Note on Using Radial Basis Functions Method for Solving Nonlinear Integral Equations

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1 Communictions in umericl Anlysis 016 o. (016) Avilble online t Volume 016, Issue, Yer 016 Article ID cn-0057, 11 Pges doi: /016/cn-0057 Reserch Article ote on Using Rdil Bsis Functions Method for Solving onliner Integrl Equtions Ahmd Golbbi 1, Omid ikn 1, Jber Rmezni Tousi (1) Deprtment of Mthemtics, Irn University of Science nd Technology,P.O.Box, ,rmk,Tehrn,Irn () Deprtment of Mthemtics, University of Mzndrn, P.O. Box , Bbolsr, Irn Copyright 016 c Ahmd Golbbi, Omid ikn nd Jber Rmezni Tousi. 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 note, the solution of nonliner integrl equtions ws discussed using rdil bsis functions (RBFs) method. This method will represent the solution of nonliner integrl eqution by interpolting the RBFs bsed on Legendre- Guss-Lobtto (LGL) nodes nd weights. Zeros of the shifted Legendre polynomils re used s the colloction points. The method is used to some exmples to illustrte the ccurcy nd the implementtion of the method. Keywords: Fredholm nd Volterr integrl equtions, RBFs, Legendre-Guss-Lobtto nodes,colloction method. 1 Introduction The theory nd ppliction of integrl eqution is n importnt subject within pplied mthemtics. Integrl equtions re used s mthemticl models for mny nd vried physicl situtions, nd integrl equtions lso occur s reformultions of other mthemticl problems such s prtil differentil equtions nd ordinry differentil equtions. For this result, we need to solve this kind of equtions. There re few numericl nd nlyticl methods to estimte the solution of the qudrtic integrl equtions such s Picrd nd Adomin decomposition method (ADM) [1], nd some other methods [, 3, 4]. The RBF method in solving integrl equtions ws initilly proposed in 006 [5, 6, 7, 8]. The relted reserch ttrcted lot of ttention recently. In this pper we propose point interpoltion mesh less method for solving the nonliner Ferdholm nd Volterr equtions. The method is bsed upon rdil bsis functions, using zeros of the shifted Legendre polynomil s the colloction points. Mny different bsis functions hve been used to estimte the solution of liner nd nonliner integrl equtions, such s orthogonl bses, wvelets nd hybrid [9]. RBFs re powerful tools in multi-vrible pproximtion [10] nd re incresingly being used in the numericl solution of prtil differentil equtions [11]. The min dvntges of the RBFs for interpolting multidimensionl scttered dt re highlighted in [1]. In recent decdes,meshless methods hve been proved to tret scientific nd engineering problems efficiently. Meshless method bsed on the colloction method hs been dominted nd very efficient. Over the lst severl decdes RBFs hve been found to be widely successful for the interpoltion of scttered dt. RBF methods re not tied to grid nd in turn belong to ctegory of methods clled meshless methods. They pply only cloud of points without ny informtion bout nodl connections. It is (conditionlly) positive definite, rottionlly nd trnsltionlly invrint. The RBF pproximtion is n extremely powerful tool for representing smooth functions in non-trivil geometries, since the method is meshfree nd cn be spectrlly ccurte[13]. Corresponding uthor. Emil ddress: omid nikn@lumni.iust.c.ir; Tel:

2 Communictions in umericl Anlysis 016 o. (016) A nonliner Ferdholm integrl eqution is defined s nd nonliner Volterr integrl eqution is defined s b y(x) = f (x) + η k(x,t)f(x,y(t))dt, x b (1.1) x y(x) = f (x) + η k(x,t)f(x,y(t))dt, x b (1.) where y(x) is n unknown function, η is constnt nd f (x) : [,b] R nd the kernel k(x,t) : S R(with S = {(x,t) : x t b)}) given functions ssumed to hve nth derivtives [14]. Though the different choices of the prmeters led to vrious problems, the method cn fford to pproximte the solution of them. The lyout of the pper is s follows. In Section, the rdil bsis functions re introduced. Section 3, reviews the Legendre-Guss-Lobtto integrtion process. Section 4, s the min prt, presents the solution nonliner integrl equtions by direct nd indirect process of RBFs. umericl illustrtive exmples re included in Section 5. A conclusion is drwn in the Section 6. Review on the principles of RBF method.1 RBF methodology The history of RBF pproximtions goes bck to 1968, when multiqudric RBFs were first used by Hrdy to represent topogrphicl surfces given sets of sprse scttered mesurements [15]. This method ws populrized in 198 by Richrd Frnke with his report on 3 of the most commonly used interpoltion methods [16]. Richrd Frnke, in 198 compred scttered dt interpoltion methods, nd concluded MQs nd TPs were best. Frnke conjectured interpoltion mtrix for MQs is invertible [16]. He lso conjectured the unconditionl non-singulrity of the interpoltion mtrix ssocited with the multiqudric rdil function, but it ws not until few yers lter tht Micchelli [17] ws ble to prove it s mentioned bove. The min feture of the MQ method is tht the interpolnt is liner combintion of trnsltions of bsis function which only depends on the Eucliden distnce from its center. This bsis function is therefore rdilly symmetric with respect to its center. Tht is how its nme rdil bsis function comes bout. The MQ method ws generlized to other rdil functions, such s the thin plte spline, the Gussin, the cubic, etc. In the 1990s reserchers becme to py ttention to the RBF method gin when Kns [18] introduced wy to use it for solving prbolic, elliptic nd (viscously dmped) hyperbolic PDEs. Results [19] on the spectrl convergence rte of MQ interpoltion followed from Mdych nd elson in 199. Definition.1. Let R + = {x R x 0} the non-negtive hlf-line nd let ϕ : R + R be continuous function with ϕ(0) 0. A rdil bsis functions on R d is function of the form ϕ( x x i ) where x,x i R d nd r = x x i, denotes the Eucliden distnce between x nd x, i s. If one chooses points {x i} i=1 in Rd then by custom s(x) = λ i ϕ( x x i ); λ i R is clled rdil bsis functions s well [0].(See Tble 1) i=1

3 Communictions in umericl Anlysis 016 o. (016) Tble 1: Deffinition of some types of RBFs me of RBF (Abbrevition) ϕ(r),r 0 Smoothness Gussin (GA) e cr Infinite Generlized Multiqudric (GMQ) (c + r ) β Infinite Inverse Multiqudric (IMQ) 1 c +r Infinite Inverse Qudrtic (IQ) (c + r ) 1 Infinite Multiqudric (MQ) c + r Infinite Hyperbolic secnt (sech) sech(cr) Infinite Cubic (CU) r 3 Piecewise Liner (LI) r Piecewise Monomil (M) r k 1 Piecewise Thin Plte Spline (TPS) r log(r) Piecewise Prmeter c is prmeter for controlling the shpe of functions which effects on the rte of convergency. The stndrd rdil bsis functions re divided into two mjor clsses [1]: Clss 1. Infinitely smooth RBFs [1]: In this clss the bsis function ϕ(r) hevily depends on the free shpe prmeter c e.g.hrdy multiqudric (MQ),Gussin (GA), inverse multiqudric (IMQ) nd inverse qudric (IQ)(See Tble 1)). Clss. Piecewise smooth RBFs [1]: The key dvntge is tht the bsis functions of this ctegory re shpe prmeter free, including Liner r, Cubic r 3, Thin plte spline (r n logr,n = 1,,3,...,) ect. The choice of shpe prmeter is n importnt tsk in pproximting functions by RBFs nd reserchers lwys hve concerned bout selecting good shpe prmeter. Optiml shpe prmeter vlues re found experimentlly nd these vlues re written for exct text problems. Theoreticlly, RBF methods re most ccurte when the shpe prmeter is smll. Mny uthors hve investigted the shpe prmeter.the implementtion of RBF methods involves solving liner system tht is extremely ill-conditioned when the prmeters of the method re such tht the best ccurcy is theoreticlly relized. Thus, in pplictions, RBF methods re not ble to produce s ccurte of results s they re theoreticlly cpble of.. Bsic knowledge bout RBFs pproximtion For scttered dt (x i,u(x i )) R d+1, the pproximtion s(x) for rel function u(x) cn be constructed by liner combintions of trnsltes of one function ϕ(. ) of one rel vrible which is centered t {x j } Rd, u(x) s(x) = λ j ϕ( x x j ), (.3) The most ttrctive feture of the RBF methods is tht the loction of the centers cn be chosen rbitrrily in the domin of interest. To determine the unknown coefficients λ j, j = 1,,...,M., we cn impose the interpoltion conditions on s(x) i.e. s(x i ) = u i ; i = 1,,..,M. This gives the liner system, This is summrized in system of equtions for the unknown coefficients λ j, λ j ϕ( x i x j ) = u(x i ), (.4) Aλ = u, (.5) where A = ϕ( x i x j ) is mtrix, λ = [λ j ] nd u = u(x i ) re 1 mtrices. The mtrix A is clled the interpoltion mtrix or the system mtrix. ote tht ϕ i (x j ) = ϕ( x i x j ) therefore we hve ϕ i (x j ) = ϕ j (x i )

4 Communictions in umericl Anlysis 016 o. (016) consequently A = A T. The interpolnt of u(x) is unique if nd only if the mtrix A is nonsingulr. It hs been discussed bout sufficient conditions for ϕ(r) to gurntee nonsingulrity of the A mtrix i.e. there is unique interpolnt of the form Eq.(.3), no mtter how the distinct dt points re scttered in ny number of spce dimensions. Micchelli [17] nd Powell [] hve shown the existence of the interpoltion. In the cses of inverse qudrtic (IQ), inverse multiqudric (IMQ), hyperbolic secnt (sech) nd Gussin (GA) the mtrix A is positive definite nd, for multiqudric (MQ), it hs one positive eigenvlue nd the remining ones re ll negtive.in the piecewise smooth cses, slight vrition of the form of Eq.(.3) will gin ensure nonsingulrity [3, 4, 5]. Theorem.1. Assume {x j } re nodes in Ω which is convex, let h = mx min x x i, when ϕ(ν) < c(1 + x Ω 1 i ν ) (l+d) for ny y(x) stisfies (ŷ(ν)/ ϕ(ν))dν < we hve, y (α) y(α) ch 1 α where ϕ(x) is RBFs nd the constnt c depends on the RBFs, d is spce dimension, l nd α re nonnegtive integer. It cn be seen tht not only RBFs itself but lso its ny order derivtive hs good convergence. Proof. See[6, 7]. 3 Legendre-Guss-Lobtto nodes nd weights Let H [ 1,1] denote the spce of lgebric polynomils of degree < P i,p j >= j+1 δ i j. Here, <... > represents the usul L [ 1,1] inner product nd re{p i } i 0 the well-known Legendre polynomils of order i which re orthogonl with respect to the weight function w(x) = 1 on the intervl [ 1,1], nd stisfy the following formule: P 0 (x) = 1, P 1 (x) = x, P i+1 (x) = i+1 i+1 xp i(x) i i+1 P i 1(x), i = 1,,3,... For convenience the solution we use RBFs with colloction nodes x i, i 1 which re the zeros of the Legendre polynomi P (x),where P (x) is derivtive P(x) of on the intervl [ 1,1]. Also we pproximte the integrl of f (x) on [ 1,1] s 1 1 f (x)dx = i=1 w i f (x i ), (3.6) where x 1 = 1 < x <... < x 1 < x = 1 re Legendre-Guss-Lobtto nodes nd w i Legendre-Guss-Lobtto weights given in [8] w i =, i = 1,..., (3.7) ( + 1)[P (x i )] thus,for rbitrry intervl [,b] b f (x)dx = b i=1 w i f (z i ), (3.8) where z = b x + b+. It is well known tht the integrtion in Eq.(3.6) is exct whenever f (x) is polynomil of degree Description of Method In this section we pply the results of the previous section to solve two the nonliner integrl equtions. The first one is fredholm integrl eqution which is solved by the RBF colloction method nd the second one is volterr integrl eqution which is solved by using RBFs. Also we pproximte its corresponding integrl by the Legendre-Guss-Lobtto points nd weights.

5 Communictions in umericl Anlysis 016 o. (016) Fredholm integrl eqution We consider the following integrl eqution of Fredholm type b y(x) = f (x) + η k(x, t)f(x, y(t))dt. (4.9) Suppose tht the one dimensionl pproximtion y(x) t n rbitrry point x by function ϕ(x), in the following form: y(x) = λ j ϕ( x x j ), (4.10) where x j, s re known s centers. The unknown coefficients λ j re to be determined by the colloction method. Then, from substituting Eq.(4.10) into Eq.(4.9) we hve, b λ i ϕ( x x j ) = f (x) + η We now collocte Eq.(4.11) t points {x i } i=1 s b λ j ϕ( x i x j ) = f (x i ) + η k(x i,t)f(x i, k(x, t)f(x, In bove eqution,we let t = b x + b+. It reduces Eq.(4.1) to the following eqution: λ j ϕ( x i x j ) = f (x i )+ η b 1 k(x i, b 1 x i + b + )F(x i, λ j ϕ( t x j ))dt. (4.11) λ j ϕ( t x j ))dt, i = 1,...,. (4.1) λ j ϕ( ( b x i + b + ) x j ))dx, (4.13) for i = 1,...,. By using the Legendre-Guss-Lobtto integrtion formul described in Eq.(3.6),we cn pproximte the integrl in Eq.(4.13) nd hence the bove eqution cn be written s follows: λ j ϕ( x i x j ) = f (x i )+ η b w j k(x i, b x i + b + )F(x i, λ j ϕ( ( b x i + b + ) x j )) (4.14) where w i,x i, i = 1,..., re weights nd nodes of the integrtion rule respectively. Eq.(4.14) genertes n set of equtions which cn be solved by mthemticl softwre for the unknowns vector λ. 4. Volterr integrl eqution The nonliner Volterr integrl eqution tke the following form: x y(x) = f (x) + η k(x, t)f(x, y(t))dt, (4.15) Let s pproximte the function y(x) in terms of rdil bsis functions, ϕ(x), s follows y(x) = Then, from substituting Eq.(4.16) into Eq.(4.15) we hve, x λ j ϕ( x x j ) = f (x) + η λ i ϕ( x x j ). (4.16) k(x, t)f(x, λ j ϕ( t x j ))dt. (4.17)

6 Communictions in umericl Anlysis 016 o. (016) Substituting the colloction points {x i } i=1 into Eq.(4.17), we obtin: xi λ j ϕ( x i x j ) = f (x j ) + η k(x i,t)f(x i, In bove eqution,we let t = x i x + x i+. It reduces Eq.(4.18) to the following eqution: λ j ϕ( x i x j ) = f (x i )+ η x i 1 k(x i, x i 1 x i + x i + )F(x i, λ j ϕ( t x j ))dt i = 1,...,. (4.18) λ j ϕ( ( x i x i + x i + ) x j ))dx, (4.19) for i = 1,...,. ow, by pplying Legendre-Guss-Lobtto integrtion formul demonstrted in Eq.(3.6), we pproximte the integrl of Eq.(4.19) s follows λ j ϕ( x i x j ) = f (x i )+ η x i w j k(x i, x i x i + x i + )F(x i, λ j ϕ( ( x i x i + x i + ) x j )) (4.0) where w i, x i, i = 1,..., re weights nd nodes of the integrtion rule respectively. Agin, we hve nonliner system of equtions tht cn be solved by mthemticl softwre for the unknowns vector λ. 5 umericl illustrtion In this section,four exmples re considered. The results of numericl experiments re compred with the exct solution in illustrtive exmples to confirm the ccurcy nd efficiency of the proposed method. We point out tht the corresponding numericl solutions re obtined using Softwre Mtlb. To study the convergence behvior of the RBFs method, we pplied the following lws: 1. The L error norm of the solution which is defined by ( L = y exct (x) y implicit (x) = 1 (y exct (x j ) y implicit (x j )) ).. The L error norm of the solution which is defined by L = y exct (x) y implicit (x) = mx j y exct (x j ) y implicit (x j ). 3. The root men squre (RMS) is defined by ( 1 RMS = 1 1 (y exct (x j ) y implicit (x j )) ). where x j, j = 1,..., re Legendre-Guss-Lobtto nodes. Exmple 5.1. As the first exmple,we consider the nonliner Fredholm integrl eqution with the exct solution y(x) = e x 1 y(x) + e (x t) [y(t)] 3 dt = e x, 0 where 0 x 1 [9].

7 Communictions in umericl Anlysis 016 o. (016) Tble : Errors in the solution of Exmple 5.1 with c = Methods L L RMS Our method e e e e e e e e e 014 Method o f [30] e e e E e e 010 Method o f [31] e e e 5 The error, L -error,l -error nd Root-Men-Squre(RMS)-error norms for = 5,10,15 with MQ-RBF re reported in Tble.As this tble illustrtes,numericl results show simplicity nd very good ccurcy of the method. The errors decreses by incresing the number of colloction points for MQ-RBFs. Accurcy of the present pproximtion is exmined in the L -error nd L -error,rms-error norms. The results of RMS-error re comprtively better thn the L -error nd L -error norms. Also, in comprison with the results of [30] nd [31], which used Sinc colloction method, the errors in ech row hve been decresed, which is good fctor in the RBFs method. Exmple 5.. ext, we consider the following nonliner Fredholm integrl eqution with the exct solution is y(x) = x y(x) = ( )x x + x t y(t)dt, (5.1) 0 where 0 x 1[31]. The error, L -error,l -error nd Root-Men-Squre(RMS)-error norms for = 5,10,15 with MQ-RBF re illustrted in Tble 3. From Tble 3, it cn be observed tht the ccurcy increses with the increse of number of colloction points. Also, we hve compred it with the Sinc-colloction method [31] nd Hr wvelet method [3]. These results verify tht our method is considerble ccurte thn the methods of [31, 3]. Tble 3: Errors in the solution of Exmple 5. with c = 1.8 Methods L L RMS Our method 5.459e e e e e e e e e 015 Method o f [31] 5.94e e e 5 Method o f [3] 5 3.3e e e 3

8 Communictions in umericl Anlysis 016 o. (016) Exmple 5.3. As the third exmple, we consider the following nonliner Volterr integrl eqution given in [33] by y(x) = 1 10 x x x 0 1 x y (t)dt where the exct solution is y(x) = x + 1. The error, L -error,l -error nd Root-Men-Squre(RMS)-error norms for = 5,10,15 with MQ-RBF re reported in Tble 4 From Tble 4, we find tht the ccurcy mesured inl,l nd RMS norm errors decreses s increse. Tble 4: Errors in the solution of Exmple 5.3 with c = L L RMS 5 5.8e e e e e e e e e 016 Exmple 5.4. Lstly, we consider the initil vlue problem with the exct solution y(x) = e x in [34] y (x) = e x e 3x + y 3 (x), y(0) = 1, 0 x 1 the bove eqution cn be converted into nonliner Volterr integrl eqution of the form y(x) = e x 1 3 e3x + x 0 y3 (t)dt where the exct solution is y(x) = e x. The error, L -error,l -error nd Root-Men-Squre(RMS)-error norms for = 5,10,15 with MQ-RBF re presented in Tble 5 Also, in comprison with the results of [35], which used Legendre wvelets method, the errors in ech row hve been decresed, which is good fctor in the RBFs method. From this tble, we find tht the ccurcy mesured inl,l nd RMS norm errors decreses s increse. Tble 5: Errors in the solution of Exmple 5.4 with c = 1.7 L L RMS 5.169e e e e e e e e e Conclusion Exct solutions for nonliner integrl equtions re not often vilble, so pproximting these solutions is very importnt. Mny uthors hve proposed different methods. In this rticle, we hve pplied meshless pproch bsed on the RBF for numericl solution of nonliner integrl equtions. The proposed method reduces n integrl eqution to system of equtions. Implementtion of our method is esy nd ccurte, this hs been verified by test exmples. The numericl results given in the previous section demonstrte the efficiency nd good ccurcy of this scheme. Acknowledgements The uthors wish to thnk the nonymous referees for their detiled reding of the mnuscript nd for mny useful comments nd excellent suggestions tht hve helped improve the mnuscript significntly. The uthors re lso very much thnkful to the Editor-in-Chief, Professor Seid Abbsbndy.

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