The Numerical Application of Dynamic Problems Involving Mass in Motion Governed by Higher Order Oscillatory Differential Equations

Real-world problems, particularly in the sciences and engineering, are often analyzed using differential equations to understand physical phenomena. Many situations involve rates of change of independent variables, represented by derivatives, which lead to differential equations. Solving higher-order ordinary differential equations typically involves reducing them to systems of first-order equations, but this approach has challenges. To overcome these and enhance numerical methods, a novel one-step block method with eight partitions was developed for the direct solution of higher-order initial value problems. This method will target issues in physics, biology, chemistry and economics. The new method was formulated using the linear block approach and numerical analysis was ensure essential and sufficient conditions. The new method addresses second-order problems like simple harmonic motion, third-order issues such as oscillatory differential equations, and fourth-order problems like thin film dynamics. The new method demonstrates faster convergence and improved accuracy compared to existing solutions for second, third, and fourth-order oscillatory differential equations.