 American Journal of Computational Mathematics, 2011, 1, 55-62 doi:10.4236/ajcm.2011.12006 Published Online June 2011 (http://www.scirp.org/journal/ajcm) Copyright © 2011 SciRes. AJCM 55Numerical Solution of Obstacle Problems by B-Spline Functions Ghasem Barid Loghmani1, Farshad Mahdifar2, Seyed Rouhollah Alavizadeh1 1Department of Mat hem at i cs, Yazd University, Yazd, Iran 2Department of Mat hem at i cs, Payame Noor University, Tehran, Iran E-mail: loghmani@yazduni.ac.ir, farshad.mehdifar@pnu.ac.ir Received February 19 , 20 11; revised April 6, 2011; accepted April 15, 2011 Abstract In this study, we use B-spline functions to solve the linear and nonlinear special systems of differential equations associated with the category of obstacle, unilateral, and contact problems. The problem can easily convert to an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. The numerical examples and computational results illustrate and guarantee a higher accuracy for this technique. Keywords: Least Square Method, Uniform B-Splines, Boundary Value Problems, Obstacle Problems 1. Introduction Variational inequality theory has become an effective and powerful tool for studying obstacle and unilateral problems arising in mathematical and engineering sci- ences. This theory has developed into an interesting branch of applicable mathematics, which contains a wealth of new ideas for inspiration and motivation to do research. It has been shown by Kikuchi and Oden  that the problem of equilibrium of elastic bodies in contact with a right foundation can be studied in the framework of variational inequality theory. Various numerical me- thods are being developed and applied to find the nume- rical solutions of the obstacle problems including finite difference techniques and spline based methods. In principle, these methods cannot be applied directly to solve the obstacle problems. However, if the obstacle function is known, one can characterize the obstacle problem by a sequence of boundary value problems without constraints via the variational inequality and a penalty function. The computational advantage of this approach is its simple applicability for solving diffe- rential equations. Such type of penalty function methods have been used quite effectively by Noor and Tirmizi , as a basis for obtaining numerical solutions for some obstacle problems. The aim of this paper is to consider the use of quadratic B-spline functions and least square method to develop a numerical method for obtaining smooth appro- ximations for the solution and its derivatives of the general form of a system of second order boundary value problem of the type. 112233(, (), ()),=()=(, (), ()),(, (), ()),=234gtut utaatautgtut utatagtut utatab  (1) 1()=ua and 2()=ub, (2) and the continuity conditions of and at 2a and 3. Here, (i), are given continuous functions, and the parameters 1uu=1,a21:[, ]iiigabRR 2, 3 and 2 are real finite constants. Linear form of such type of systems arise in the study of one dimensional obstacle, unilateral, moving and free boundary value problems, [1,3-10] and the references therein. In general, it is not possible to obtain the analytic form of the solution of (1)-(2) for arbitary choices of i(, , ) gtuu, ( ), so we resort to some numerical methods for ob- taining an approx imate solution of (1)-(2). =1i, 2,3In 1981, Villaggio  used the classical Rayleigh- Ritz method for solving a special form of (1), namely, 30,0 and 44()= 3() 1,44ttutut t   (3) (0) =0u and ()=0u, (4) and the continuity conditions of and u at u4 and 56 G.B. Loghmani ET AL. 34. Later, Noor and Khalifa  have solved problem (3)-(4) using collocation method with cubic splines as basis functions. Similar conclusions were pointed out by Noor and Tirmizi , Al-Said  and Al-Said et al. , where second and fourth order finite difference and spline methods were used to solve a special linear form of problem (1), namely, 2323 (), and ()= () ()(), ftataatbut gtutft rata   (5) 1()=ua and 2()=ub. (6) where, the functions ()ft and ()gt are continuous on [a, b] and [, 3], respectively and the continuity conditions of and at 2 and 3 is assumed, and is a real finite constants parameter. On the other hand, Al-Said [3,15,16] has developed and analyzed quadratic and cubic spline methods for solving (5)-(6) and compared his numerical results with other available results given in [2,12]. It was shown in  that the cubic spline method gives much better results than those produced by other methods (including the fourth order Nemerov method). 2auaua arIn 2003, Khan and Aziz  have solved problem (1)-(2) using parametric cubic spline technique and have shown that the their method gives approximations which are better than that produced by Al-Said method . Then in 2005, Siraj-ul-Islam, Aslam Noor, Tirmizi and Azam Khan  have established and analyzed optimal smooth approximations for systems of second order boundary value problems of the form (5)-(6) with qua- dratic non-polynomial splines. Also in 2006, similar methods were pointed out by Siraj-ul-Islam and Tirmizi  who developed a class of methods based on cubic non-polynomial splines for problem (5)-(6). The ob- tained results in [18,19] are very encouraging and non- -polynomial spline methods perform better than other existing methods [2,3,12-17] of the same order. Owing to importance of problem (5)-(6) in physics, the existence and uniqueness of solution to this problems has been related with one dimensional second order obstacle boundary value problems. Moreover, existence and uni- queness theorem for obstacle problems has been studied by variational inequalities theory and demonstrate by Friedman  and also by Kinderlehrer and Stampacchia , see, for more examples [1,6,10]. But in general form (1)-(2), it has been no very discussion. In this paper, we shall solve the general problem numerically (1)-(2) by scaling functions , for and (). Our pre- sentation finds a sequence of functions of the form ()ki t}Nk1 =2, 1, 0,,321ki {kv132 1,=2()= ()kkiivtc tki, which satisfy the exact boundary conditions. Also, up to an error k, the function satisfies the differential equation, where kv0k as . k2. Statement of the Method Consider, the general system of differential equations of the type 1122334[()], =()=[ ()],[()], =Guta at autGutat aGutatab 23 (7) with the general boundary conditions [()]=iiUut, (8) =1, 2iand the continuity conditions of and at 2a and 3, where i and i are second-order linear and boundary operators, respectively, and ’s are operators defined in the forms uuaG UiU2211=1 =1[()]= () ()jjiijijjjUutuau b, =1, 2iwhere, ij, ij and i are real constants. Since, least square mehtod for system of the differential Equations (7)-(8) lead to complicated large scale and can not ensure existence and uniqueness of solution to this problems. Therefore, we will be study the system of two-point second-order boundary value pr o blems of the type 112233(, (), ()),=()=(, (), ()),(, (), ()), =234gtut utaatautgtut utatagtut utatab  1()=ua and 2()=ub, and the continuity conditions of and at 2a and 3. Let () be con- tinuous functions. We convert the problem to an optimal control problem uu2, a21:[ ,]iiigabRR=1, 3i312=1 ()(, (), ())min aiiauiiutgtututdt  and two-point boundary conditions 1()=ua and 2()=ub. The actual solution of (1)-(2) is a function v such that   322([, ])1=112, , =0=, =iLaaiiivtgtvtvtva vb . Copyright © 2011 SciRes. AJCM G.B. Loghmani ET AL. 57For all >0, the method finds an approximate solution v satisfying 322([ ,])1=112()(, (), ())<()=, ()=. iLaaiiivtgtvt vtva vb  The sketch of the method is delineated as follows: Consider uniform quadratic B-spline function [20,21] (Figure 1). 222220<13(2)(3)1< 21()=2(3)2< 30othertttttBt ttwise (9) For simplicity, the break up point of the interval [, ] are taken at ab234aba and 33=4aba to develop the numerical method for approximating solu- tion of a system of differential Equations (1)-(2). For a fix natural number , we divided the interval [, ] into () equal subinterval using the control points, 3kab312k=(2)itai h=2,i, , , (), 2=ta 1,,31321 =ktb112kwhere 1=32kbah, (with attenti on t o gri d points, , ). 32322 =kta 2332 2=kta 3We define, 1,232()=( )kki tB taibaki, () 1=2, 1,,321ki where is a scaling function and , ki (, ) are translations and dilations of as prescribed in [22-26]. 2B, 13k1=2,,321ki B2Let 132 1, =2()= ()kkiivtc t, where the coefficients are determined from the conditions {}ic1()=kva, 2()=kvb, and the following least square problem: 322([, ])1=1 , , min kikkLaaciiiivtgtvtvt  The minimization problem is equivalent to the following nonlinear system: Figure 1. Uniform quadratic B-spline function.  322([, ])1=1112, , =0 (=2, 1,,321)()=, ()=.kikkLaaiiiikkkvtgtvtvtciva vb  ， 3. Convergence Analysis In this section, we analyzed new method in the special case of system of one-order boundary value problems of the type (1) with boundary condition 1()=ua. How- ever, consider the optimal control problem (, (), ())min bauftutut dt (10) and satisfying the boundary condition u1()=ua 1 (11) where . We supp oses 1=aaf is in the form 112223334(, (), ()), =(, (), ()),(, (), ())= (, (), ()), =ftut utaa taftututftututataftut utatab (12) 2(, (), ()):=[()(, ())]iiftut ututgtut, 13i. Consider the uniform linear B-spline function [20,27, 28] (Figure 2), 1,0<1()= 2,1<2,0otherwisttBtt t,e. For simplicity, suppose 23=4aba and 33=4aba such that divided the interval [, ] into () equal subinterval using the control points, ab2k=(1)itai h, , , (), 1=ta21=ktb=1,,21 kiwhere =2kbah and , (with attention to grid 2kCopyright © 2011 SciRes. AJCM 58 G.B. Loghmani ET AL. Figure 2. Uniform linear B-spline function (hat function). points, , ). 2221=kta21k23321 =ktaLet , where ,=1()= ()ikiiutc t,1():=ki tB 2()ta i=1, 0,ikba, . ( 1, 2,,2 1k,ki's are translations and dilations of linear spline (hat function) . 1BThen (2)(2)21 212'1, ,=1 =1(2)(2)32 132 13'2, ,2(2) (2)=2 1=2 1233(2)=3 21 (, (), ())=(, (), ())(, (), ())(, bakkaiki ikiaiikkaiki ikiakkiikbakiftut ut dtftctct dtftct ctdft 121',,(2)=3 21(), ())kiki ikikict cttdtd (13) Furthermore, we know that the support of uniform -degree B-splines , () are into the [,] (Remark that hat functions is 1). On the other hand, for all fix value of , just consecutive terms of the sequence of is nonzer o. Then, we have for : dth01=BdBt()jdBt1=0, 1, ,(1)jt1{, ()dBt2kj, }jdB01(),ddB, ()t,(2)2,23(2) (2),(2)3() 0, [, ], 221,() 0, [, ], 122 and 3221,() 0, [, ], 1322.kikkkikkkkikttaa ittaaiittab i    So, from (13), we obtain 321211',,=1=1= 1101 21 (, (), ())=(, (), ())=(, , ,,)bakkajjikiikiajji ikftut ut dtftc t c tdFcc cct1 (14) Therefore, in view of (14), the optimal problem (10)-(11) reduces to the problem 101 21, , ,...,10 121(, , ,, )min kcccckFcc cc (15) subject to 1=()=ua c. (16) For finding , ,…, , we have to solve the system 0c1c21kc=0jFck*, () Let =0, ,2 1j1c, , *0c**121,,kcc be an approximate solution of (15)-(16) and set 21**,=1()= ()kkikiutc ti. (17) We assume integral operator define the following by the form L():=(, (), ())baLuf tututdt*. suppose that there exists a solution of (10) satisfying (11). (We assume .) M. Ahmadinia and G. B. Loghmani  shows that under reasonable conditions, converges to as . ([, ])uCab*()Lu*()Lu*()kLukTheorem. Let f in (12) have the property that for all >0 there exists >0 such that |(, , )ftuv 11(, , u v)|0 there exists and k such that 2kh*)(Lu0()0 there exists an approximate solution for the optimal control problem (10)-(11) such that the difference between the value of and the value is at most *ku*()kLu *(Lu). Following the steps of the proof of above theorem we obtain the following corollary. Corollary 1. All derivatives of the approximate solution converges to the related derivatives of the exact solution. Remark 1. If the problem involves the higher derivative uB, we will use the uniform quadratic spline function 2 in (9). Here, 2 is a left continuous step function. In this method, dB ised when the regularity of dB is minial. That is, if the problem involves uB usm only, thedBn  (1d) st be chosen to be a step function; if it involves umu and u, then d B (d2) Copyright © 2011 SciRes. AJCM G.B. Loghmani ET AL. 59ust b]me chosen to be a step function, etc. 4. Applications To illustrate the application of the method developed in the previous sections, we suggested penalty functions technique for solving one dimension obstacle problems and deduction of existence and uniqueness solution of a systems (1)-(2). So, we consider the following second order obstacle boundary value problem about finding such that u()(),on =[0, ]()(),on =[0, ]ut ftut t (())(()())=0,on =[0, (0)=0 and ()=0,uftut tuu   (18) where f is a given continuous force acting on the beam and ()t is the elastic obstacle. Problem (18) describes the equilibrium configuration of an elastic beam, pulled at the ends and lying over an elastic obstacle. We study problem (18) in the framework of variational inequality approach. To do so, we define the set as K10K= () : on vH v 1()H, which is a closed convex set in 0, where 10()H is a Sobolev space, which is in fact a Hilbert space. For the definitions of the spaces , see [1,29]. Here, can be the following form 10()H10()H1100()=():()=0 ()=0lim limttHzHzt zt where is the space of absolutely continuous functions on the interval [0, 1()H] such that their first and second derivatives belonging to . It can be easily shown that the energy functional associated with the obstacle problem (18), for all is 2()LKv2200[]= ()()2()() =(, )2<, >dvIvvtdtftvdtav vfvtdt. (19) where 22220(, )=()()du dvau vdtdt dt (20) and 0<, >=()()fvftvt(,audt. (21) Also it can be easily shown that defined by (20) is bilinear, symmetric and positive (in fact, coercive [5,7]) and the functional )vf defined by (21) is a linear continuous functional. It is well known [1,5,7,29] that th e m i n i mum of the functional u[]Iv defined by (19) on the closed convex set in can be cha- racterized by the variational inequality K v10()H(, )<,>au vufu for all . (22) KvThus, we conclude that the obstacle problem (18) is equivalent to solving the variational inequality problem (22). This equivalence has been used to study the existence of a unique solution of (18), [1,4,5]. Now using the idea of Lewy and Stampacchia , problem (22) can be written as {} )uu(u=f , 0