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Abstract
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The finite difference method is an important numerical method for solving both ordinary differential equations and partial differential equations. However, in recent years, this method has been losing ground to other numerical methods, such as the finite element method and boundary element method. One of the reasons for this trend is the difficulty of obtaining approximations of derivatives by finite differences in high precision and ease of use in practical problems. This article begins with a review of finite differences. The paper presents a way to deduce finite difference formulas that approximate the derivatives, mainly high-order and very precise formulas using numerical operators. In addition, this paper introduces many finite difference formulas used in the approximation of derivatives with order values of errors that far exceed those existing in other methods. The paper exposes how recent computational advances in hardware and software favor and facilitate the use of the finite difference method, especially in more complex practical problems involving differential equations. Finally, some finite difference equations obtained in a problem of determination of stresses in a thin annular disk are tested using polar coordinates and a second derivative boundary condition. As supplementary material of the paper, it follows many of the approximations of derivatives by finite differences in PDF and text format, following the table model proposed in the paper to store the finite difference equations. Thus, the article provides a large form of finite difference equations for approximating derivatives, which can be used in various applications.
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