A numerical approach to nonlinear two-point boundary value problems for ODEs

Transcription

1 Computer and Mathematic with Application ) A numerical approach to nonlinear two-point boundary value problem for ODE S. Cuomo, A. Maraco Department of Mathematic and Application R. Caccioppoli, Univerity of Naple Federico II, Via Cintia, 80126, Naple, Italy Received 19 July 2007; accepted 24 October 2007 Abtract In thi paper we propoe a numerical approach to olve ome problem connected with the implementation of the Newton type method for the reolution of the nonlinear ytem of equation related to the dicretization of a nonlinear two-point BVP for ODE with mixed linear boundary condition by uing the finite difference method. c 2007 Elevier Ltd. All right reerved. Keyword: Nonlinear boundary value problem; Finite difference method; Green function; Nonlinear ytem; Newton method 1. Introduction The theory of the boundary value problem i an extremely important and intereting area of reearch in differential equation ee [1 36]). In thi paper we conider the following two-point boundary value problem, which occur in applied mathematic, theoretical phyic, engineering, control and optimization theory y = f x, y, y ), l a ya) m a y a) = v a, x [a, b], 1) l b yb) + m b y b) = v b, where f C [a, b] R 2, R ) i a nonlinear function, l a, m a, l b, and m b are given non-negative contant and v a, v b aigned contant. We face thi problem by uing analytical and/or numerical approximation method ince, generally, the olution cannot be exhibited in a cloed form even when it exit. Uually, the adopted integration method for 1), are the finite difference method [9,7,8,32,33,35,34], the hooting method [9,7,8,31,30], the monotone iterative method [15, 18,20,23,21,22], and the quailinearization method [13,17,19,24,25,27,28]. The analyi contained in Section 2 put in evidence the advantage and the open problem of all thee method. A a conequence, we are forced to ue the finite difference method ince we are able to overcome the open problem of thi approach in many cae. It i well Correponding author. addree: alvatore.cuomo@dma.unina.it S. Cuomo), maraco@unina.it A. Maraco) /$ - ee front matter c 2007 Elevier Ltd. All right reerved. doi: /j.camwa

2 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) known that the finite difference method reduce the problem 1) to the following dicrete problem D 2 y k = f x k, y k, Dy k ), k = 1,..., n 1 l a y 0 m a Dy 0 = v a, l b y n + m b Dy n = v b, 2) where h = b a)/n i the tep ize of the grid point x k = a + kh, y k = y x k ), and Dy k, D 2 y k are the centered difference quotient, for k = 0,..., n. The problem connected with the reolution of ytem 2) are: It i eential to determine under which condition problem 2) admit a olution converging to the olution yx) of problem 1) ince the property of exitence and uniquene of the olution doe not necearily tranfer from the continuou problem 1) to the dicrete problem 2). Sytem 2), when it i nonlinear, can be olved by uing Newton-like method. Thee iterative method generate a equence yk ν } ν N converging to y k provided that uitable initial data yk 0, k = 0,..., n are aigned. About the firt item, we recall that in 1974 Gaine in [1] proved that the dicrete BVP 2) could admit puriou olution which become unbounded and irrelevant for the correponding continuou problem 1) when the grid ize h tend to zero. Moreover, in [2] Agarwal exhibited an example in which the continuou problem ha a olution, wherea the dicrete problem doe not. Finally, Henderon and Thompon in [4,3,5,6] extended the exitence reult of Gaine to the cae of nonlinear boundary condition G y a, y b ), y a, b)) y = 0, proving convergence theorem for the olution of the dicrete problem when the grid ize h goe to zero. However, it i not eay to verify the hypothee of the cited theorem of exitence, uniquene and convergence ince they refer to dicrete Green function, the lower and upper olution, the degree theory, and a uitable compatibility condition on G. Differently, in Numerical Analyi, the convergence propertie of a finite difference cheme i, generally, baed on the concept of conitency and tability ee [8]). Generally, it ha to be noted that the convergence theorem of the BVP with mixed linear or nonlinear boundary condition refer to the convergence propertie of the linearized difference cheme around an iolated olution of the BVP. However, in [7], the convergence propertie are directly proved for the finite difference cheme 2). In any cae, the convergence of the finite difference method i trongly conditioned by the tart point y 0 π = y0 0,..., y0 n ) we have choen in the Newton-like method ued for the olution of the dicrete problem in the unknown y π = y 0,..., y n ), for any choice of the meh π : a = x 0 < x 2 < < x n = b. We remark that if y 0 π i not cloe to y π, then the promied convergence could not be reached, and even if there i eventually a convergence of the nonlinear iteration, it could not be realized before the end of the iterative proce. In order to olve thi lat important problem, we propoe a cheme in which a family of BVP A j, j = 0,...,, i conidered uch that, for j = 0, the correponding problem can be analytically integrated, wherea, for j =, we are again faced with the original problem ee [9] for a particular cla of the BVP with linear boundary condition). The propoed numerical cheme find the tart point y 0 π ufficiently cloe to the analytic exact olution of the conidered BVP provided that the following condition are atified: 1) An exitence and uniquene theorem mut hold both for the aigned problem and the whole cla of problem A j, j = 0,..., 1. Moreover, a contant χ > 0 exit, uch that: yx, j) yx) 1 j ) χ, x [a, b], where yx, j) i the unique olution of A j. 2) The finite difference method ha to be valid both for the aigned problem and the cla of the problem A j, j = 0,..., 1. In thi paper, we propoe ufficient condition to make requet 1) and 2) atified for the problem 1). Moreover, the propoed cheme will be formally extended to BVP with nonlinear boundary condition without verifying that condition 1) and 2) are atified. Thi paper i organized in five ection including the introduction. In Section 2, the uually ued numerical and analytical method for the BVP 1) are critically decribed. Moreover, we recall ome known convergence reult of the finite difference method. In Section 3 the propoed cheme i widely decribed for the BVP 1) and the condition

3 2478 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) to verify that the requet 1) and 2) are expoed. In the Section 4 the numerical cheme i formally extended to BVP with fully nonlinear boundary condition. Finally, in Section 5 ome meaningful application are preented. Although the content of thi paper have been focued on econd-order equation, variou computational experiment how that the method efficiently operate alo in higher dimenion. Therefore, an intereting reearch project look at the generalization of the method to PDE, a long a the pace domain can be properly dicretized, ee [37,38], o that the boundary value problem with boundary value both for the time and pace variable) can be tranformed into a boundary value problem for a ytem of ordinary differential equation [39]. 2. Survey of known reult for BVP In thi ection we decribe ome analytical and numerical method ued to find an approximate olution of the BVP for ODE. In particular, we decribe in detail the finite difference method highlighting the open problem and recalling ome known convergence reult. The monotone iterative method, originally introduced by Picard [15], are baed on the idea of building equence of approximated olution which converge monotonically to the olution of the BVP 1). Generally, thee method ue the lower and upper olution, 1 generating equence of approximation α n } n N and β n } n N defined by a uitable iterative cheme. In [21] and [22] the author conider the Dirichlet problem y = f x, y, y ), x [a, b], 3) ya) = yb) = 0, under uitable hypothei on f, propoing for it the following iterative cheme α n + λα n = f x, α n 1, α n) + λαn 1, α n a) = α n b) = 0, α n + 2k α n α n 1 + λα n = f x, α n 1, α n 1) + λαn 1, α n a) = α n b) = 0, where k, λ are uitable contant, and α 0 i a lower olution of 3). We remark that approach 4) doe not explicitly give computable approximation α n ince the nonlinear function f depend on α n, wherea 5) i a linear problem. In [20], under uitable hypothee on f, the Neumann problem y = f x, y, y ), x [a, b], y a) = y 6) b) = 0, i examined introducing the equence α n } n N and β n } n N defined by α n + λα n = f x, α n 1, α n 1) + λαn 1, β n + λβ n = f x, β n 1, β n 1) + λβn 1, 7) α n a) = α n b) = 0, β n a) = β n b) = 0, where α 0 and β 0 are the lower and upper olution of 6) with α 0 β 0. Moreover, the equence α n } n N and β n } n N are monotone increaing and decreaing, repectively, and they converge punctually to olution u and v of 6) and are uch that α n u v β n for all n. The ue of thee method exhibit many difficultie. The lower and upper olution of BVP repreent the tart point of the iterative cheme, but there i no clue to finding them. Moreover, the iterative cheme do not upply contructive algorithm to approximate the olution of the BVP. If thi happen ee [23] for periodic problem), then it i neceary to give an efficient numerical algorithm to approximate the olution by monotone equence. In thi cae the error analyi i only poible a poteriori. The quailinearization method QLM) wa originally introduced by Bellman and Kalaba [16] a a generalization of the Newton Raphon method. Thi method generate equence of approximate olution of linear problem which 4) 5) 1 We recall that a function α β) C 2 [a, b]) i a lower upper) olution for 1) if α x) f x, α x), α x) ) 0 f x, βx), β x) ) β x) 0) x [a, b].

4 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) quadratically, often monotonically, converge to a olution of the original nonlinear BVP provided that the function involved are convex [17,19]. The aim of QLM i to olve the nonlinear BVP 1) a a limit of the following equence of linear BVP ) + 1 ) y ) r+1 y) r f y ) x, yr, y r ), x [a, b], y r+1 = f x, y r, y r =0 l a y r+1 a) m a y r+1 a) = v a, l b y r+1 b) + m b y r+1 b) = v b, where y 0) = y, y 1) = y and f y ) = f/ y ). We remark that the zeroth approximation y 0 i baed on mathematical or phyical conideration, and only a ufficiently good initial gue y 0 generate a rapid convergence of the method. For thee reaon, the QLM give excellent reult when applied to nonlinear BVP in phyic [12,29]. The generalized quailinearization method GQL) adopt the technique of lower and upper olution combined with QLM and it generate lower and upper monotone equence whoe element are the olution of the correponding linear problem. Both the equence converge rapidly to the olution of the conidered BVP [13,24,25,27,28]. The hooting method conit in reducing the BVP to a family F of initial value problem IVP) for the ame equation, where F i choen in uch a way to contain the olution of the given BVP [9,7,8]. If we denote by y x, r) the olution of the following IVP y = f x, y, y ), x [a, b], F : y a, r) = v a + m a r) /l a, 8) y a, r) = r, the problem i reduced to finding r = r which olve the equation φ r) l b y b, r ) + m b y b, r ) v b = 0. 9) In other word, in the hooting method we olve 8) for different choice of r until condition 9) i atified. We remark that each evaluation of 9) involve again the olution of the IVP 8). Then, we cannot hope to exactly evaluate φ r). Therefore, to approximate the root of 9) we hould try to ue very rapidly converging iterative cheme uch a Newton method. Finally, to obtain a general code for olving nonlinear BVP by the hooting method, it i neceary to ue a library routine for olving nonlinear equation and a tandard IVP olver. We note that the IVP in 8) could be ill-conditioned, even if the BVP i well-conditioned. In thi cae, the hooting method i untable. In the preence of the nonlinear BVP, there i another potential trouble: when the hooting method tart from wrong initial value r m, the IVP olution y x, r m ) could exit only in [a, c], where c < b. In uch cae, it i not known how to correct r m, becaue the nonlinear iteration of 9) cannot be completed, and therefore Newton method or any of it variant) fail ee [31,30] and the reference therein). Finally, we expoe in detail the finite difference method [9,7,8]. It i well know that the fundamental tep of thi method can be decribed a follow: 1) The interval [a, b] i ubtituted by a dicrete et of it point x k, k = 0,..., n, where x 0 = a and x n = b, i.e. we introduce the meh π : a = x 1 < x 2 < < x n = b. 2) Intead of the function yx), depending on the continuou variable x, a grid function y π = y 0,..., y n ), where y k = y x k ), of the dicrete variable x k i conidered. The derivative appearing in the differential equation and in the boundary condition are approximated by uitable algebraic expreion Dy k, D 2 y k containing the unknown value y k. 3) The reulting finite ytem S D 2 y k = f x k, y k, Dy k ), k = 1,..., n 1 l a y 0 m a Dy 0 = v a, 10) l b y n + m b Dy n = v b, in the unknown numerical value y k i olved in order to approximate yx) with the function interpolating the value x k, y k ), k = 0,..., n. We recall that the approximate formulae of the derivative are often given by the Lagrangian interpolating polynomial [32], the Chebyhev polynomial [33] or by the finite difference expreion of order four [35], ix and eight [34].

5 2480 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) In order to highlight the problem connected with the implementation of the finite difference method it i neceary to recall ome definition. We define the differential operator Ly = y + f x, y, y ) 11) and a correponding difference operator L π y i = y i+1 2y i + y i 1 h 2 + f x i, y i, y ) i+1 y i 1, 12) 2h where h i the tep ize of the uniform meh π. The difference operator L π i aid to be conitent of order p with L if, for every mooth function vx), poitive contant c and h 0 exit uch that for all mehe π with h h 0, we have that T π v = T i v ch p, where T i v = L π v x i ) Lv) x i ) i the local truncation error. Moreover, from Taylor theorem, we can derive the relation T i v = h2 [ v 4) ξ 1 ) 2 f z xi, v i, v ξ 2 ) ) ] v ξ 3 ), 13) 12 for any grid function v C 4 [a, b]) and ξ i [ x i 1, x i+1 ], i = 1, 2, 3. Then, Lπ ha a econd-order accuracy in approximating L for any function v having continuou fourth derivative on [a, b] ee [7,36]). Finally, the cheme 10) i aid to be converging to the olution y = yx) of the BVP 1) if for any grid function v a contant h 0 > 0 exit uch that for all the mehe π with h h 0 we have lim v x i) y x i ) = 0. h 0 Another important concept of the Numerical Analyi i the tability of the difference cheme 10) which require that the invere of the difference operator, including the boundary condition, i uitably bounded. Moreover, for both the linear and nonlinear BVP, the conitency and the tability of the finite difference cheme aure the convergence ee [8]). The definition and proof of tability for the difference cheme D 2 y k = f x k, y k, Dy k ), k = 1,..., n 1 14) y 0 = v a, y n = v b, relative to the Dirichlet problem y = f x, y, y ), ya) = v a, yb) = v b, are contained in the following theorem. Theorem 1. If we aume that 1) f x, y, z) C 1 [a, b] R 2, R ) 2) K 1, K 2, L 2 uch that 0 < K 1 f y K 2, f z L 2, on [a, b] R 2, x [a, b], 15) then, for all h uch that hl 2 2, the cheme 14) i table in the ene that for any two grid function v = v n }, w = w n } we have that v w M max v 0 w 0, v n w n ) + L π v L π w }, where M = max 1, 1/K 1 ). Moreover, for any function v C 4 [a, b]) it i v y MT π y, and the olution of 14) can be computed by a Newton-like method.

6 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) Remark 2. The above theorem require that the function f i mooth, f z i bounded and f y i bounded and poitive. Thee aumption are uually the hypothee of the exitence and uniquene theorem of BVP 15). Neverthele, it i worthwhile to note that in the application we frequently face with mooth function that do not atify one or more of thee condition. Remark 3. We have already oberved that the convergence of the finite difference cheme, which i aured by the Theorem 1, i ubjected to a good choice of the tart point y 0 π = y0 1,..., y0 n ). However, how to chooe the initial gue y 0 π cloe to the exact olution remain an open problem. Uually, for olving 14) the tart point y0 i are choen in uch a way that the point P i x i, yi 0 ) belong to the egment of extrema A a, ya ) and B b, y b ). In other cae the choice yi 0 = 0 for all i i preferred. However, a it i proved in thi paper, the choice of the initial gue i trictly connected to the aigned BVP and a wrong choice could compromie the reult obtained with the implementation of the finite difference method. Generally, a unified theory of tability and convergence for the nonlinear BVP with mixed linear or nonlinear boundary condition, i given in term of firt-order ytem of ODE ee [7,8]). In addition, the conitency and the order of accuracy are defined a before, wherea the tability i defined only in the proximity of an iolated olution. The known reult that the convergence i aured by the conitence and tability i adopted in [8], where the author prove the convergence theorem under the aumption that the linearized finite difference cheme, around the unknown) iolated olution, i conitent and table ee [8], p. 207). We rather prefer to follow Keller approach ee [7]) in which the conitency and convergence of the finite difference cheme i proved by the following theorem it ha been reformulated in order to fit the formulation of BVP 1)). Theorem 4. Let f x, y, z) atify in R = x, y, z) : x [a, b], y, z < }, the condition 1) f x, y, z) C 1 R, R), ) 2) K uch that 0 1 f y f z K, 3) σ := m a l b + m b l a 0, 4) for ome λ 0, 1) and m = σ 1 max m al b, m a m b, l a l b, m b l a }, K b a ln 1 + λ ). m 16) Then, both the BVP 1) and, for all h, the finite difference problem 10) have olution which are unique. In particular, the olution of the difference equation 10) i the limit of the equence y ν } π defined by ν D 2 y ν+1) k = f x k, yk ν, ) Dyν k, k = 1,..., n 1 l a y ν+1) 0 m a Dy ν+1) 0 = v a, l b y n ν+1) + m b Dy n ν+1) = v b, where y 0 } π are arbitrary. Moreover, if f x, y, z) C 2 R, R), then the olution yx) of the BVP 1) and the approximate olution y π x) defined by the difference problem 10) atify the condition y x k ) y π x k ) = O h 2). Even in thi cae, the Remark 2 and 3 till hold. In the hypothee of the Theorem 1 and 4, the finite difference method i conitent and convergent and the problem 15) and 1) can be olved with the accuracy order O h 2). The next ection i devoted to the decription of a particular numerical cheme for olving the problem of the tarting point. 3. The propoed cheme for BVP with linear boundary condition In order to overcome the problem related to the implementation of the finite difference method for 1), we firt earch for a function y x) uch that f x, y x), y x)) 0, and/or la y a) m a y a) v a 0, l b y b) + m b y b) v b 0, 17)

7 2482 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) and then we introduce the following family of BVP A j, j = 0,...,, y = f x, y, y ) 1 j ) f x), x [a, b], A j : l a ya) m a y a) v a = 1 j ) g 1, l b yb) + m b y b) v b = 1 j ) g 2, 18) where f x) = f x, y x), y x)), g 1 = l a y a) m a y a) v a, g 2 = l b y b) + m b y b) v b. Let u uppoe that each BVP 18) admit a olution which can be evaluated by the finite difference method. Thi mean that Theorem 1 and 4 for the problem 15) and 1), repectively, hold. If yx, j) denote the olution of the BVP A j, for j = 0,...,, then the problem A 0 admit only the known olution y x, 0) = y x), wherea the problem A coincide with the original problem 1). To each A j in 18) i aociated the following finite ytem S j D 2 y k = f x k, y k, Dy k ) 1 j ) f x k ), k = 1,, n 1 S j : l a y 0 m a Dy 0 v a = 1 j ) g 1, 19) l b y n + m b Dy n v b = 1 j ) g 2. The ytem S 0 admit the exact olution yx, 0) = y x), for j = 1, the nonlinear ytem S 1 can be olved with an iterative method that ue the olution y x k ), k = 0,..., n, a tart point of Newton-like method. Similarly, the problem S j, j > 1, can be olved uing the approximate olution obtained at the tep j 1. Finally, for j =, the nonlinear ytem S, correponding to BVP 1), i olved by uing the approximate olution evaluated at the tep j = 1. The only aim of thi procedure i to aign the tarting value y 0 π of the equence whoe limit i the required olution of 19) for j =. We preume that the choen tarting point are very cloe to the value y k, j = y x k, j) which, for j =, repreent approximate value of the olution yx) of the BVP 1) at the point x k. Thi conjecture i imply verified for the problem 1) when Green function are ued. Indeed, it i well known that if f C [a, b] R 2, R ) and the following condition hold l a m b + l b m a 0, or l a + l b > 0, m a + m b > 0, 20) then the BVP 1) ha a olution yx) in the form ee f.i. [10,1,11]) yx) = b a Gx, τ) f τ, y τ), y τ) ) dτ + φx), 21) where φx) i the olution of the BVP y = 0, 1) 2,3, and Green function G x, τ) i Gx, τ) = 1/c) ux)v τ), a τ x b, 1/c) u τ) vx), a x τ b, 22) where c = ux)v x) vx)u x), and ux), vx) are two linearly independent olution of the following problem, repectively y = 0, y = 0, l a ya) m a y a) = 0, l b yb) + m b y x [a, b]. b) = 0, In particular, the following exitence reult for the BVP 1) hold. Propoition 5. Let f C [a, b] R 2, R ) be bounded on [a, b] R 2, then, the BVP 1) ha a olution, whenever 20) hold.

8 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) It i eay to believe that if an exitence theorem for the BVP 1) i valid, then any other BVP in 18) ha a olution yx, j) cloe to the olution yx) of 1). In fact, the olution yx, j), j = 0,..., 1, of any BVP 18) write b [ yx, j) = Gx, τ) f τ, y τ), y τ) ) 1 j ) ] f τ) dτ + φ j x), 23) a where φ j x) i the olution of the BVP y = 0, 18) 2,3, and Gx, τ) i Green function defined in 22). With imple but tediou computation it i eay to prove that φ j x) = j φx) 1 j ) Ax B), j = 0,..., 1, 24) where φx) i the olution of the BVP y = 0, 1) 2,3, and l b va A = l avb l b m a + l a [b a)l b + m b ], B = bl b + m b ) va al a m a ) vb, l b m a + l a [b a)l b + m b ] va = l a y a) m a y a), v b = l b y b) + m b y b). By uing 23), 21) and 24) the following relation hold yx, j) yx) 1 j ) ) N M 0 + L + H, 25) where N = max Gx, τ)b a), x,τ [a,b] M0 L = max x [a,b] φx), H = max x [a,b] = max f x), x [a,b] Ax B. Remark 6. For the Dirichlet BVP 15) the relation 17) become f x, y x), y x)) y a) v 0, and/or a 0, y b) v b 0, and 18) write y = f x, y, y ) 1 j ) f x), A j : ya) v a = 1 j ) g 1, yb) v b = 1 j ) g 2, x [a, b], 26) 27) where g 1 = y a) v a, g 2 = y b) v b. The Green function 22) write b x) τ a) /b a), a τ x b, Gx, τ) = b τ) x a) /b a), a x τ b, and, if we put A = y a) y b)) /b a), B = by a) ay b)) /b a) in 24), the relation 25) hold with N = b a) 2 /8. Moreover, the Propoition 5 become Propoition 7. Let f C [a, b] R 2, R ) and for x, y 1, z 1 ), x, y 2, z 2 ) [a, b] R 2, we have f x, y 1, z 1 ) f x, y 2, z 2 ) k 1 y 1 y 2 + k 2 z 1 z 2, where k 1, k 2 are poitive contant uch that b a) 2 b a) k 1 + k 2 < Then, the BVP 15) ha one and only one olution.

9 2484 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) Remark 8. If we aume that f C [a, b] R 2, R ) i bounded in [a, b] R 2, then every BVP 15) ha a olution ee [10] p. 9). Remark 9. Uually, it i imple to verify the condition 17). In mot cae it i ufficient to chooe y x) = cont. For example, in [9] the numerical cheme 18) i implemented under the hypothei that condition 17) i atified by the zero function. 4. The propoed cheme for BVP with nonlinear boundary condition In thi ection, we conider the following fully nonlinear BVP y = f x, y, y ), x [a, b], G ya), yb)), y a), y b) )) = 0, 28) where G = g 1, g 2 ), g i C [a, b] R 2, R ) are nonlinear function, a well a it dicrete approximation D 2 y k = f x k, y k, Dy k ), k = 1,..., n 1 G y 0, y n ), Dy 0, Dy 1 )) = 0, 29) where the meaning of the adopted ymbol i elf-evident. A in the previou ection, we conider the following family of BVP A j, j = 0,...,, y = f x, y, y ) 1 j ) f x), x [a, b], A j : g 1 ya), yb)), y a), y b) )) = 1 j ) g 1, 30) g 2 ya), yb)), y a), y b) )) = 1 j ) g 2, where f x) = f x, y x), y x)), g i = g i y a), y b)), y a), y b))), i = 1, 2, and y = y x) i uch that f x, y x), y x)) 0, and/or g1 y a), y b)), y a), y b))) 0, g 2 y a), y b)), y a), y b))) 0. 31) Once again, the problem A 0 admit only the exact olution y x), wherea the problem A coincide with the original problem 28). To each A j in 30) the following finite ytem S j i aociated D 2 y k = f x k, y k, Dy k ) 1 j ) f x k ), k = 1,..., n 1 S j : g 1 y 0, y n ), Dy 0, Dy n )) = 1 j ) g 1, 32) g 2 y 0, y n ), Dy 0, Dy n )) = 1 j ) g 2, which could be olved by uing Newton-like method baed on tarting value which are known from the tep j = 1. The conideration reported in Remark 9 hold for the condition 31). For the problem 28), the cheme 30) give only a uggetion about the tarting point of the Newton method. Indeed, the only theorem which enure the convergence of the finite difference method and which do not refer to the linearization of 28) cloe to an iolated olution, are publihed in [4]. However, it i very difficult to verify the hypothee on which thee convergence and exitence theorem for the problem 28) are baed [14]. In the next ection we will how ome application of the propoed method for fully nonlinear BVP for which we know an analytical olution or an exitence and uniquene theorem.

10 5. The numerical imulation S. Cuomo, A. Maraco / Computer and Mathematic with Application ) In thi ection we propoe few numerical imulation implemented by the notebook NBoundaryD.nb and NBoundaryM.nb written by Mathematica, for olving the problem 15), 1) and 28), repectively. The ue of Mathematica i uggeted by the need of implementing the numerical cheme propoed in thi paper, deputing to the routine FindRoot the tak to numerically olve the finite difference ytem 27), 19) and 32) with the mot appropriate iterative method Newton, Brent, ecant, etc). Moreover, in thee program we expre the firt and the econd derivative in any BVP 15), 1) and 28) by the formulae Dy k = y k+1 y k 1 2h + O D 2 y k = y k+1 2y k + y k 1 h 2 h 2), + O h 2). In order to keep the accuracy order O h 2), we write the finite difference cheme 10) and 29) in the form D 2 y k = f x k, y k, Dy k ), k = 0,..., n y 0 y 1 l a y 0 m a = v a, 2h y n+1 y n l b y n + m b = v b, 2h D 2 y k = f x k, y k, Dy k ), k = 0,..., n y0 y 1 G y 0, y n ),, y )) n+1 y n = 0, 2h 2h where x 1 = a h, x n+1 = b + h. Simulation 1. We conider the following Dirichlet BVP y = co y in y + 2y + co y 1) = 0, y1) = 0, 1 x 2) in 2x) 2 ) x , x [ 1, 1] which admit the unique olution y = x 2 1. Since f y = 2 + in y in y, f y = co y co y, the hypothee of Theorem 1 are atified and the finite difference method converge for all h 2, or equivalently for n 1. Moreover, by uing the computing notebook NBoundaryD.nb for n = 5, we have the following table of the approximation of the analytic olution 33) 34) 35) 36) Starting olution Max Abolute Error for = 1 Max Abolute Error for = 2 y x)= y x)= y x)= x y x)= in 2x y x)= x ) Simulation 2. In thi imulation the limit and the capacitie of Theorem 1 and 4 are dicued. The function y = x 4x 2 1 ) i the only olution of the following BVP with mixed linear boundary condition ee Propoition 5) ) y = in y co y + co x 4x 2 1 in 1 12x 2) + 24x, x [ 1/2, 1/2] y 1/2) y 1/2) = 2, y1/2) + y 38) 1/2) = 2. Since f y = in y, f y = co y, the hypothee of Theorem 4 are atified for K = 1, m = 1/2, and for any λ : 1/2 + e/2 λ < 1 the relation 16) i verified. Then, the finite difference method converge for all tep h to

11 2486 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) the unique olution that they can be computed with an iterative Newton-like method by tarting with arbitrary initial gue. By uing the notebook NBoundaryM.nb, we experimentally find the reult of Theorem 4, a it i hown in the following table Starting olution y x)= 0 y x)= x Max-Min Ab. Error n = Max-Min Ab. Error n = Max-Min Ab. Error n = ) We ee that the approximate olution doe not change in a ignificant way on increaing the number > 1 of iteration. If we ubtitute the boundary condition 38) 2 with the following one y 1/2) = 2, y1/2) + y 1/2) = 2. 40) then there not exit any value of λ 0, 1) uch that 16) i valid, although the hypothee of the Propoition 5 are atified. NBoundaryM.nb how that the finite difference method upplie reult cloe to thoe of Table 39), or better till it give reult with a lower minimum abolute error ee Table 41)). Finally, the independence of the initial approximation in FindRoot till perit. Starting olution y x)= 0 y x)= x Max-Min Ab. Error n = Max-Min Ab. Error n = Max-Min Ab. Error n = ) It i not poible to apply the Theorem 1 to the Dirichlet problem for the Eq. 38) 1 with the following boundary condition y 1/2) = 0, y1/2) = 0. However, the right-hand ide in 38) 1 i bounded in [a, b] R 2 o that the problem ha at leat a olution which the notebook NBoundaryD.nb give with high accuracy ee Table 42)) Starting olution y x)= 0 y x)= x Max-Min Ab. Error n = Max-Min Ab. Error n = Max-Min Ab. Error n = ) Simulation 3. We conider the following problem which arie in the tudy of finite deflection of an elatic tring under a tranvere load y = 1 + a 2 y ) 2), x [0, 1] 43) y0) = 0, y1) = 0, which admit the unique olution ) / co a x 1/2) yx) = ln a 2. co a/2 For a = 1/7, in [26] it i proved that the dicrete problem, aociated to 43), ha a olution y π atifying 0 y π x k ) 4 xk 2, k = 0,..., n for h ufficiently mall. By uing the notebook NBoundaryD.nb for

12 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) Fig. 1a. Graphic of y = yx, j). Fig. 1b. Abolute error in the knot. Fig. 2a. Graphic of y = yx, j). Fig. 2b. Abolute error in the knot. n = 20, = 4, y x) = 0, we have the following reult ee Fig. 1a and 1b). By running again the notebook NBoundaryD.nb for y x) = x/8 we get the reult in Fig. 2a and 2b.

13 2488 S. Cuomo, A. Maraco / Computer and Mathematic with Application ) Fig. 3a. Graphic of yx, j). Fig. 3b. Graphic of yx, j). Simulation 4. In [14] and [4], the author prove that both the following fully nonlinear BVP and it dicrete problem admit a olution y = in x 2 co x y 2) in y y ) 5, x [0, 1] [ y 0) y 0) + y 2 1) y 1) ) ] / 2 10 = 0, 44) y 0) + y 0) + 6y1) + y 1) + in [ y 0) y 1) ] = 0. Moreover, the olution yx) and y π x k ) of the above problem are uch that π/2 yx), y π x k ) π/2, 2 y x), y π x k) 2, k = 0,..., n. By reorting to the program NBoundaryM.nb, for n = 40, = 4, we obtain the following reult Fig. 3a and 3b) where in the firt cae we choe y x) = 0 and in the econd one y x) = x/8. Moreover, comparing the numerical reult we have obtained in the above computation, we can oberve that the difference between them i of the order Reference [1] R. Gaine, Difference equation aociated with boundary value problem for econd order nonlinear ordinary differential equation, SIAM J. Numer. Anal ) [2] R.P. Agarwal, On multipoint boundary value problem for dicrete equation, J. Math. Anal. Appl ) [3] H.B. Thompon, Topological method for ome boundary value problem, J. Comput. Math. Appl ) [4] J. Henderon, H.B. Thompon, Difference equation aociated with fully nonlinear boundary value problem for econd order ordinary differential equation, J. Differential Equation Appl. 7 2) 2001) [5] J. Henderon, H.B. Thompon, Exitence of multiple olution for econd-order dicrete boundary value problem, Comput. Math. Appl ) 2002) [6] H.B. Thompon, Exitence of multiple olution for finite difference approximation to econd-order boundary value problem, Nonlinear Anal. 53 1) 2003) [7] H.B. Keller, Numerical Method for Two-Point Boundary-Value Problem, Blaidell Publihing Co. Ginn and Co., Waltham, Ma.-Toronto, Ont.-London, 1968, viii+184 pp. [8] U.M. Acher, R.M.M. Mattjeij, R.D. Ruel, Numerical Solution of Boundary Value Problem for Ordinary Differential Equation, in: Claic in Applied Mathematic, vol. 13, Society for Indutrial and Applied Mathematic SIAM), Philadelphia, PA, ISBN: , 1995, xxvi+595 pp. [9] A. Maraco, A. Romano, Scientific Computing with Mathematica: Mathematical Problem for Ordinary Differential Equation, in: Modeling and Simulation in Science, Engineering and Technology, Birkhäuer Boton, Inc., Boton, MA, ISBN: , 2001, xiv+270 pp.

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