Solving Linear Sixth-Order Boundary Value Problems by Using Hyperbolic Uniform Spline Method

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1 International Journal of Mathematical Modelling & Computations Vol. 03, No. 03, 013, Solving Linear Sixth-Order Boundary Value Problems by Using Hyperbolic Uniform Spline Method J. Dabounou a, O. El Khayyari a, and A. Lamnii a a Faculty of Science and Technology, University Hassan first, Settat, Morocco. Abstract. In this paper, a numerical method is developed for solving a linear sixth order boundary value problem (6VBP ) by using the hyperbolic uniform spline of order 3 (lower order). There is proved to be first-order convergent. Numerical results confirm the order of convergence predicted by the analysis. Received:06 May 013, Revised:17 July 011, Accepted: August 013. Keywords: Boundary value problem, Interpolation, Hyperbolic uniform spline. Index to information contained in this paper 1 Introduction Hyperbolic B-splines of order 3 3 Hyperbolic B-splines interpolation 4 Hyperbolic B-splines solutions of 6 BVP 5 Numerical results 6 Conclusion 1. Introduction When an infinite horizontal layer of fluid is heated from below and is under the action of rotation, instability sets in when this instability is an ordinary convection, the ordinary differential equation is a sixth order ordinary differential equation. Much attention have been given to solve the sixth-order boundary value problems, which have application in various branches of applied sciences. These problems are generally arise in the mathematical modeling of viscoelastic flows [4]. A spline has been widely applied for the numerical solutions of some ordinary and partial differential equations in the numerical analysis. Many authors have used numerical and approximate methods to solve sixth-order BVPs. Some of the details about the numerical methods can be found in references [9 11]. In a series of paper [, 3], Caglar et al. solved a two, three, five and six order BVPs by using third, fourth and sixth-degree splines. Lamnii et al. [6, 7] discussed a boundary-value Corresponding author. omar-hayari@hotmail.fr c 013 IAUCTB

2 170 J. Dabounou et al./ IJM C, (013) problems based on spline interpolation and quasi-interpolation with second order convergence. The numerical analysis literature contains little on the solution of boundary value problems by using the hyperbolic B-splines, generally we find the splines used in the above mentioned papers are all non hyperbolic B-splines with higher degrees, which effect the computational efficiency in pratical application. This motivates us to use hyperbolic B-splines of order 3 (lower order) to solve these problems. In this paper we study a method based on the hyperbolic B-splines of order 3 for constructing numerical solutions to six-order boundary value problems (6BVPs) of the form: with boundary conditions: y (6) (θ) + f(θ)y(θ) = g(θ), (1) y(a) = a 0, y (a) = a 1, y (a) = a, y(b) = b 0, y (b) = b 1, y (b) = b, () where f(θ) and g(θ) are given continuous functions defined in the bounded interval [a, b], a i (i = 0, 1, ), and b i (i = 0, 1, ) are real constants. The rest of paper is organized as follows. In Section, we give a explicit representation of B-splines of order 3, for more details see [1, 8]. The interpolation hyperbolic B-splines is developed in Section 3. Solution and the convergence analysis is presented in Section 4. Numerical examples are presented in Section 5.. Hyperbolic B-splines of order 3 In this section, we briefly give a explicit representation of Uniform Hyperbolic B- splines of order 3 (UH B-splines) and we give the interesting properties of UH B-splines of order 3, for more details see [1, 8]. To do this, we need the following notations. Suppose be an intergr such that 3. Let m = and h = Put θ = θ 1 = θ0 = a, θi = a + ih, i = 1...m 3, θm = θm 1 = θ m = b, b a m. the set of nots that subdivide the interval I = [a, b] uniformly. The hyperbolic tension splines space of order 3 is defined as follows (3) V = {s C 1 (I) : s [θ i,θ i+1 ] Γ 3 } where Γ 3 = span{1, cosh(θ), sinh(θ)}. The dimension of V is m and the third-order hyperbolic B-splines are given by: for i = 0, 1,..., m 5, ( sinh( θ θ i )), θ i θ < θi+1 ; cosh(h ν i, (θ) = C ) cosh(θi+ θ) cosh(θ θ i+1 ), θ i+1 θ < θ i+ ( ; sinh( θ i+3 θ )), θ i+ θ < θi+3 ; 0, otherwise.

3 J. Dabounou et al./ IJM C, (013) with the respective left and right hand side boundary hyperbolic B-splines are ν, (θ) = C { 4 ( sinh( θ+a+h ) ), θ 0 θ < θ 1 0, otherwise. 1 + cosh(h ) cosh(h θ + a) cosh( a + θ), θ0 θ < θ1 ν 1, (θ) = C ( sinh( a+h θ ) ), θ 1 θ < θ 0, otherwise. ( sinh( θ+h b ) ), θ m 4 θ < θm 3 ν m 4,(θ) = C 1 + cosh(h ) cosh(b θ) cosh(θ + h b), θm 3 θ < θm 0, otherwise. ν m 3,(θ) = C { 4 ( sinh( θ+h b ) ), θ m 3 θ < θ m 0, otherwise. where C = 1 4(sinh( h )). The hyperbolic B-splines of order 3 possess all the desirable properties of classical polynomial B-splines, see [8]. In this paper, we limit ourselves to list some of them ν i, (θ) is supported on the interval [θi, θ i+1 ]; Positivity : ν i, (θ) 0, θ [θi, θ i+1 ]; 3 Partition of unity: ν i, (θ) = 1. i= Table 1. The values of ν i, (θ) and ν i, (θ) at the nots. θ i θ i+1 θ i+ θ i+3 else ν i, (θ) 0 1 ν i, (θ) 0 1 tanh( h ) tanh( h ) 0 0 In Figure 1, we present the graphs of the hyperbolic B-spline ν i, of order 3, for i =,, 5, with = 3 and [a, b] = [0, 1] Figure 1. Hyperbolic B-spline ν i, of order 3, for = 3 and [a, b] = [0, 1].

4 17 J. Dabounou et al./ IJM C, (013) Hyperbolic B-splines interpolation In this section, we will construct an approximate of y (6) (θj ) by using Taylor series expansion. According to Schoenberg-Whitney theorem (see [? ]), for a given function y(θ) sufficiently smooth there exists a unique hyperbolic spline s(θ) = 3 i= satisfying the interpolation conditions: µ i ν i, (θ) V s(θ j ) = y(θ j ), j = 0, 1,, m ; (4) s (a) = y (a), s (b) = y (b). (5) s (a) = y (a), s (b) = y (b). (6) For j = 0, 1,, m 3, let m j = s (θ j ) and for j = 1,,, m 3, let M j = s(θ j + h ) s(θj ) + s(θ j h ) h. By using the Taylor series expansion we have: m j = s (θj ) = y (θj ) h4 y(5) (θj ) + O(h 6 ); (7) M j = y (θj ) h y(4) (θj ) h4 y(6) (θj ) + O(h 6 ); (8) Now we can apply M j to construct y (3) (θ j ) and y(4) (θ j ) for j =, 3,, m 4, y (5) (θ j ) and y(6) (θ j ) for j = 3, 4,, m 5, as follows, where the errors obtained by Taylor series expansion. M j+1 M j 1 = y (3) (θj ) + 1 h 4 h y(5) (θj ) + O(h 4 ); (9) M j+1 M j + M j 1 h = y (4) (θj ) h4 y(6) (θj ) + O(h 6 ); (10) M j+ M j+1 + M j 1 M j h 3 = y (5) (θj ) + O(h ); (11) M j+ 4M j+1 + 6M j 4M j 1 + M j h 4 = y (6) (θj ) + O(h ); (1) By Table 1 and the equations (9), (10), (11) and (1), we get:

5 J. Dabounou et al./ IJM C, (013) Table. Approximation values of y(θ j ),y (θ j ), and y (θ j ). Approximate value Representation in µ j Order y(θ j ) s(θ j ) µ j +µ j 1 O(h 4 ) y (θ j ) m j µ j 1 µ j tanh( h ) O(h 4 ) y (θ j ) M j = s j+1 s j +s j 1 h µ j 3 µ j µ j 1 +µ j h O(h ) Table 3. Approximation values of y (3) (θ j ), and y(4) (θ j ). Approximate value Representation in µ j Order y (3) (θ j ) M j+1 M j 1 h µ j 4 +µ j 3 +µ j µ j 1 µ j +µ j+1 4h 3 O(h ) y (4) (θ j ) M j+1 M j +M j 1 h µ j 4 3µ j 3 +µ j +µ j 1 3µ j +µ j+1 h 4 O(h 4 ) Table 4. Approximation values of y (5) (θ j ), and y(6) (θ j ). Approximate value Representation in µ j Order y (5) (θ j ) M j+ M j+1 +M j 1 M j h 3 µ j 5 +3µ j 4 µ j 3 5µ j +5µ j 1 +µ j 3µ j+1 +µ j+ 4h 5 O(h ) y (6) (θ j ) M j+ 4M j+1 +6M j 4M j 1 +M j h 4 µ j 5 5µ j 4 +9µ j 3 5µ j 5µ j 1 +9µ j 5µ j+1 +µ j+ h 6 O(h ) 4. Hyperbolic B-splines solutions of 6 BVP Let s(θ) = 3 i= 3 i= µ i ν i, (θ) be the approximate solution of (1) and s(θ) = µ i ν i, (θ) be the approximate spline of s(θ). Discretize (1) at θ j for j = 3,, m 5, we get : y (6) (θ j ) + f(θ j )y(θ j ) = g(θ j ), j = 3, 4,, m 5. (13) Now, by using Table and 3, the equation (13) becomes µ j 5 5µ j 4 + 9µ j 3 5µ j 5µ j 1 + 9µ j 5µ j+1 + µ j+ µ j + µ j 1 h 6 +f j = g j +O(h ) (14) where f j = f(θj ) and g j = g(θj ). Consequently, (µ j 5 5µ j 4 +9µ j 3 5µ j 5µ j 1 +9µ j 5µ j+1 +µ j+ )+f j h 6 (µ j +µ j 1 ) = g j h 6 +O(h8 ) (15) By dropping O(h 8 ) from (15), we yield a linear system with m 7 linear equations in m unnowns µ j, j =, 1,, m 3. So seven more equations are needed. On the other hand, by using the sixth boundary conditions (), we get { µ = a 0 ; µ 1 µ = a 1 tanh( h ).

6 174 J. Dabounou et al./ IJM C, (013) Thus, { µ = a 0 ; µ 1 = a 0 + a 1 tanh( h ). (16) By using a similar technique, we get: Thus, { µm 3 = b 0 ; µ m 3 µ m 4 = b 1 tanh( h ). { µm 3 = b 0 µ m 4 = b 0 b 1 tanh( h ) (17) To build the three other equations we propose the following formulas y j 1 + 4y j + y j+1 = 3 h (y j+1 y j 1 ) + O(h 4 ) (18) and y j y j + y j+1 = 1 (y j+1 y j + y j 1 ) + O(h 4 ) (19) h which can be easily demonstrated using a Taylor series expansion. By formulas (18) and (19), Table and 3, we can construct an approximate formulae for y (6) (a), as follows y (6) (a) = 10M 5 61M M 3 86M + 50M 1 45M 0 1Ω h h 4 + O(h ) (0) with Ω h = 3 h (m m 0 ). By using M 0 = a, m 0 = a 1 and tanh( h ) h for smaller h, and by turning (0), the coefficients are determined as follows 88µ 0 +3µ 1 395µ +31µ 3 71µ 4 +10µ 5 = h 6 g 0 h 6 f 0 a 0 50µ +536µ 1 +90h a 7ha 1 +O(h 8 ) (1) The two other equations we need are an approximate of y (6) (θ ) and y(6) (θ m 4 ), by using M 0 = a, M m = b and Table 4, we yield : y (6) (θ ) = M 4 4M 3 + 6M 4M 1 + a h 4 + O(h ) () and y (6) (θ m 4) = b 4M m 3 + 6M m 4 4M m 5 + M m 6 h 4 + O(h ) (3) By turning (), (3) and (13), the coefficients are determined as follows 6µ 0 5µ 1 +9µ 5µ 3 +µ 4 +f h 6 (µ 1 +µ 0 ) = h 6 g +4µ 10µ 1 h a +O(h 8 ) (4)

7 J. Dabounou et al./ IJM C, (013) and µ m 9 5µ m 8 + 9µ m 7 5µ m 6 6µ m 5 + h 6 f m 4(µ m 5 + µ m 6) = h 6 g m 4 + 4µ m 3 10µ m 4 h b + O(h 8 ) (5) Tae (1),(4),(15) and (5), we get m 4 linear equations with µ i, i = 0, 1,, m 5, as unnowns since µ, µ 1, µ m 4 and µ m 3 have been yielded from (16)and (17). Let C = [µ 0, µ 1,, µ m 5] T, C = [ µ 0, µ 1,, µ m 5] T, D = [d 0, d 1,, d m 5] T, E = [e 0, e 1,, e m 5] T and using equations (15), (16), (17) and (1), we get (A + h 6 F B)C = D + E; (6) (A + h 6 F B) C = D, (7) where A and B are the following (m 4) (m 4) matrix: A =, B = and where D, F and E are the following matrix D = h 6 g 0 h 6 f 0a 0 50µ + 536µ h a 7ha 1 h 6 g + 4µ 10µ 1 h a h 6 g 3 µ + 5µ 1 h 6 g 4 µ 1 h 6 g 5. h 6 g m 7 h 6 g m 6 µ m 4 h 6 g m 5 µ m 3 + 5µ m 4 h 6 g m 4 + 4µ m 3 10µ m 4 h b F = diag(0, f, f 3,, f m 4) and E = [O(h 8 ), O(h8 ),, O(h8 )]T. After solving the linear system (7), µ i, i = 0, 1,, m 5, µ = µ, µ 1 = µ 1, µ m 4 = µ m 4, and µ m 3 = µ m 3 will be used together to get the approximation spline solution s(θ) = µ i ν i, 3 (θ). i= In order to determine the bound of C C, we need the following lemma. Lemma 4.1 The matrix A is inversible. Proof,

8 176 J. Dabounou et al./ IJM C, (013) It suffices to prove that for all D = [d 0, d 1,, d m 5] T A D = 0, we have D = 0. Indeed, If we put R m 4 such that z(θ) = 7 j= d j+ ν j, (θ) + 3 j=m 6 0 ν j, (θ), then z (6) (θ i ) = 0, for all i = 5, 6,, m 7. On the other hand, using the fact that z is hyperbolic spline function of C 1, we deduce that z(θ) = α + β cosh(θ) + γ sinh(θ) in [θ 5, θ 6 ]. From z(6) (θ 5 ) = 0 and z (6) (θ 6 ) = 0, we have, { β cosh(θ 5 ) + γ sinh(θ 5 ) = 0; β cosh(θ 6 ) + γ sinh(θ 6 ) = 0; we deduce β = 0 and γ = 0. Consequently, z (6) (θ) = 0 and z (θ) = 0 in all the interval [θ 5, θ 6 ]. In a same way, we have in all the other subintervals of [θ 5, θ m 7 ], z (6) (θ) = 0 and z (θ) = 0. Consequently, we have z (θ5 ) = 0 z (θ6 ) = 0. z (θm 8 ) = 0 z (θm ) = 0 7 thus, d 6 d 5 tanh( h = 0 ) d 7 d 6 = 0 tanh( h ). d m 7 d m 8 tanh( h d m 6 d m 7 tanh( h ) = 0 ) = 0 so d 5 = d 6 = d 7 = = d m 7 = d m 6, we have also 88d 0 + 3d 1 395d + 31d 3 71d d 5 = 0 6d 0 5d 1 + 9d 5d 3 + d 4 = 0 9d 0 5d 1 5d + 9d 3 5d 4 + d 5 = 0 5d 0 + 9d 1 5d 5d 3 + 9d 4 5d 5 + d 6 = 0. d m 10 5d m 9 + 9d m 8 5d m 7 5d m 6 + 9d m 5 = 0 d m 9 5d m 8 + 9d m 7 5d m 6 + d m 5 = 0 finally we have d 0 = d 1 = d =... = d m 5 = 0 which in turn gives D = 0. (8) Proposition 4. If we assume that h 6 A 1 B F 1, then there exists

9 J. Dabounou et al./ IJM C, (013) a constant K which depends only of the functions f and g such that C C Kh. Proof From (6), (7) and Lemma 1, we have C C = (I + h 6 A 1 BF ) 1 A 1 E. Since E = O(h 8 ), there exists a constant K 1 such that E K 1 h 8. Consequently C C K 1 h 8 A 1 (I + h 6 A 1 BF ) 1. Using the inequality h 6 A 1 B F 1, and B =, we deduce that C C h 7 K A h 6 h K 1(b a) h. A 1 B F F Hence, s(θ) s(θ) m C C 3 Therefore, we get i= ν i, (θ) = C C O(h ). y(θ) s(θ) y(θ) s(θ) + s(θ) s(θ) O(h 4 ) + O(h ) O(h ). 5. Numerical results To test our method, we considered three examples of sixth-order boundary value problems (6BVPs) of the form ((),(1)). Example 5.1 We consider the following boundary-value problem y (6) + xy = (4 + 11x + x 3 )e x, x [0, 1]; y(0) = 0, y (0) = 1, y (0) = 0, y(1) = 0, y (1) = e, y (1) = 4e, (9) The exact solution is y(x) = x(1 x)e x. As an example, we give in Figure, the graph of the exact solution and the graph of hyperbolic-spline solution with = 5. In Table 5, we give the maximum absolute errors computed at various points of the interval [0, 1], for problem (9), and the convergence order. Table 5. Maximum absolute error and order of convergence for Problem (9) Error 6.501e e e e e e-003 Order

10 178 J. Dabounou et al./ IJM C, (013) ine solution act solution Figure. Exact solution and hyperbolic-spline solution for = 5. Example 5. Consider the following boundary-value problem y (6) y = 6(1 x) sinh(x) 1 cosh(x), x [0, 1]; y(0) = 0, y (0) = 1, y (0) =, y(1) = 0, y (1) = cosh(1), y (1) = (cosh(1) + sinh(1)), (30) The exact solution is y(x) = x(1 x) cosh(x). Result has been shown for different values of in Table 6. Table 6. Maximum absolute error and order of convergence for Problem (30) Error 4.433e e e e e e-003 Order Example 5.3 Consider the following boundary-value problem y (6) y = 6 cosh(x), x [0, 1]; y(0) = 0, y (0) = 1, y (0) =, y(1) = 0, y (1) = sinh(1), y (1) = cosh(1),. (31) The exact solution is y(x) = (1 x) sinh(x). Result has been shown for different values of in Table 7. Table 7. Maximum absolute error and order of convergence for Problem (31) Error 3.887e e e e e e-003 Order

11 J. Dabounou et al./ IJM C, (013) Conclusion Numerical results confirm the order of convergence predicted by the analysis. Experimental results demonstrate that our method is more effective for the problems where the exact solution is hyperbolic (see Tables 6 and 7). The construction of this type of approximants requires the solution of linear systems of lower order compared with the methods given in the literature, see for example [6, 7]. So, its extension to general linear and nonlinear boundary value problems using the hyperbolic (tension) B-splines of order more than 4, 5, is under progress. References [1] Amirfahrian M., Nouriani H., Interpolation by Hyperbolic B-spline Functions, International Journal of Mathematical Modelling & Computations, Vol. 01, No. 03 (011), [] Çaǧlar H., Özer M., Çaǧlar N., The numerical solution of the one-dimensional heat equation by using third degree B-spline functions, Chaos, Solitons & Fractals, 38 (008) [3] Çaǧlar N., Çaǧlar H., B-spline solution of singular boundary value problems. Appl. Math. Comput. 18 (006) [4] Davies A. R., Karageorghis A., Phillips T. N., Spectral Galerin methods for the primary two-point boundary value problem in modelling viscoelastic flows, Int. J. Numer. Methods Engng., 6 (1988) [5] de Boor C., A Practical Guide to Spline, Springer-Verlag, [6] Lamnii A., Mraoui H., sbibih D., Tijini A., zidna A., Spline Solution Of Some Linear Boundary Value Problems, Applied Mathematics E-Notes, 8 (008) [7] Lamnii A., Mraoui H., Sbibih D., Tijini A., Sextic spline solution of nonlinear fifth-order boundary value problems, Mathematics and Computers in Simulation, 77 (008) [8] Lü Y., Wang G., Yang X., Uniform hyperbolic polynomial B-spline curves, Computer Aided Geometric Design, 19 (00) [9] Noor M. A., Mohyud-Din S. T., Variational iteration technique for solving linear and nonlinear sixth order boundary value problems, Int.J. Comput. Appl. Math., (007) [10] Siddiqi S., Aram G., Septic spline solution of sixth-order boundary value problems, J. Comput. Appl.Math., 15 (008) [11] ul. Islam S., Tirmizi I. A., Haq F., Taseer S. K., Family of numerical methods based on nonpolynomial splines for solution of contact problems, Comm. Nonl. Sci. Numer. Simul., 13 (008)

12 180 J. Dabounou et al./ IJM C, (013)