A fourth-order orthogonal spline collocation method to fourth-order boundary value problems

Abstract

In this article, we use orthogonal spline collocation methods (OSCM) for the fourth-order linear and nonlinear boundary value problems (BVPs). Cubic monomial basis functions and piecewise Hermite cubic basis functions are used to approximate the solution. We establish the existence and uniqueness solution to the discrete problem. We discuss dynamics of the stationary Swift–Hohenberg equation for different values of α. Finally, we perform some numerical experiments and using grid refinement analysis, we compute the order of convergence of the numerical method. Comparative to existing methods, we show that the OSCM gives optimal order of convergence for $\left|\right|u-u_h|{|}_{L^p},p=2,\infty$ norms and superconvergent result for $\left|\right|u\prime -u{\prime }_h|{|}_{L^{\infty }}$-norm at the knots with minimal computational cost.

Keywords:

• Orthogonal cubic spline collocation methods (OCSCM)
• fourth-order linear and nonlinear boundary value problem
• Swift–Hohenberg equation
• cubic monomial basis functions
• piecewise Hermite cubic basis functions
• almost block diagonal (ABD) matrix

Acknowledgments

The authors thank Prof. Graeme Fairweather and Prof. Amiya Kumar Pani for many fruitful discussions. The authors also thank Mr. Rahul Nath for helping in the computation.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by grants from National Boardfor Higher Mathematics (Grant No. 2/48(36)/2016/NBHM/4529).