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A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

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A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems
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  Research Article  A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems Mohammed AL-Smadi, 1 Omar Abu Arqub, 2 and Ahmad El-Ajou 2 󰀱  Applied Science Department, Ajloun College, Al-Balqa Applied University, Ajloun 󰀲󰀶󰀸󰀱󰀶, Jordan 󰀲 Department of Mathematics, Al-Balqa Applied University, Salt 󰀱󰀹󰀱󰀱󰀷, Jordan Correspondence should be addressed to Omar Abu Arqub; o.abuarqub@bau.edu.joReceived 󰀱󰀲 September 󰀲󰀰󰀱󰀳; Accepted 󰀱󰀱 February 󰀲󰀰󰀱󰀴; Published 󰀲󰀵 March 󰀲󰀰󰀱󰀴Academic Editor: Hak-Keung LamCopyright © 󰀲󰀰󰀱󰀴 Mohammed AL-Smadi et al. Tis is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work is properly cited.Teobjectiveothispaperistopresentanumericaliterativemethodorsolvingsystemso󿬁rst-orderordinarydifferentialequationssubjecttoperiodicboundaryconditions.TisiterativetechniqueisbasedontheuseothereproducingkernelHilbertspacemethodin which every unction satis󿬁es the periodic boundary conditions. Te present method is accurate, needs less effort to achieve theresults, and is especially developed or nonlinear case. Furthermore, the present method enables us to approximate the solutionsand their derivatives at every point o the range o integration. Indeed, three numerical examples are provided to illustrate theeffectiveness o the present method. Results obtained show that the numerical scheme is very effective and convenient or solvingsystems o 󿬁rst-order ordinary differential equations with periodic boundary conditions. 1. Introduction Systems o ordinary differential equations with periodicboundary value conditions, the so-called periodic boundary  value problems (BVPs), are well known or their applicationsin sciences and engineering [󰀱–󰀵]. In this paper, we ocus on 󿬁nding approximate solutions to systems o 󿬁rst-orderperiodic BVPs, which are a combination o systems o 󿬁rst-order ordinary differential equations and periodic boundary conditions. In act, accurate and ast numerical solutions o systems o 󿬁rst-order periodic BVPs are o great importancedue to their wide applications in scienti󿬁c and engineeringresearch.Numericalmethodsarebecomingmoreandmoreimpor-tant in mathematical and engineering applications, simply not only because o the difficulties encountered in 󿬁ndingexact analytical solutions but also because o the ease withwhich numerical techniques can be used in conjunctionwith modern high-speed digital computers. A numericalprocedure or solving systems o 󿬁rst-order periodic BVPsbasedontheuseoreproducingkernelHilbertspace(RKHS)method is discussed in this work.Among a substantial number o works dealing withsystems o 󿬁rst-order periodic BVPs, we mention [󰀶–󰀱󰀰]. Te existence o solutions to systems o 󿬁rst-order periodicBVPs has been discussed as described in [󰀶]. In [󰀷], the authorshavediscussedsomeexistenceanduniquenessresultso periodic solutions or 󿬁rst-order periodic differentialsystems. Also, in [󰀸] the authors have provided the existence,multiplicity, and nonexistence o positive periodic solutionsor systems o 󿬁rst-order periodic BVPs. Furthermore, theexistence o periodic solutions or the coupled 󿬁rst-orderdifferential systems o Hamiltonian type is carried out in [󰀹].Recently, the existence o positive solutions or systems o 󿬁rst-orderperiodicBVPsisproposedin[󰀱󰀰].Formoreresultson the solvability analysis o solutions or systems o 󿬁rst-order periodic BVPs, we reer the reader to [󰀱󰀱–󰀱󰀵], and or numerical solvability o different categories o BVPs, one canconsult [󰀱󰀶–󰀱󰀹]. Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 135465, 10 pageshttp://dx.doi.org/10.1155/2014/135465  󰀲 Journal o Applied MathematicsInvestigation about systems o 󿬁rst-order periodic BVPsnumerically is scarce. In this paper, we utilize a methodicalway to solve these types o differential systems. In act,we provide criteria or 󿬁nding the approximate and exactsolutions to the ollowing system:  1  (󽠵) = 􍠵 1  󰀨󽠵, 1  (󽠵), 2  (󽠵),...,   (󽠵)󰀩, 2  (󽠵) = 􍠵 2  󰀨󽠵, 1  (󽠵), 2  (󽠵),...,   (󽠵)󰀩, ...    (󽠵) = 􍠵   󰀨󽠵, 1  (󽠵), 2  (󽠵),...,   (󽠵)󰀩, (󰀱)subject to the periodic boundary conditions  1  (0) =  1  (1), 2  (0) =  2  (1), ...    (0) =    (1), (󰀲)where  󽠵 ∈ [0,1] ,   󽠵  ∈ 󝠵 22 [0,1]  are unknown unctions tobe determined,  􍠵 󽠵 (󽠵, V  1 , V  1 ,..., V   )  are continuous terms in 󝠵 12 [0,1]  as  V  󽠵  =  V  󽠵 (󽠵) ∈ 󝠵 22 [0,1] ,  0 ≤ 󽠵 ≤ 1 ,  −∞ <  V  󽠵  <∞  in which   = 1,2,..., , and  󝠵 12 [0,1] ,  󝠵 22 [0,1]  are tworeproducing kernel spaces. Here, we assume that (󰀱) subjecttotheperiodicboundaryconditions(󰀲)hasauniquesolutionon  [0,1] .Reproducing kernel theory has important applications innumerical analysis, differential equations, integral equations,probability and statistics, and so orth [󰀲󰀰–󰀲󰀲]. In the last years, extensive work has been done using RKHS methodwhich provides numerical approximations or linear andnonlinear equations. Tis method has been implemented inseveral operator, differential, integral, and integrodifferentialequations side by side with their theories. Te reader iskindly requested to go through [󰀲󰀳–󰀳󰀵] in order to know  more details about RKHS method, including its history, itsmodi󿬁cation or use, its applications, and its characteristics.Te rest o the paper is organized as ollows. In the nextsection,tworeproducingkernelspacesaredescribedinorderto ormulate the reproducing kernel unctions. In Section 󰀳,some essential results are introduced and a method or theexistenceosolutionsor(󰀱)and(󰀲)isdescribed.InSection 󰀴, we give an iterative method to solve (󰀱) and (󰀲) numerically. Numerical examples are presented in Section 󰀵. Section 󰀶 ends this paper with brie conclusions. 2. Construct of Reproducing Kernel Functions In this section, two reproducing kernels needed are con-structed in order to solve (󰀱) and (󰀲) using RKHS method. Beore the construction, we utilize the reproducing kernelconcept. Troughout this paper,  C  is the set o complexnumbers,   2 [,] =  | ∫ 􍠵󝠵   2 (󽠵)󽠵 < ∞} , and   2 =  |∑ ∞=1  (  ) 2 < ∞} . De󿬁nition 󰀱  (see [󰀲󰀳]). Let    be a nonempty abstract set. Aunction  : × →  C isareproducingkernelotheHilbertspace    i (󰀱) or each  󽠵 ∈  ,  (⋅,󽠵) ∈  ,(󰀲) or each  󽠵 ∈   and   ∈  ,  ⟨(⋅),(⋅,󽠵)⟩ = (󽠵) . Remark 󰀲.  Condition  (2)  in De󿬁nition 󰀱 is called “the repro-ducing property,” which means that the value o the unction   at the point  󽠵  is reproducing by the inner product o   (⋅) with  (⋅,󽠵) . A Hilbert space which possesses a reproducingkernel is called a RKHS.o solve (󰀱) and (󰀲) using RKHS method, we 󿬁rst de󿬁ne and construct a reproducing kernel space  󝠵 22 [0,1]  in whichevery unction satis󿬁es the periodic boundary condition (0) = (1) . Afer that, we utilize the reproducing kernelspace  󝠵 12 [0,1] . De󿬁nition 󰀳.  Te inner product space  󝠵 22 [0,1]  is de󿬁ned as 󝠵 22 [0,1] = (󽠵) | ,  areabsolutelycontinuousreal-valuedunctions on  [0,1] ,  ,  ,  ∈  2 [0,1] , and  (0) = (1)} . Onthe other hand, the inner product and the norm in  󝠵 22 [0,1] are de󿬁ned, respectively, by  ⟨, V  ⟩  22 =  1 󲈑 =0  () (0) V  () (0) + 󲈫 10   () V   (),  (󰀳)and  ‖‖  22 = √⟨,⟩  22 , where  , V   ∈ 󝠵 22 [0,1] .It is easy to see that  ⟨, V  ⟩  22 satis󿬁es all the require-ments or the inner product. First,  ⟨,⟩  22 ≥ 0 . Second, ⟨, V  ⟩  22 = ⟨ V  ,⟩  22 . Tird,  ⟨, V  ⟩  22 = ⟨, V  ⟩  22 . Fourth, ⟨ + , V  ⟩  22 = ⟨, V  ⟩  22 + ⟨, V  ⟩  22 , where  , V  , ∈ 󝠵 22 [0,1] .It thereore remains only to prove that  ⟨,⟩  22 = 0  i andonly i    = 0 . In act, it is obvious that when   = 0 , then ⟨,⟩  22 = 0 . On the other hand, i   ⟨,⟩  22 = 0 , then by (󰀳), we have  ⟨,⟩  22 = ∑ 1=0  ( () (0)) 2 + ∫ 10  (  ()) 2  = 0 ;thereore,  (0) =   (0) = 0  and    () = 0 . Ten, we canobtain   = 0 . De󿬁nition 󰀴  (see [󰀲󰀳]). Te Hilbert space  󝠵 22 [0,1]  is calleda reproducing kernel i, or each 󿬁xed  󽠵 ∈ [0,1] ,there exist  (󽠵,) ∈ 󝠵 22 [0,1]  (simply     () ) such that ⟨(),  ()⟩  22 = (󽠵)  or any   () ∈ 󝠵 22 [0,1]  and   ∈[0,1] .An important subset o the RKHSs is the RKHSs asso-ciated with continuous kernel unctions. Tese spaces havewide applications, including complex analysis, harmonicanalysis, quantum mechanics, statistics, and machine learn-ing. Teorem 󰀵.  Te Hilbert space  󝠵 22 [0,1]  is a complete repro-ducing kernel and its reproducing kernel function   () can bewritten as    󰀨󰀩=  1  (󽠵) +  2  (󽠵) +  3  (󽠵) 2 +  4  (󽠵) 3 ,  ≤ 󽠵, 1  (󽠵) +  2  (󽠵) +  3  (󽠵) 2 +  4  (󽠵) 3 ,  > 󽠵, (󰀴)  Journal o Applied Mathematics 󰀳 where    (󽠵)  and     (󽠵),  = 1,2,3,4  , are unknown coefficientsof     ()  and will be given in the following proof.Proof.  Te proo o the completeness and reproducingproperty o   󝠵 22 [0,1]  is similar to the proo in[󰀲󰀴]. Now, let us 󿬁nd out the expression orm o the reproing kernel unction    ()  in the space 󝠵 22 [0,1] . Trough several integration by parts, we have ∫ 10    () 3   () = ∑ 1=0  (−1) 1−  () () 3−    ()| =1=0  +∫ 10  () 4   () . Tus, rom (󰀳), we can write ⟨(),  ()⟩  22 = ∑ 1=0   () (0)[    (0) + (−1)   3−    (0)] +  ∑ 1=0 (−1) 1−  () (1) 3−    (1)  +  ∫ 10  () 4   () . Since   () ∈ 󝠵 22 [0,1] , it ollows that    (0) =   (1) ; also since (󽠵) ∈ 󝠵 22 [0,1] , it ollows that  (0) = (1) . Ten 󲟨󰀨󰀩,   󰀨󰀩󲟩  22 =  1 󲈑 =0  () (0)󰁛     (0) + (−1)   3−     (0)󰁝+  1 󲈑 =0 (−1) +1  () (1) 3−     (1)+ 󲈫 10 󰀨󰀩 4    󰀨󰀩 +  1  ((0) − (1)). (󰀵)But on the other aspect as well, i    2   (1) = 0 ,   (0) +  3   (0) +  1  = 0 ,   1   (0) −  2   (0) = 0 , and  3   (1) +  1  = 0 , then (󰀵) implies that  ⟨(),  ()⟩  22 =∫ 10  () 4   () . Now, or any   󽠵 ∈ [0,1] , i     ()  satis󿬁es  4    󰀨󰀩 = −󰀨󽠵 − 󰀩,   dirac-delta unction ,  (󰀶)then  ⟨(),  ()⟩  22 = (󽠵) . Obviously,    ()  is thereproducing kernel unction o the space  󝠵 22 [0,1] . Next, wegive the expression orm o the reproducing kernel unction   () . Te characteristic ormula o (󰀶) is given by    4 = 0 .Ten the characteristic values are   = 0  with multiplicity  4 . So, let the expression orm o the reproducing kernelunction    ()  be as de󿬁ned in (󰀴). On the other hand, or(󰀶),let   () satisytheequation      (󽠵+0) =      (󽠵−0) ,  = 0,1,2 .Integrating  6   () = −(󽠵−) rom 󽠵− to 󽠵+ withrespectto  andletting  → 0 ,wehavethejumpdegreeo    5   ()  at   = 󽠵  given by    3   (󽠵+0)− 3   (󽠵−0) = −1 .Trough the last descriptions, the unknown coefficients o (󰀴) can be obtained. However, by using MAPLE  13  sofwarepackage, the representation orm o the reproducing kernelunction    ()  is provided by     󰀨󰀩 = 󰁻􀁻􀁻󐁻􀁻􀁻{148 􀀨󽠵 3 􀀨6 + 3 −  2 􀀩 + 3󽠵 2 􀀨−6 − 3 2 􀀩 + 6󽠵􀀨2 +  +  2 􀀩 − 8􀀨−6 +  3 􀀩􀀩,  ≤ 󽠵,148 􀀨48 + 6󽠵􀀨2 − 3 +  2 􀀩 + 3󽠵 2 􀀨2 − 3 +  2 􀀩 − 󽠵 3 􀀨8 − 6 − 3 2 +  3 􀀩􀀩,  > 󽠵,  (󰀷)Tis completes the proo. De󿬁nition 󰀶   (see [󰀲󰀵]). Te inner product space  󝠵 12 [0,1]  isde󿬁ned as  󝠵 12 [0,1] = (󽠵) |   is absolutely continuous real- valued unction on  [0,1]  and    ∈  2 [0,1]} . On the otherhand,theinnerproductandthenormin 󝠵 12 [0,1] arede󿬁ned,respectively, by   ⟨(󽠵), V  (󽠵)⟩  12 = (0) V  (0) + ∫ 10    (󽠵) V   (󽠵)󽠵 ,and  ‖‖  12 = √⟨,⟩  12 , where  , V   ∈ 󝠵 12 [0,1] . Teorem󰀷 (see[󰀲󰀵]).  TeHilbertspace 󝠵 12 [0,1] isacompletereproducing kernel and its reproducing kernel function    () can be written as  x   󰀨󰀩 = 1 + ,  ≤ 󽠵,1 + 󽠵,  > 󽠵.  (󰀸)Reproducing kernel unctions possess some importantproperties such as being symmetric, unique, and nonnega-tive. Te reader is asked to reer to [󰀲󰀳–󰀳󰀵] in order to know  more details about reproducing kernel unctions, includingtheir mathematical and geometrical properties, their typesandkinds,andtheirapplicationsandmethodocalculations. 3. Formulation of Linear Operator In this section, the ormulation o a differential linear oper-ator and the implementation method are presented in thereproducing kernel space  󝠵 22 [0,1] . Afer that, we constructan orthogonal unction system o the space  󝠵 22 [0,1]  basedon the use o the Gram-Schmidt orthogonalization processin order to obtain the exact and approximate solutions o (󰀱)and (󰀲) using RKHS method.First, as in [󰀲󰀳–󰀳󰀵], we transorm the problem into a differential operator. o do this, we de󿬁ne a differentialoperator    as   : 󝠵 22 [0,1] → 󝠵 12 [0,1]  such that  (󽠵) =  (󽠵) . As a result, (󰀱) and (󰀲) can be converted into the equivalent orm as ollows:  󽠵  (󽠵) = 􍠵 󽠵  󰀨󽠵, 1  (󽠵), 2  (󽠵),...,   (󽠵)󰀩, 󽠵  (0) −  󽠵  (1) = 0,  (󰀹)where 0 ≤ 󽠵 ≤ 1 and  = 1,2,..., inwhich  󽠵 (󽠵) ∈ 󝠵 22 [0,1] and  􍠵 󽠵 (󽠵, V  1 , V  1 ,..., V   ) ∈ 󝠵 12 [0,1]  or  V  󽠵  =  V  󽠵 (󽠵) ∈ 󝠵 22 [0,1] , −∞ <  V  󽠵  < ∞ , and  0 ≤ 󽠵 ≤ 1 . It is easy to show that    is  󰀴 Journal o Applied Mathematicsa bounded linear operator rom the space  󝠵 22 [0,1]  into thespace  󝠵 12 [0,1] .Initially, we construct an orthogonal unction system o  󝠵 22 [0,1] . o do so, put    (󽠵) =    (󽠵)  and    (󽠵) =  ∗   (󽠵) ,where  󽠵  } ∞=1  is dense on  [0,1]  and   ∗ is the adjoint operatoro   .Intermsothepropertiesoreproducingkernelunction   () , one obtains  ⟨ 󽠵 (󽠵),  (󽠵)⟩  22 = ⟨ 󽠵 (󽠵), ∗   (󽠵)⟩  22 =⟨ 󽠵 (󽠵),  (󽠵)⟩  12 =  󽠵 (󽠵  ) ,   = 1,2,... ,   = 1,2,..., .For the orthonormal unction system    (󽠵)} ∞=1  o thespace  󝠵 22 [0,1] , it can be derived rom the Gram-Schmidtorthogonalization process o     (󽠵)} ∞=1  as ollows:    (󽠵) =   󲈑 =1      (󽠵),  (󰀱󰀰)where     are orthogonalization coefficients and are given as    = 1 1 ,  or   =  = 1,   = 1󲈚   2 − ∑ −1=1  󐀨󲟨  ,  󲟩  22 󐀩 2 ,  or   =  ̸=1,   = − 1√   2 − ∑ −1=1  󰀨  󰀩 2−1 󲈑 = 󲟨  ,  󲟩  22   , or   > . (󰀱󰀱)Clearly,    (󽠵) =  ∗   (󽠵) = ⟨ ∗   (󽠵),  ()⟩  22 =⟨  (󽠵),    ()⟩  12 =     ()| =  ∈ 󝠵 22 [0,1] . Tus,    (󽠵) can be written in the orm    (󽠵) =     ()| =  , where    indicates that the operator    applies to the unction o    . Teorem 󰀸.  If   󽠵  } ∞=1  is dense on  [0,1]  , then    (󽠵)} ∞=1  is acomplete function system of the space  󝠵 22 [0,1] .Proof.  For each 󿬁xed   󽠵 (󽠵) ∈ 󝠵 22 [0,1] , let  ⟨ 󽠵 (󽠵),  (󽠵)⟩  22 =0 . In other words, one can write  ⟨ 󽠵 (󽠵),  (󽠵)⟩  22 =⟨ 󽠵 (󽠵), ∗   (󽠵)⟩  22 = ⟨ 󽠵 (󽠵),  (󽠵)⟩  12 =  󽠵 (󽠵  ) = 0 . Notethat  󽠵  } ∞=1  is dense on  [0,1] ; thereore   󽠵 (󽠵) = 0 . It ollowsthat   󽠵 (󽠵) = 0 ,   = 1,2,..., , rom the existence o    −1 . So,the proo o the theorem is complete. Lemma 󰀹.  If    󽠵 (󽠵) ∈ 󝠵 22 [0,1]  , then there exist positiveconstants   {󽠵} such that   ‖ ()󽠵  (󽠵)‖   ≤  {󽠵} ‖ 󽠵 (󽠵)‖  22  ,   = 0,1  ,  = 1,2,...,  , where  ‖ 󽠵 (󽠵)‖   =  max 0≤≤1 | 󽠵 (󽠵)| .Proof.  For any   󽠵, ∈ [0,1] , we have   ()󽠵  (󽠵) =⟨ 󽠵 (),    ()⟩  22 . By the expression orm o the kernelunction    () , it ollows that  ‖    ()‖  22 ≤  {󽠵}  . Tus, | ()󽠵  (󽠵)| = |⟨ 󽠵 (󽠵),    (󽠵)⟩  22 | ≤ ‖    (󽠵)‖  22 ‖ 󽠵 (󽠵)‖  22 ≤ {󽠵}  ‖ 󽠵 (󽠵)‖  22 .Hence, ‖ ()󽠵  (󽠵)‖   ≤  max =0,1  {󽠵}  }‖ 󽠵 (󽠵)‖  22 ,  = 0,1 ,   = 1,2,..., .Te internal structure o the ollowing theorem is asollows: 󿬁rstly, we will give the representation orm o theexact solutions o (󰀱) and (󰀲) in the orm o an in󿬁nite series in the space  󝠵 22 [0,1] . Afer that, the convergence o approximate solutions   󽠵, (󽠵)  to the exact solutions   󽠵 (󽠵) ,  = 1,2,..., , will be proved. Teorem󰀱󰀰.  Foreach  󽠵  ,  = 1,2,..., inthespace 󝠵 22 [0,1]  ,the series  ∑ ∞=1 ⟨ 󽠵 (󽠵),  (󽠵)⟩  (󽠵)  is convergent in the sense of the norm of   󝠵 22 [0,1] . On the other hand, if   󽠵  } ∞=1  is dense on [0,1]  , then the following hold: (i)  the exact solutions of   (󰀹)  could be represented by   󽠵  (󽠵)=  ∞ 󲈑 =1 󲈑 =1   􍠵 󽠵  󰀨󽠵  , 1  󰀨󽠵  󰀩, 2  󰀨󽠵  󰀩,...,   󰀨󽠵  󰀩󰀩   (󽠵), (󰀱󰀲)(ii)  the approximate solutions of   (󰀹)  󽠵,  (󽠵)=   󲈑 =1 󲈑 =1   􍠵 󽠵  󰀨󽠵  , 1  󰀨󽠵  󰀩, 2  󰀨󽠵  󰀩,...,   󰀨󽠵  󰀩󰀩   (󽠵), (󰀱󰀳) and    ()󽠵, (󽠵)  ,   = 0,1  , are converging uniformly to theexactsolutions  󽠵 (󽠵) andtheirderivativesas  → ∞  ,respectively.Proof.  For the 󿬁rst part, let   󽠵 (󽠵)  be solutions o (󰀹) in the space  󝠵 22 [0,1] . Since   󽠵 (󽠵) ∈ 󝠵 22 [0,1] , ∑ ∞=1 ⟨ 󽠵 (󽠵),  (󽠵)⟩  (󽠵)  is the Fourier series expansionabout normal orthogonal system    (󽠵)} ∞=1 , and  󝠵 22 [0,1]  isthe Hilbert space, then the series  ∑ ∞=1 ⟨ 󽠵 (󽠵),  (󽠵)⟩  (󽠵)  isconvergent in the sense o   ‖ ⋅ ‖  22 . On the other hand, using(󰀱󰀰), it easy to see that  󽠵  (󽠵) =  ∞ 󲈑 =1 󲟨 󽠵  (󽠵),   (󽠵)󲟩  22    (󽠵)=  ∞ 󲈑 =1 󲈑 =1   󲟨 󽠵  (󽠵),   (󽠵)󲟩  22    (󽠵)=  ∞ 󲈑 =1 󲈑 =1   󲟨 󽠵  (󽠵), ∗    (󽠵)󲟩  22    (󽠵)=  ∞ 󲈑 =1 󲈑 =1   󲟨 󽠵  (󽠵),   (󽠵)󲟩  12    (󽠵)  Journal o Applied Mathematics 󰀵 =  ∞ 󲈑 =1 󲈑 =1    󲟨􍠵 󽠵  󰀨󽠵, 1  (󽠵), 2  (󽠵),...,   (󽠵)󰀩,   (󽠵)󲟩  12    (󽠵)=  ∞ 󲈑 =1 󲈑 =1   􍠵 󽠵  󰀨󽠵  , 1  󰀨󽠵  󰀩, 2  󰀨󽠵  󰀩,...,   󰀨󽠵  󰀩󰀩   (󽠵). (󰀱󰀴)Tereore, the orm o (󰀱󰀲) is the exact solutions o (󰀹). For the second part, it is easy to see that by  Lemma 󰀹, or any   󽠵 ∈[0,1] , 󽮔󽮔󽮔󽮔󽮔 ()󽠵,  (󽠵) −  ()󽠵  (󽠵)󽮔󽮔󽮔󽮔󽮔= 󽮔󽮔󽮔󽮔󽮔󽮔󽮔􂟨 󽠵,  (󽠵) −  󽠵  (󽠵), ()  (󽠵)􂟩  22 󽮔󽮔󽮔󽮔󽮔󽮔󽮔≤      (󽠵)  22  󽠵,  (󽠵) −  󽠵  (󽠵)  22 ≤  {󽠵}  󽠵,  (󽠵) −  󽠵  (󽠵)  22 , (󰀱󰀵)where   = 0,1  and   {󽠵}  are positive constants. Hence, i  ‖ 󽠵, (󽠵) −  󽠵 (󽠵)‖  22 → 0  as   → ∞ , the approximatesolutions   󽠵, (󽠵)  and   ()󽠵, (󽠵) ,   = 0,1 ,   = 1,2,..., ,are converged uniormly to the exact solutions   󽠵 (󽠵)  andtheir derivatives, respectively. So, the proo o the theorem iscomplete.We mention here that the approximate solutions   󽠵, (󽠵) in (󰀱󰀳) can be obtained directly by taking 󿬁nitely many termsin the series representation or   󽠵 (󽠵)  o (󰀱󰀲). 4. Construction of Iterative Method In thissection,aniterativemethodoobtainingthesolutionso (󰀱) and (󰀲) is represented in the reproducing kernel space  󝠵 22 [0,1]  or linear and nonlinear cases. Initially, wewill mention the ollowing remark about the exact andapproximate solutions o (󰀱) and (󰀲). In order to apply the RKHS technique to solve (󰀱) and(󰀲), we have the ollowing two cases based on the algebraicstructure o the unction  􍠵 󽠵 ,   = 1,2,..., . Case 󰀱.  I (󰀱) is linear, then the exact and approximate solu-tionscanbeobtaineddirectlyrom(󰀱󰀲)and(󰀱󰀳),respectively. Case 󰀲.  I (󰀱) is nonlinear, then in this case the exact andapproximatesolutionscanbeobtainedbyusingtheollowingiterative algorithm.  Algorithm 󰀱󰀱.  According to (󰀱󰀲), the representation orm o the solutions o (󰀱) can be denoted by   󽠵  (󽠵) =  ∞ 󲈑 =1  {󽠵}     (󽠵),  = 1,2,...,,  (󰀱󰀶)where   {󽠵}  = ∑ =1    􍠵 󽠵 (󽠵  , 1,−1 (󽠵  ), 2,−1 (󽠵  ),... ,  ,−1 (󽠵  )) . In act,   {󽠵}  in (󰀱󰀶) are unknown; onewill approximate   {󽠵}  using known   {󽠵}  . For numericalcomputations, one de󿬁nes the initial unctions   󽠵,0 (󽠵 1 ) = 0 ,put   󽠵,0 (󽠵 1 ) =  󽠵 (󽠵 1 ) , and de󿬁ne the   -term approximationsto   󽠵 (󽠵)  by   󽠵,  (󽠵) =   󲈑 =1  {󽠵}     (󽠵),  = 1,2,...,,  (󰀱󰀷)where the coefficients   {󽠵}  o     (󽠵) ,   = 1,2,..., ,   =1,2,..., , are given as  {󽠵}1  =  11 􍠵 󽠵  󰀨󽠵 1 , 1,0  󰀨󽠵 1 󰀩, 2,0  󰀨󽠵 1 󰀩,..., ,0  󰀨󽠵 1 󰀩󰀩, 󽠵,1  (󽠵) =  {󽠵}1   1  (󽠵), {󽠵}2  =  2 󲈑 =1  2 􍠵 󽠵  󰀨󽠵  , 1,−1  󰀨󽠵  󰀩, 2,−1  󰀨󽠵  󰀩,..., ,−1  󰀨󽠵  󰀩󰀩, 󽠵,2  (󽠵) =  2 󲈑 =1  {󽠵}     (󽠵), ...  {󽠵}  =   󲈑 =1   􍠵 󽠵  󰀨󽠵  , 1,−1  󰀨󽠵  󰀩, 2,−1  󰀨󽠵  󰀩,..., ,−1  󰀨󽠵  󰀩󰀩, 󽠵,  (󽠵) =  −1 󲈑 =1  {󽠵}     (󽠵). (󰀱󰀸)Here, we note that, in the iterative process o (󰀱󰀷), we canguaranteethattheapproximations  󽠵, (󽠵) satisytheperiodicboundary conditions (󰀲). Now, the approximate solutions  󽠵, (󽠵)  can be obtained by taking 󿬁nitely many terms in theseries representation o    󽠵, (󽠵)  and  󽠵,  (󽠵)=   󲈑 =1 󲈑 =1   􍠵 󽠵  󰀨󽠵  , 1,−1  󰀨󽠵  󰀩, 2,−1  󰀨󽠵  󰀩,..., ,−1  󰀨󽠵  󰀩󰀩   (󽠵), = 1,2,...,. (󰀱󰀹)Now, we will proo that   󽠵, (󽠵)  in the iterative ormula(󰀱󰀷) are converged to the exact solutions   󽠵 (󽠵)  o (󰀱). Inact, this result is a undamental in the RKHS theory and itsapplications. Te next two lemmas are collected in order toprove the prerecent theorem. Lemma 󰀱󰀲.  If   ‖ 󽠵, (󽠵) −  󽠵 (󽠵)‖  22 → 0  ,  󽠵   →   as  → ∞  , and   􍠵 󽠵 (󽠵, V  1 , V  2 ,..., V   )  is continuous in  [0,1]
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