An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

Abstract

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

“…The proposed method is fourth order convergence in space and (2 − )-th order convergence in time. Various numerical schemes based on B-spline collocation methods have been applied to solve a wide variety of problems, see [24][25][26][27][28][29][30][31][32][33][34]. Besides B-spline collocation techniques, there are other methods that can be used to solve classic or fractional differential equations, such as finite difference methods [6,7,14], finite element method [35], boundary element methods [36,37], meshless methods [38,39] etc.…”

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

“…We notice that the method based on B-spline collocation methods have been used to solve a wide variety of problems, for example, see previous studies. [16][17][18][19][20][21][22][23][24][25][26] The rest of the the paper is organized as follows: In Section 2, we first describe time semi discretization of problem (1)-(4), and then describe spatial discretization based on an optimal B-spline collocation approach. The unconditional stability and convergence of the proposed scheme are presented in Section 3.…”

“…We notice that the method based on B‐spline collocation methods have been used to solve a wide variety of problems, for example, see previous studies. 16‐26 …”

An efficient numerical method based on exponential B‐spline basis functions for solving a class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

“…To the best of our knowledge, the present method has not been discussed in the numerical analysis literature for the solution of the problem under consideration. The numerical method based on B‐spline collocation method has been successfully applied to solve a wide variety of boundary value problem 39‐42 …”