Function Quadrature Using Quartic Spline Interpolation

This research paper explores the application of quartic splines for numerical integration, particularly in situations with curved boundaries in two-dimensional space. The paper dives into the concept of quartic splines, which are piecewise polynomial functions where each piece is a polynomial of the fourth degree. These splines are designed to ensure continuity of the function itself, its first, second, and third derivatives across the boundaries between the pieces. The core technique involves using quartic splines to approximate the curved boundary of a two-dimensional domain. This allows for the calculation of integrals defined over that domain by transforming it into a simpler shape (often a rectangle) where traditional numerical integration methods like Gauss-Legendre quadrature can be applied effectively. The paper validates the method by applying it to various curved domains and comparing the obtained integral values with known analytical solutions (if available). The absolute errors are calculated and tabulated, demonstrating the accuracy of the approach. The paper showcases the efficiency of the method by solving numerical integration problems from existing literature. These examples serve as evidence that the method is effective and can handle real-world scenarios. A significant advantage of this method is its ability to handle situations where only coordinate data for the boundary is available. Quartic splines can accurately reconstruct the curved boundary from these coordinates, enabling the integration process.