Efficient Numerical Approaches for Solving Differential Equations with Applications in Science and Engineering
Keywords:
Numerical methods, ordinary differential equations, partial differential equations, stability analysis, convergence, Runge–Kutta method, finite difference methodAbstract
Differential equations constitute the mathematical backbone of numerous models in physics, engineering, chemistry, and biological sciences. These equations are often used to describe systems involving change, transport, diffusion, and interaction processes. However, obtaining closed-form analytical solutions for such equations is usually difficult or impossible when the governing models are nonlinear, coupled, or subject to complicated initial and boundary conditions. As a result, numerical methods have become essential tools for approximating the solutions of ordinary and partial differential equations with sufficient accuracy for practical applications.
In this paper, a detailed study of efficient numerical approaches for solving differential equations is presented. Classical numerical techniques including the Taylor series method, Picard iteration method, Runge–Kutta schemes, and finite difference methods are examined both theoretically and computationally. The derivation of these methods is briefly discussed, followed by an analysis of their accuracy, convergence, and stability properties. Representative problems arising in science and engineering are solved to demonstrate the practical performance of each method. Comparative results are presented in terms of error behavior, computational efficiency, and robustness of the numerical solutions. The study highlights that higher-order methods provide significant improvements in accuracy, while stability considerations play a dominant role in the numerical solution of time-dependent partial differential equations. The findings of this work aim to guide researchers and practitioners in selecting suitable numerical methods for real-world scientific and engineering problems.
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