Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations

Many problems in computing, sciences, and engineering’s applications represented by the initial value problems (IVPs) and boundary value problems (BVPs) of ordinary differential equations (ODEs) cannot be solved explicitly. Analytical approximation methods in the form of polynomial or piecewise polynomial functions have been developed to address such problems. Among existing studies, Bézier curve and Ball curve consisting of the same geometrical structure are commonly used to approximate analytical solutions. However, Bézier curve required higher degree to find their best control points compared to Ball curve, which increased computational burden, thus affecting the accuracy. Moreover, the least squares method (LSM) is incorporated within the Bézier curve algorithm to minimize errors during the determination of control points. Therefore, by taking advantages of this strength, this study intends to incorporate Ball curve with LSM. The main objective of the study is to develop approximation methods base on the Ball curve using LSMfor both linear and nonlinear IVPs and BVPs of ODEs. The best control points of Ball curve are determined by minimising the sum of the squared residuals of the control functions. Following this, the convergence analysis of the proposed methods is conducted theoretically and numerically. Three Ball curves with LSM have been developed to solve eleven (11) numerical examples of various order of ODEs: DP Ball curves (DPBC), Said Ball curves (SBC), and Wang Ball curves (WBC). Numerical results show that all three methods performed better compared to the existing methods in terms of accuracy, with SBC as the best among the three. Furthermore, uniqueness and existence theorem are also proven for each method to verify their corresponding numerical result. In conclusion, Ball curves with LSM provided better accuracy for solving IVPs and BVPs of ODEs, with potential application in animation, computer aided graphic design, fluid dynamic, and network.