Solving Initial Value Problems for Ordinary Differential Equations Using Fourth-Order Adams Explicit Formula

The fourth-order Adams explicit formula provides a computationally effective approach for solving initial value problems in ordinary differential equations (ODEs). Implementation typically begins by transforming higher-order ODEs into a system of first-order equations through variable substitution. The algorithm then employs a multi-step method using four previous solution points (y_n, y_n-1, y_n-2, y_n-3) to predict the next value y_n+1 through weighted function evaluations. Key computational aspects include: - Utilizing previously computed function values at equidistant steps to construct the integration formula - Applying the explicit Adams-Bashforth coefficients for weighted summation of derivative terms - Maintaining numerical stability through appropriate step size selection This method delivers enhanced accuracy for various ODE problems, particularly initial value scenarios where solution history is available. The technique finds practical application in simulating physical system dynamics, such as trajectory computation through Newton's equations of motion, where it efficiently models continuous system evolution with reduced computational overhead compared to single-step methods.