Seeking accurate solutions to complex equations? This paper explores the use of cubic splines to approximate solutions to fourth-order differential equations with boundary values, offering a practical approach for numerical analysis. The study highlights a reduction of the approximation to a five-term recurrence relationship, simplifying the computational process. Focusing on specific cases, the research demonstrates a direct link between the cubic spline approximation and a finite difference representation, defining a local truncation error to enhance precision. This connection provides valuable insights into the error bounds of the approximation method. This research offers a tool for solving boundary value problems commonly encountered in engineering and physics. By providing a simplified and accurate approximation method, it aids in simulations, modeling, and analysis across various scientific and technical domains.
Published in Communications of the ACM, this paper fits the journal’s scope by presenting a computational technique for solving complex mathematical problems. The use of cubic splines and the analysis of truncation errors are relevant to numerical analysis and computer-based simulations, aligning with the journal’s focus on computational methods.