A Novel Approximation Method for Solving Ordinary Differential Equations Using the Representation of Ball Curves

Numerical methods are frequently developed to investigate concepts for approximately solving ordinary differential equations (ODEs). To achieve minimal error and higher accuracy in approximate solutions, researchers have focused on developing algorithms using various numerical techniques. This study proposes the application of Ball curves, specifically the Said–Ball curve, for estimating solutions to higher-order ODEs. To obtain the best control points of the Said–Ball curve, the least squares method is used. These control points are calculated by minimizing the residual error through the sum of the squares of the residual functions. To demonstrate the proposed method, several boundary value problems are presented, and their performance is compared with existing methods in terms of error accuracy. The numerical results indicate that the proposed method improves error accuracy compared to existing studies, including those employing Bézier curves and the steepest descent method.