Finite Difference Approaches to Ordinary Differential Equations

Abstract

Finite difference methods constitute a core class of numerical techniques for approximating solutions of ordinary differential equations (ODEs). This paper presents a rigorous and systematic treatment of finite difference schemes for first- and second-order ODEs, emphasizing theoretical properties and computational performance. Forward, backward, and central difference discretizations are analyzed in terms of consistency, stability, and convergence. Formal theorems with proofs are provided for linear problems. Hypotheses concerning accuracy and convergence order are empirically tested through computational experiments on benchmark initial and boundary value problems. Numerical results confirm theoretical predictions, demonstrating first-order convergence for Euler-based schemes and second-order convergence for central difference formulations. The study reinforces the foundational role of finite difference methods in scientific computing while highlighting accuracy–stability trade-offs relevant to modern applications.