Untitled

Abstract

B-splines have been widely used to approximate solutions to differential equations. In this paper, a class of singular boundary value problems are treated by using extended cubic uniform B-spline approximations. The advantage of using an extended cubic B-spline rather the ordinary B-spline is that it introduces one additional free parameter. For a number of examples where exact solutions are known, the solutions obtained using the extended B-splines are found to be better approximations than those obtained using ordinary B-splines.

A new extended B-spline approximation technique for second order singular boundary value problems arising in physiology

“…where V k and v k (x) are used for approximate and exact solutions respectively. Moreover, the numerical results are compared with CPCBSM [2], cubic basis spline method (CBSM) [4], extended cubic B-spline method (ECBM) [10], quartic B-spline method (QBSM) [11], new CBSM (NCBSM) [12], variational iteration method (VIM) [26], Green's functions associated with improved form of Adomian decomposition method (GIADM) [30], adomian decomposition method (ADM) [31], semi numerical method (SNM) [31], RKM [33], and IDTM [35]. The resulting equation is also known as Bessel's equation of order zero.…”

“…Abukhaled et al [2] used B-spline functions with chebyshev polynomials (CPCBSM) for solving SBVPs. Goh et al [10,11] investigated the approximate solution of linear SBVPs with the help of quartic and extended uniform B-spline functions. The authors in [14], proposed a new quartic B-spline approximation technique for numerical investigation of third order SBVPs.…”

Extended cubic B-spline method for solving a linear system of second-order boundary value problems

“…Similarly, for , cubic B-spline and extended cubic B-spline curves have the same properties: symmetry, geometric invariability, and convex hull Goh et al. (2011). The values of and its derivatives , at the nodal points are tabulated in Table.…”

Extended cubic B-spline method for solving a system of non-linear second-order boundary value problems

“…The quasilinearization method is utilized to reduce the given nonlinear problem (1.1) to a sequence of linear problems. The method used is adopted from [1,15]. From (1.1), we have…”

“…The cubic B-spline was developed to obtain an extended cubic B-spline with degree 4, 5, and 6 in [29]. Extended cubic B-spline method (ECBSM) was developed to solve linear and singular two point boundary value problems involving ordinary differential equations in [1,15]. The same approach was also used to solve a linear system of second order boundary value problems by the present authors [17].…”