Hybrid B-spline collocation method with particle swarm optimization for solving linear differential problems

B-spline collocation methods were developed to provide simpler numerical solutions for differential problems. Over the years, various types of B-splines have been established, including the cubic B-spline collocation method (CBSM), cubic trigonometric B-spline collocation method (CTBSM), extended cubic B-spline collocation method (ECBSM), and cubic hybrid B-spline collocation method (CHBSM). Among these methods, CHBSM has been shown to produce the most accurate approximations due to the presence of a free parameter, $\gamma$, which allows for greater flexibility in the basis functions. However, the accuracy of the CHBSM is highly dependent on the value of $\gamma$, which must be optimized for improved results. While traditional brute-force optimization methods can achieve minimal errors, they often require significant computational time and effort. Therefore, this study has proposed using particle swarm optimization (PSO) to efficiently determine the optimal $\gamma$ value for the CHBSM. The optimized CHBSM (OCHBSM) was tested on four examples of linear two-point boundary value problems (BVPs), including a linear BVP system. For comparison, the well-established CBSM and CTBSM were also applied to the same problems. The numerical results were analyzed and compared with analytical solutions revealing that the OCHBSM provided the most accurate approximations among the methods tested. Moreover, an average improvement percentage of 99.83% was achieved across all examples, indicating that our method outperforms the compared methods significantly.