Step 3 Since nonliner integrl equtions reduce to sequence of liner integrl equtions vi the qusilineriztion technique, we do not need to use ny solution techniques of nonliner integrl equtions As liner eqution system, if rnk (W r ) = rnk (W r ; H r ) = n + 1, then solution of this system is uniquely determined The system cn lso be solved by the Guss elimintion, generlized inverse, LU nd QR fctoriztion methods 3 Convergence nd error nlysis Definition 31 Error is denoted by e n (x) = y(x) y r+1 (x) such tht y(x) is n exct solution nd y r+1 (x) = B n (y r+1 ; x) is generlized Bernstein pproximte solution Then bsolute nd men errors cn be numericlly computed t the colloction points respectively by e n (x s ) = y (x s ) y r+1 (x s ), e men = 1 n+1, n e n (x s ) Definition 32 Let E r (x s ) = y r+1 (x s ) y r (x s ) be error on the colloction points x s [, b] for the r-th itertion function Then bsolute nd mximum errors cn be expressed s follows: s=0 E r (x s ) = y r+1 (x s ) y r (x s ), nd E mx = mx x s [,b] E r (x s ) ( Theorem 33 (Uniqueness Theorem) Let g C [, b] nd f, v C [, b] 2) stisfy the Lipschitz condition with respect to the lst vribles nd T : C [, b] C [, b] be mpping where Ty (x) = g (x, y (x)) λ 1 f(x, t, y(t))dt λ 2 v (x, t, y(t)) dt Then (11) hs unique solution whenever 0 < α < 1, α = L g + (b ) [ λ 1 L f + λ 2 L v ]

5 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), Proof Since g, f nd v stisfy the Lipschitz condition with respect to the lst vribles y nd y, we hve the following inequlity for y, y C [, b]: Ty (x) Ty (x) g (x, y (x)) g (x, y (x)) + λ 1 f(x, t, y(t)) f(x, t, y (t)) dt + λ 2 v (x, t, y(t)) v (x, t, y (t)) dt L g y (x) y (x) + λ 1 L f y (t) y (t) dt + λ 2 L v y (t) y (t) dt Considering the definition of mximum norm for one vrible function, the inequlity becomes Ty Ty L g y y + λ 1 L f y y dt + λ 2 L v y y dt L g y y + λ 1 L f (b ) y y + λ 2 L v (b ) y y [L g + (b ) ( λ 1 L f + λ 2 L v )] y y for ll x, t [, b] Denoting α = L g + (b ) [ λ 1 L f + λ 2 L v ], the inequlity is Ty Ty α y y Under the condition 0 < α < 1, by Bnch fixed-point theorem, (11) hs unique solution nd this completes the proof Theorem 34 Suppose tht y 0 (x) nd y r (x) re the first nd r-th itertion functions, g C 2 [, b], f nd v C 2 ([, b] 2) Then the following qudrtic convergence nd error estimte hold E r σ E r 1 2 nd E r (σ E 1 ) 2r σ such tht σ = M 2+(b )[ λ 1 L 1 + λ 2 L 2 ] 2[M 1 (b )( λ 1 K 1 + λ 2 K 2 )] positive constnt If the quntity E 1 < 1, then lim r E r = 0 Proof Applying the qusilineriztion technique to the following equlity we hve g (x, y r+1 ) g (x, y r ) = λ 1 {f (x, t, y r+1 ) f (x, t, y r )} dt + λ 2 {v (x, t, y r+1 ) v (x, t, y r )} dt, g y (x, y r ) (y r+1 y r ) = {g (x, y r ) g (x, y r 1 ) g y (x, y r 1 ) (y r y r 1 )} + λ 1 {f (x, t, y r ) f (x, t, y r 1 ) f y (x, t, y r 1 ) (y r y r 1 )} dt + λ 2 {v (x, t, y r ) v (x, t, y r 1 ) v y (x, t, y r 1 ) (y r y r 1 )} dt + λ 1 f y (x, t, y r ) (y r+1 y r ) dt + +λ 2 v y (x, t, y r ) (y r+1 y r ) dt (31) Since g C 2 [, b], f nd v C 2 ([, b] 2), the following equtions cn be written from men vlue theorem

6 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), g (x, y r ) g (x, y r 1 ) g y (x, y r 1 ) (y r y r 1 ) = 1 2 (y r y r 1 ) 2 g yy (x, α), f (x, t, y r ) f (x, t, y r 1 ) f y (x, t, y r 1 ) (y r y r 1 ) = 1 2 (y r y r 1 ) 2 f yy (x, t, β), v (x, t, y r ) v (x, t, y r 1 ) v y (x, t, y r 1 ) (y r y r 1 ) = 1 2 (y r y r 1 ) 2 v yy (x, t, γ), where y r 1 < α, β, γ < y r Substituting these equtions into (31), the eqution becomes g y (x, y r ) (y r+1 y r ) = 1 2 (y r y r 1 ) 2 g yy (x, α) + λ 1 + λ (y r y r 1 ) 2 v yy (x, t, γ) dt + λ 1 + λ 2 v y (x, t, y r ) (y r+1 y r ) dt 1 2 (y r y r 1 ) 2 f yy (x, t, β) dt f y (x, t, y r ) (y r+1 y r ) dt By definition of the mximum norm for one nd two vribles functions, the following inequlity is obtined g y y r+1 y r 1 2 y r y r 1 2 g yy λ 1 y r y r 1 2 f yy (b ) λ 2 y r y r 1 2 v yy (b ) + λ 1 f y y r+1 y r (b ) + λ 2 v y y r+1 y r (b ) Denoting nd the bove inequlity becomes g y = mx g y (x, y r (x)) = M 1, x [,b] g yy = mx g yy (x, α (x)) = M 2, x [,b] f y = v y = f yy = v yy = mx f y (x, t, y r ) = K 1, x,t [,b] mx v y (x, t, y r ) = K 2, x,t [,b] mx f yy (x, t, z) = L 1, x,t [,b] mx v yy (x, t, ) = L 2, x,t [,b] mx (x ) = b, x [,b] M 1 y r+1 y r M 2 2 y r y r L 1 2 λ 1 (b ) y r y r L 2 2 λ 2 (b ) y r y r λ 1 K 1 (b ) y r+1 y r + λ 2 K 2 (b ) y r+1 y r This inequlity cn be rerrnged s { } M2 + λ 1 L 1 (b ) + λ 2 L 2 (b ) E r E r [M 1 λ 1 K 1 (b ) λ 2 K 2 (b )],

7 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), E r σ E r 1 2, M 2+ λ 1 L 1 (b )+ λ 2 L 2 (b ) 2[M 1 λ 1 K 1 (b ) λ 2 K 2 (b )] where σ = positive constnt This eqution shows tht convergence is qudrtic if there is convergence t ll We rewrite this eqution s By the use of substitutions, the new inequlity is E r σ E r 1 2, E r 1 σ E r 2 2, E 2 σ E 1 2 ) 2 [ ] E r σ E r 1 2 (σ σ E r 2 2 σ σ 2r 2 E r 3 2r = [σ E 1 ] 2r /σ If the quntity σ E 1 < 1, then lim r E r = 0 It completes the proof 4 Numericl results Two exmples including nonliner integrl equtions re considered for nlyzing the pplicbility nd ccurcy of the proposed Bernstein colloction method The numericl results obtined on the colloction points + (b )s n, s = 0, 1,, n re presented nd compred with the other methods Exmple 41 Consider the following nonliner Volterr integrl eqution: y(x) = 2 e x + e x t y 2 (t)dt, 0 x 1 0 Exct solution of the bove eqution is y(x) = 1 Let y 0 (x) = 2 e x be initil pproximtion function In Figure 1 nd Tble 1, the bsolute errors E r (x) by considering the Bernstein colloction method obtined t the colloction points s n, s = 0, 1,, n re given for n = 3 nd itertions r = 1, 2, 3, 4 Besides, in Figure 2, the bsolute errors e n (x) of the proposed method re presented for different vlues n = 2, 3, 4 nd 5-th itertion Figure 1: The numericl results of E r (x) errors for n = 3

8 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), Tble 1: The numericl results of E r (x) errors for n = 3 n = 3 r = 1 r = 2 r = 3 r = Figure 2: The numericl results of e n (x) errors for 5-th itertion Figure 3: The comprison of numericl results of e n (x) errors for n = 4 nd 5-th itertion Tble 2: The comprison of the e n (x) errors of Exmple 41 Lgrnge Colloction Method [13] Presented Method x n = 2 n = 3 n = 4 n = 4 r = 2 r = 5 r = 2 r = 5 r = 2 r = 5 r = 2 r = The bsolute error results of the proposed method re compred with the results given by Mleknejd nd Njfi [13] in Tble 2 nd Figure 3 The numericl results of the Bernstein colloction method re obtined t the colloction points s n, s = 0, 1,, n by considering the first itertion function y 0(x) = 2 e x Besides, Mleknejd nd Njfi hve presented colloction method bsed on qusilineriztion technique nd Lgrnge bsis polynomils for the first itertion function y 0 (x) = 1 e x According to Tble 2, the presented method hs better numericl solutions thn the other method for r = 2 nd r = 5 The

9 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), best results re lso obtined for n = 2 by the presented method, therefore, it is not necessry to enlrge n since the rounding error increses while n increses Exmple 42 Consider the following nonliner Fredholm-Volterr integrl eqution: y(x) = 2x+7 7x (x + t)y 2 (t)dt + (x t)y(t)dt, 1 x 1 1 Exct solution of the bove eqution is y(x) = 2x Let y 0 (x) = 0 be the first itertion function 1 Tble 3: The numericl results of e men errors for Exmple 42 n r = 2 r = 3 r = 4 r = 5 r = Men errors of the presented method with incresing n re given t the colloction points x s = 1 + 2s n, s = 0, 1,, n in Tble 3 We cn sy tht the numericl results of proposed method converge more rpidly for incresing itertion r 5 Conclusions In this study, colloction method bsed on the generlized Bernstein polynomils hs been developed for the numericl solution of nonliner Fredholm-Volterr integrl equtions itertively by using the qusilineriztion technique The qudrtic convergence nd error bound of the Bernstein colloction method hve been nlyzed Exmples 41 nd 42 support tht the proposed method derived itertively converges more rpidly for incresing itertions r The dvntge of the proposed method is not complicted nd esy pplicbility to the nonliner equtions, becuse the nonliner equtions re reduced to liner equtions vi the qusilineriztion So the method does not need ny solution methods of nonliner equtions Acknowledgment This work is supported by the Scientific Reserch Project Coordintion Unit of Pmukkle University with number 2017KRM Some results of the present work hve been reported t the Interntionl Conference on the Theory, Methods nd Applictions of Nonliner Equtions, held t the University of Texs A&M, Kingsville in 2012 References [1] R P Agrwl, Y M Chow, Itertive methods for fourth order boundry vlue problem, J Comput Appl Mth, 10 (1984), [2] B Ahmd, R Ali Khn, S Sivsundrm, Generlized qusilineriztion method for nonliner functionl differentil equtions, J Appl Mth Stochstic Anl, 16 (2003), [3] A Akyuz Dscioglu, N Isler, Bernstein colloction method for solving nonliner differentil equtions, Mth Comput Appl, 18 (2013), [4] A Akyüz-Dşcıoğlu, N İşler Acr, C Güler, Bernstein colloction method for solving nonliner Fredholm-Volterr integrodifferentil equtions in the most generl form, J Appl Mth, 2014 (2014), 8 pges 1 [5] A C Bird, Jr, Modified qusilineriztion technique for the solution of boundry-vlue problems for ordinry differentil equtions, J Optimiztion Theory Appl, 3 (1969),

10 N İşler Acr, A Dşcıoğlu, J Nonliner Sci Appl, 10 (2017), [6] R E Bellmn, R E Klb, Qusilineriztion nd nonliner boundry-vlue problems, Modern Anlytic nd Computionl Methods in Science nd Mthemtics, Americn Elsevier Publishing Co, Inc, New York, (1965) 1 [7] Z Drici, F A McRe, J V Devi, Qusilineriztion for functionl differentil equtions with retrdtion nd nticiption, Nonliner Anl, 70 (2009), [8] R T Frouki, V T Rjn, Algorithms for polynomils in Bernstein form, Comput Aided Geom Design, 5 (1988), [9] N İşler Acr, A Akyüz-Dşcıoğlu, A colloction method for Lne-Emden Type equtions in terms of generlized Bernstein polynomils, Pioneer J Mth Mth Sci, 12 (2014), [10] K I Joy, Bernstein polynomils, On-Line Geometric Modeling Notes, Visuliztion nd Grphics Reserch Group, Deprtment of Computer Science, University of Cliforni, Dvis, (2000), [11] E S Lee, Qusilineriztion nd invrint imbedding, First edition, Acdemic Press, New York, (1968) 1 [12] G G Lorentz, Bernstein polynomils, Second edition, Chelse Publishing Co, New York, (1986) 1 [13] K Mleknejd, E Njfi, Numericl solution of nonliner Volterr integrl equtions using the ide of qusilineriztion, Commun Nonliner Sci Numer Simul, 16 (2011), , 2, 41 [14] V B Mndelzweig, F Tbkin, Qusilineriztion pproch to nonliner problems in physics with ppliction to nonliner ODEs, Comput Phys Comm, 141 (2001), [15] S G Pndit, Qudrticlly converging itertive schemes for nonliner Volterr integrl equtions nd n ppliction, J Appl Mth Stochstic Anl, 10 (1997), [16] M G Phillips, Interpoltion nd pproximtion by polynomils, CMS Books in Mthemtics/Ouvrges de Mthmtiques de l SMC, Springer-Verlg, New York, (2003) 1 [17] J I Rmos, Piecewise qusilineriztion techniques for singulr boundry-vlue problems, Comput Phys Comm, 158 (2004), [18] P-G Wng, Y-H Wu, B Wiwtnpphee, An extension of method of qusilineriztion for integro-differentil equtions, Int J Pure Appl Mth, 54 (2009),