Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

• Published in: Applied mathematics and computation Vol. 217; no. 16; pp. 6872 - 6882
• Main Authors: Liu, Li-Bin, Liu, Huan-Wen, Chen, Yanping
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.04.2011; Elsevier
• Subjects: Accuracy; Approximations and expansions; Boundaries; Boundary value problems; Convergence; Difference scheme; Differential equations; Exact sciences and technology; High accuracy; Mathematical analysis; Mathematical models; Mathematics; Neumann-type boundary condition; Nonlinearity; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Ordinary differential equations; Partial differential equations, boundary value problems; Quartic spline; Sciences and techniques of general use; Second-order boundary-value problem; Splines; High accuracy; Difference scheme; Second-order boundary-value problem; Quartic spline; Neumann-type boundary condition; Polynomial; Second order equation; Differential equation; Boundary condition; Partial differential equation; Spline approximation; Numerical approximation; Non linear equation; Numerical analysis; Boundary value problem; Applied mathematics
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.01.047

In this paper, a new difference scheme based on quartic splines is derived for solving linear and nonlinear second-order ordinary differential equations subject to Neumann-type boundary conditions. The scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary, which is superior to the well-known Numerov’s scheme with the accuracy being fourth order. Convergence analysis of the present method for linear cases is discussed. Finally, numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method.