Quartic B-Spline Collocation Method for Fifth Order Boundary Value Problems

Abstract

A finite element method involving collocation method with quartic B-splines as basis functions have been developed to solve fifth order boundary value problems. The fifth order and fourth order derivatives for the dependent variable are approximated by the central differences of third 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 two non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions 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.

An Enhanced Quartic B-spline Method for a Class of Non-linear Fifth-Order Boundary Value Problems

“…Example (4.1) has been solved by the sextic spline method (SSM) in [5][6][7][8], non-polynomial sextic spline method (NSSM) in [9][10][11][12], cubic spline method (CSM) in [13], quartic spline method (QSM) in [14][15][16], and finite-difference method (FDM) in [27]. We collect their respective maximum absolute errors in Table 3.…”

“…It can provide numerical approximations of y (k) (x) (k = 0, 1, 2, 3, 4) with O(h 2 ) errors. In [14,15], quartic spline is used. However, the methods can only provide the numerical solutions at the isolated knots with O(h 2 ) errors.…”

“…Furthermore, the methods can not provide the numerical derivatives. Obviously, these methods in [13][14][15] need to be improved. Later, we studied a new and different quartic spline method in [16].…”

Collocation Method for Fifth Order Boundary Value Problems by Using Quintic B-splines

“…Lamnii et al [14] developed the sextic spline collocation method to solve a special case of fifth order boundary value problems. Kasi Viswanadham and Showri Raju [15] developed the quartic B-spline collocation method to solve a general fifth order boundary value problem. Kasi Viswanadham and Murali [16] presented quintic B-spline Galerkin method to solve a special case of fifth order boundary value problem.…”

Numerical Solution of Fourth Order Boundary Value Problems by Petrov-Galerkin Method with Cubic B-splines as basis Functions and Quintic B-Splines as Weight Functions

“…Over the years, many researchers have worked on boundary value problems by using different methods for numerical solutions [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. So far various numerical methods such as Cubic spline method [2], Iterative methods [3], Perturbed collocation method [4], Modified decomposition method [5], Decomposition method [6], Differential transform method [7,8], A higher order B-spline collocation method [9], Homotopy perturbation method [10], Variational iteration technique [11], Sinc-Galerkin method [12], Spline techniques [13][14][15][16], Galerkin method with quintic B-splines [17], B-spline collocation method [18], Cubic B-spline collocation method [19], Galerkin method with cubic B-splines [20] have been employed to solve fourth order boundary balue problems. So far, fourth order boundary value problems have not been solved by using Petrov-Galerkin method with cubic B-splines as basis functions and quintic B-splines as weight functions .…”