This study integrated a metaheuristic optimization method, specifically the whale optimization algorithm (WOA), to enhance the positivity and monotonicity-preserving interpolation methods by optimizing the free shape parameter. While this free parameter offers flexibility in modifying the shape of the curve, improper selection can lead to visually unpleasing results. By applying an optimization method, this study introduced a more efficient approach to determine the optimal parameter value. To achieve an optimally smooth curve, three different smoothness metrics, including arc length, strain energy, and curvature variation energy, were minimized as objective functions. The resulting curves were then compared to identify the most effective smoothness metric. Results demonstrated that WOA effectively optimized the free shape parameters, and the curvature variation energy was proven to be the best smoothness metric as it produced the smoothest interpolation. Applications of this technique were demonstrated in preserving positivity for COVID-19 death cases and ensuring monotonicity of cumulative rainfall measurements.

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.

Functional data analysis (FDA) is a method used to analyze data represented in its functional form. The method is particularly useful for exploring both curve and longitudinal data in both exploratory and inferential contexts, with minimal constraints on the parameters. In FDA, the choice of basis function is crucial for the smoothing process. However, traditional basis functions lack flexibility, limiting the ability to modify the shape of curves and accurately represent abnormal details in modern and complex datasets. This study introduced a novel and flexible data smoothing technique for interpreting functional data, employing the beta spline introduced by Barsky in 1981. The beta spline offers flexibility due to the inclusion of two shape parameters. The proposed methodology integrated the roughness penalty approach and generalized cross-validation (GCV) to identify the optimal curve that best fitted the data, ensuring appropriate parameters were considered for transforming data into a functional form. The effectiveness of the approach was assessed by analyzing the GCV color grid chart to determine the optimal curve. In contrast to existing methodologies, the proposed method enhanced flexibility by incorporating the beta spline into the smoothing procedure. This approach was anticipated to effectively handle various forms of time series data, offering improved interpretability and accuracy in data analysis, including forecasting.

Bézier curves are essential for data interpolation. However, traditional Bézier curves often fail to detect special features that may exist in a data set, such as monotonicity or convexity, leading to invalid interpolations. This study aims to improve the deficiency of Bézier curves by imposing monotonicity or convexity-preserving conditions on the shape parameter and control points. For this purpose, the quintic trigonometric Bézier curves with two shape parameters are used. These techniques constrain only one of the shape parameters, leaving the other free to provide users with more freedom and flexibility in modifying the final curve. To guarantee smooth interpolation, the curvature profiles of the curves are analyzed, which aids in selecting the optimal shape parameter values. The effectiveness of the developed schemes was evaluated by implementing real-life data and data obtained from the existing schemes. Compared with the existing schemes, the developed schemes produce low-curvature interpolation curves with unnoticeable wiggles and turns. The proposed methods also work effectively for both nonuniformly spaced data and negative-valued convex data in real-life applications. When the shape parameter is correctly chosen, the developed interpolants exhibit continuous curvature plots, assuring $C^2$ continuity.