Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems. 1 Introduction

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1 ISSN print, online International Journal of Nonlinear Science Vol No.3,pp Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems K.N.S. Kasi Viswanadham, Y. Showri Raju Department of Mathematics, National Institute of Technology Warangal INDIA. Received 14 November 2011, accepted 29 October 2012 Abstract: A finite element method involving collocation method with Cubic B-splines as basis functions has been developed to solve fourth order boundary value problems. The fourth order and third order derivatives for the dependent variable are approximated by the central differences of second order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on four linear and six non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of solutions of sequence of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature. Keywords: collocation method; cubic B-splines; basis functions; fourth order boundary value problems; absolute error 1 Introduction Fourth order ordinary differential equationsodes appear in several branches of applied mathematics and engineering. For example, fourth order ODEs are used to describe deformable systems. These systems include arches, beams, load bearing members like street lights in electrical engineering to robotic arms in other multi-purpose engineering systems elastic members serve as key elements for shedding or transmitting loads. Because of the pervasive presence of deformable systems in the development and application of latest technologies, there has been a continuous interest in this area of research. Solving such type of boundary value problems analytically is possible only in very rare cases. In this paper, we consider the general fourth order linear boundary value problem a 0 xy 4 x + a 1 xy x + a 2 xy x + a 3 xy x + a 4 xyx = bx, c < x < d 1 subject to boundary conditions yc = A 0, yd = B 0, y c = A 1, y d = B 1, 2 A 0, B 0, A 1, B 1 are finite real constants and a 0 x, a 1 x, a 2 x, a 3 x, a 4 x and bx are all continuous functions defined on the interval c, d]. The existence and uniqueness of the solution for these types of problems has been discussed in Agarwal1]. Over the years, there are several authors who worked on these types of boundary value problems by using different methods. For example, El-Gamel et al.2] used Sinc-Galerkin method to solve fourth order boundary value problems. Quintic splines have been used to solve a fourth order boundary value problem 3 5], as Usmani6] used Quartic splines to solve a fourth order boundary value problem. Noor and Mohyud-din7] used variational iteration method to solve fourth order boundary value problems. Onyejekwe8] used green element method to solve the fourth order ordinary differential equations. Taiwo9] used cubic spline collocation tau method to solve fourth order linear ordinary differential equations. The objective of this paper is to present a simple technique in solving a fourth order boundary value problem 1-2. In the present paper, cubic B-splines as basis functions have been used to solve the boundary value problems of the type 1-2. Corresponding author. address:kasi nitw@yahoo.co.in Copyright c World Academic Press, World Academic Union IJNS /673

2 K. Viswanadham, Y. Showri Raju: Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems 337 In section 2 of this paper, the justification for using the collocation method has been mentioned. In section 3, the definition of cubic B-splines has been described. In section 4, description of the collocation method with cubic B-splines as basis functions has been presented and in section 5, solution procedure to find the nodal parameters is presented. In section 6, numerical examples of both linear and non-linear boundary value problems are presented. The solution of a nonlinear boundary value problem has been obtained as the limit of solutions of sequence of linear boundary value problems generated by quasilinearization technique 10]. Finally, the last section is dealt with conclusions of the paper. 2 Justification for using collocation method In finite element method FEM the approximate solution can be written as a linear combination of basis functions which constitute a basis for the approximation space under consideration. FEM involves variational methods like Ritz s approach, Galerkin s approach, least squares method and collocation method etc. The collocation method seeks an approximate solution by requiring the residual of the differential equation to be identically zero at N selected points in the given space variable domain N is the number of basis functions in the basis 11]. That means, to get an accurate solution by the collocation method one needs a set of basis functions which in number match with the number of collocation points selected in the given space variable domain. Further, the collocation method is the easiest to implement among the variational methods of FEM. The collocation method with cubic B-splines as basis functions has been used to solve a second order boundary value problem 12]. For the case of a single differential equation, it is shown in Douglas and Dupont 13] that the cubic B-splines yield fourth order accurate results. Hence this motivated us to use the collocation method to solve a fourth order boundary value problem of type 1-2 with cubic B-splines. 3 Definition of cubic B-splines The existence of the cubic spline interpolate sx to a function in a closed interval a,b] for spaced knots need not be evenly spaced a = x 0 < x 1 < x 2 <... < x < x n = b is established by constructing it. The construction of sx is done with the help of the cubic B-Splines. Introducing six additional knots x 3, x 2, x 1, x n+1, x n+2 and x n+3 such that x 3 < x 2 < x 1 < x 0 and x n < x n+1 < x n+2 < x n+3. Now the cubic B-splines, given in Cox14] and Boor15], are defined by i+2 x r x 3 + B i x = π r=i 2 1 if x x i 2, x i+2 ] x r 0 otherwise x r x 3 + = { x r x 3, if x r x 0, if x r x and πx = i+2 r=i 2 x x r. It can be shown the set {B 1 x, B 0 x,..., B n x, B n+1 x} forms a basis for the space S 3 π of cubic polynomial splines 12]. Schoenberg16] has proved that the cubic B-splines are the unique non-zero splines of smallest compact support with knots at x 3 < x 2 < x 1 < x 0 <... < x n < x n+1 < x n+2 < x n+3. 4 Description of the method To solve the boundary value problem 1-2 by the collocation method with cubic B-splines as basis functions, we define the approximation for yx as yx = n+1 j= 1 α j B j x 3 IJNS homepage:

3 338 International Journal of Nonlinear Science, Vol , No.3, pp α j s are the nodal parameters to be determined. In the present method, the internal mesh points are selected as the collocation points. In collocation method, the number of basis functions in the approximation should match with the number of selected collocation points 11]. Here the number of basis functions in the approximation is n + 3, as the number of selected collocation points is n 1. So, there is a need to redefine the basis functions into a new set of basis functions which in number match with the number of selected collocation points. The procedure for redefining the basis functions is as follows: Using the cubic B-splines described in section 3 and the Dirichlet boundary conditions of 2, we get the approximate solution at the boundary points as 1 yc = yx 0 = α j B j x 0 = A 0 4 and yd = yx n = j= 1 n+1 j= α j B j x n = B 0 5 Eliminating α 1,α n+1 from the equations 3,4 and 5, we get the approximation for yx as w 1 x = yx = w 1 x + n α j P j x 6 j=0 A 0 B 1 x 0 B B 0 1x + B n+1 x n B n+1x B j x B jx 0 B 1 x 0 B 1x, for j = 0, 1 P j x = B j x, for j = 2, 3,..., n 2 B j x B jx n B n+1 x n B n+1x, for j = n 1, n. Using the Neumann boundary conditions of 2 to the approximation yx in 6, we get A 1 = y x 0 = w 1x 0 + α 0 P 0x 0 + α 1 P 1x 0 7 B 1 = y x n = w 1x n + α P x n + α n P nx n. 8 Now, eliminating α 0, α n from the equations 6,7 and 8, we get the approximation for yx as yx = wx + α j Bj x 9 wx = w 1 x + A 1 w 1x 0 P 0 x 0 P 0 x + B 1 w 1x n P nx P n x n and P j x P j x 0 P 0 B x 0 P 0x, for j = 1 j x = P j x, for j = 2, 3,..., n 2 P j x P j x n P nx n P nx, for j = n 1. } Now the number of new basis functions { Bj x, j = 1, 2,..., n 1 in the approximation for yx in 9 is matching with the number of selected collocation points. Since the approximation for yx in 9 is a cubic approximation, let us approximate y and y 4 at the selected collocated points with central differences as y i = y i+1 y i 1 and y 4 i = y i+1 2y i + y i 1 h 2 10 IJNS for contribution: editor@nonlinearscience.org.uk

4 K. Viswanadham, Y. Showri Raju: Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems 339 y i = yx i = wx i + α j Bj x i 11 Now applying collocation method to 1, we get a 0 x i y 4 i + a 1 x i y i + a 2 x i y i + a 3 x i y i + a 4 x i y i = bx i for i = 1, 2,..., n 1 12 Using 10 and 11 in 12, we get a 0 x i h 2 w x i a 1x i { α j Bj xi+1 2 w x i + w x i a 2 x i w x i + α j Bj xi } α j Bj xi + w x i 1 + { }] α j Bj xi+1 w x i 1 + α j Bj xi 1 ] ] + a 3 x i w x i + α j Bj xi ] α j Bj xi 1 ] + a 4 x i wx i + α j Bj x i = bx i for i = 1, 2,..., n Rearranging the terms and writing the system of equations 13 in matrix form, we get Aα = B 14 A = a ij ]; a ij = B a0 x i j xi 1 h 2 a 1x i + B a0 x i j xi+1 h 2 + a 1x i + B j xi 2 a 0x i h 2 + B j xi a 2 x i + B j xi a 3 x i + B j x i a 4 x i for i = 1, 2,..., n 1, j = 1, 2,..., n 1 15 B = B i ]; B i = bx i w a0 x i x i 1 h 2 a 1x i + w x i 2 a 0x i h 2 + w a0 x i x i+1 h 2 + a 1x i ] + w x i a 2 x i + w x i a 3 x i + wx i a 4 x i for i = 1, 2,..., n 1 16 and α = α 1, α 2,..., α ] T. 5 Solution procedure to find the nodal parameters The basis function B i x is defined only in the interval x i 2, x i+2 ] and outside of this interval it is zero. Also at the end points of the interval x i 2, x i+2 ] the basis function B i x vanishes. Therefore, B i x is having non-vanishing values at the mesh points x i 1, x i, x i+1 and zero at the other mesh points. The first two derivatives of B i x also have the same nature at the mesh points as in the case of B i x. Using these facts, we can say that the matrix A defined in 15 is a five band matrix. Therefore, the system of equations 14 is a five band system in α i s. The nodal parameters α i s can be obtained by using band matrix solution package. IJNS homepage:

5 340 International Journal of Nonlinear Science, Vol , No.3, pp Numerical results To test the efficiency of the proposed method, four linear and six non-linear problems are presented. Numerical results for each problem are presented in a table and compared with the exact solutions available in the literature. Example 1 Consider the linear boundary value problem y 4 = y + y + e x x 3, 0 < x < 1 17 subject to y0 = 1, y 0 = 0, y1 = 0, y 1 = e. The exact solution is y = 1 xe x. The proposed method is tested on this problem the domain 0,1] is divided into 10 equal subintervals. Numerical results for this problem are shown in Table 1. The maximum absolute error obtained by the proposed method is Table 1: Numerical results for Example E E E E E E E E E E E E E E E E E E-06 Table 2: Numerical results for Example E E E E E E E E E E E E E E E E E E-06 Example 2 Consider the linear boundary value problem y 4 + 4y = 1, 1 < x < 1 18 subject to y 1 = y1 = 0, y 1 = y sinh 2 sin 2 1 = 4cosh 2 + cos 2. sinh 1 sin 1 sinh x sin x + cosh 1 cos 1 cosh x cos x ] The exact solution is y = The proposed method is cosh 2 + cos 2 tested on this problem the domain -1,1] is divided into 10 equal subintervals. Numerical results for this problem are shown in Table 2. The maximum absolute error obtained by the proposed method is Example 3 Consider the linear boundary value problem y 4 + xy = e x 8 + 7x + x 3, 0 < x < 1 19 subject to y0 = 0, y 0 = 1, y1 = 0, y 1 = e. The exact solution is y = x1 xe x. The proposed method is tested on this problem the domain 0,1] is divided into 10 equal subintervals. Numerical results for this problem are shown in Table 3. The maximum absolute error obtained by the proposed method is Example 4 Consider the linear boundary value problem y 4 y = 42x cos x + 3 sin x, 1 < x < 1 20 subject to y 1 = y1 = 0, y 1 = y 1 = 2 sin 1. The exact solution is y = x 2 1 sin x. The proposed method is tested on this problem the domain -1,1] is divided into 10 equal subintervals. Numerical results for this problem are shown in Table 4. The maximum absolute error obtained by the proposed method is IJNS for contribution: editor@nonlinearscience.org.uk

6 K. Viswanadham, Y. Showri Raju: Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems 341 Table 3: Numerical results for Example E E E E E E E E E E E E E E E E E E-06 Table 4: Numerical results for Example E E E E E E E E E E E E E E E E E E-06 Example 5 Consider the nonlinear boundary value problem y 4 = y 2 x x 9 4x 8 4x 7 + 8x 6 4x x 48, 0 < x < 1 21 subject to y0 = y 0 = 0, y1 = y 1 = 1. The exact solution is y = x 5 2x 4 + 2x 2. This nonlinear boundary value problem is converted into a sequence of linear boundary value problems generated by quasilinearization technique 10] as y 4 n+1 2y n]y n+1 = x x 9 4x 8 4x 7 + 8x 6 4x x 48 y n ] 2, n = 0, 1, 2, subject to y n+1 0 = y n+1 0 = 0, y n+11 = y n+1 1 = 1. Here y n+1 is the n + 1 th approximation for y. The domain 0,1] is divided into 10 equal subintervals and the proposed method is applied to the sequence of problems 22. Numerical results for this problem are presented in Table 5. The maximum absolute error obtained by the proposed method is Table 5: Numerical results for Example E E E E E E E E E E E E E E E E E E-06 Table 6: Numerical results for Example E E E E E E E E E E E E E E E E E E-07 Example 6 Consider the nonlinear boundary value problem y 4 = sin x + sin 2 x y ] 2, 0 < x < 1 23 subject to y0 = 0, y 0 = 1, y1 = sin 1, y 1 = cos 1. The exact solution is y = sin x. This nonlinear boundary value problem is converted into a sequence of linear boundary value problems generated by quasilinearization technique 10] as y 4 n+1 + 2y n]y n+1 = sin x + sin2 x + yn ]2, n = 0, 1, 2, subject to y n+1 0 = 0, y n+1 0 = 1, y n+11 = sin 1, y n+1 1 = cos 1. Here y n+1 is the n + 1 th approximation for y. The domain 0,1] is divided into 10 equal subintervals and the proposed method is applied to the sequence of problems 24. Numerical results for this problem are presented in Table 6. The maximum absolute error obtained by the proposed method is IJNS homepage:

7 342 International Journal of Nonlinear Science, Vol , No.3, pp Example 7 Consider the nonlinear boundary value problem y 4 6e 4y = x 4, 0 < x < 1 25 subject to y0 = 0, y 0 = 1, y1 = ln 2, y 1 = 0.5. The exact solution is y = ln1 + x. This nonlinear boundary value problem is converted into a sequence of linear boundary value problems generated by quasilinearization technique 10] as y 4 n e 4y n ]y n+1 = x 4 + e 4y n 24y n + 6], n = 0, 1, 2, subject to y n+1 0 = 0, y n+1 0 = 1, y n+11 = ln 2, y n+1 1 = 0.5. Here y n+1 is the n + 1 th approximation for y. The domain 0,1] is divided into 10 equal subintervals and the proposed method is applied to the sequence of problems 26. Numerical results for this problem are presented in Table 7. The maximum absolute error obtained by the proposed method is Table 7: Numerical results for Example E E E E E E E E E E E E E E E E E E-00 Table 8: Numerical results for Example E E E E E E E E E-07 Example 8 Consider the nonlinear boundary value problem y 4 y 2 + yy = 4x 2 + e x 1 + x 2 4x, 0 < x < 1 27 subject to y0 = 1, y 0 = 1, y1 = 1 + e, y 1 = 2 + e. The exact solution is y = x 2 + e x. This nonlinear boundary value problem is converted into a sequence of linear boundary value problems generated by quasilinearization technique 10] as y 4 n+1 + y n]y n+1 2y n]y n+1 + y n ]y n+1 = 4x 2 + e x 1 + x 2 4x y n 2 + y n y n, n = 0, 1, 2, subject to y n+1 0 = 1, y n+1 0 = 1, y n+11 = 1 + e, y n+1 1 = 2 + e. Here y n+1 is the n + 1 th approximation for y. The domain 0,1] is divided into 10 equal subintervals and the proposed method is applied to the sequence of problems 28. Numerical results for this problem are presented in Table 8. The maximum absolute error obtained by the proposed method is Example 9 Consider the nonlinear boundary value problem y 4 e x y yy = e x 2x 1, 0 < x < 1 29 subject to y0 = 1, y 0 = 1, y1 = 2 e, y 1 = 2 e. The exact solution is y = 2x e x. This nonlinear boundary value problem is converted into a sequence of linear boundary value problems generated by quasilinearization technique 10] as y 4 n+1 + y n e x ]y n+1 y n ]y n+1 = 2xe x e x y n y n, n = 0, 1, 2, subject to y n+1 0 = 1, y n+1 0 = 1, y n+11 = 2 e, y n+1 1 = 2 e. Here y n+1 is the n + 1 th approximation for y. The domain 0,1] is divided into 10 equal subintervals and the proposed method is applied to the sequence of problems 30. Numerical results for this problem are presented in Table 9. The maximum absolute error obtained by the proposed method is IJNS for contribution: editor@nonlinearscience.org.uk

8 K. Viswanadham, Y. Showri Raju: Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems 343 Table 9: Numerical results for Example E E E E E E E E E E E E E E E E E E-07 Table 10: Numerical results for Example E E E E E E E E E E E E E E E E E E-05 Example 10 Consider the nonlinear boundary value problem y 4 + x2 1 + y 2 = 721 5x + x 2 5x x x 2 6, 0 < x < 1 31 subject to y0 = 0, y 0 = 0, y1 = 0, y 1 = 0. The exact solution is y = x 3 1 x 3. This nonlinear boundary value problem is converted into a sequence of linear boundary value problems generated by quasilinearization technique 10] as y 4 n+1 2x2 y n 1 + yn 2 2 y n+1 = 2x2 yn yn 2 2 x2 1 + yn x + 5x 2 + x x x 2, n = 0, 1, 2, subject to y n+1 0 = y n+1 0 = y n+11 = y n+1 1 = 0. Here y n+1 is the n + 1 th approximation for y. The domain 0,1] is divided into 10 equal subintervals and the proposed method is applied to the sequence of problems 32. Numerical results for this problem are presented in Table 10. The maximum absolute error obtained by the proposed method is Conclusions In this paper, we have developed a collocation method with cubic B-splines as basis functions to solve fourth order boundary value problems. Here we have taken internal mesh points as the collocation points. The cubic B-spline basis set has been redefined into a new set of basis functions which in number match with the number of selected collocation points. The proposed method is applied to solve several number of linear and non-linear problems to test the efficiency of the method. The numerical results obtained by the proposed method are in good agreement with the exact solutions available in the literature. The objective of this paper is to present a simple method to solve a fourth order boundary value problem and its easiness for implementation. References 1] R. P. Agarwal. Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore ] M. E. Gamel and A. I. Zayed. Sinc -Galerkin method for solving nonlinear boundary value problems. Comput. Math. Appl., : ] S. S. Siddiqi and G. Akram. Quintic spline solutions of fourth order boundary value problems. Int. J. Numer. Anal. Model., 52008: ] K. N. S. K. Viswanadham, P. M. Krishna and R. S.Koneru. Numerical Solutions of Fourth Order Boundary Value Problems by Galerkin Method with Quintic B-splines. Internatinal Journal of Nonlinear Science, : ] K. N. S. K. Viswanadham and P. M. Krishna. Quintic B-spline Collocation Method for Fourth Order Boundary Value Problems. Proceedings of 54th Congress of Indian Society Of Theoretical And Applied MechanicsAn International MeetINDIA., : IJNS homepage:

9 344 International Journal of Nonlinear Science, Vol , No.3, pp ] R. A. Usmani. The use of quartic splines in the numerical solution of a fourth order boundary value problem. Journal of Computational and Applied Mathematics, : ] M. A. Noor and S. T. Mohyud-din. An efficient method for fourth order boundary value problems. Comput. Math. Appl., : ] O. O. Onyejekwe. A green element method for fourth order ordinary differential equations. Advances in Engineering Software, : ] O. A. Taiwo and O. M. Ogunlaran. Numerical solution of fourth order linear ordinary differential equations by cubic spline collocation tau method. J. Math. Stat., 42008: ] R. E. Bellman and R. E. Kalaba. Quasilinearization and Nonlinear Boundary value problems. American Elsevier, New York ] J. N. Reddy. An introduction to the Finite Element Method. Tata Mc-GrawHill Publishing Company Limited, 3rd Edition, New Delhi ] P. M. Prenter. Splines and Variational Methods. John-Wiley and Sons, New York ] J. J. Douglas and T. Dupont. Galerkin methods for parabolic equations with nonlinear boundary conditions. Numerical Mathematics, : ] M. G. Cox. The numerical evaluation of b-splines. Jour. Inst. Mathematics and Appl., Stat, : ] C. de Boor. A Practical Guide to Splines. Springer-Verlag ] I. J. Schoenberg. On Spline Functions. MRC Report 625, University of Wisconsin IJNS for contribution: editor@nonlinearscience.org.uk