Generation, evaluation and documentation of numerical derivative expressions with finite difference approaches

Finite difference equations are widely used mathematical tools for a variety of scientific and engineering problems for which analytical solutions cannot be applied. One of the influential application areas in engineering disciplines is numerical derivative calculations. In this thesis, using finite difference approaches, a software application is developed that performs the derivation of numerical differentiation expressions, their evaluation for a specific set of points, and their documentation of the operation steps. Application components can be integrated into Java programming environments to support numerical derivative calculations. Derivative data can be either read from a formal file or sent from another application environment. After these data are analyzed with an LL(k) parser, a data structure called abstract syntax tree is created for Taylor series expansions. A different arrangement of the syntax tree is made depending on the type of finite difference approximation chosen in accordance with the direction of each point. One of the solutions is obtained by applying the Gauss elimination method to the system of linear equations with infinite solutions, which emerge with tree arrangements. The derived steps of the expressions determined according to the order of the derivative can be transferred to a PDF document, and the evaluation result can be done to other applications.