NUMERIČKO RJEŠAVANJE JEDNODIMENZIONALNE TALASNE JEDNAČINE METODOM KONAČNIH RAZLIKA KORIŠTENJEM SOFTVERA MATHEMATICA
NUMERICAL SOLUTION OF ONE-DIMENSIONAL WAVE EQUATIONS BY FINITE DIFFERENCE METHOD USING MATHEMATICA SOFTWARE

Summary/Abstract: Partial differential equations (PDEs) play an important role in modeling numerous natural phenomena and technical processes. These equations most often describe how physical quantities such as temperature, pressure, density, and electromagnetic fields change in space and time. Analytical solution of PDEs is not always possible, which requires the use of various numerical methods that allow the determination of so-called approximate solutions. The paper hypothesizes the efficiency of the finite difference method within the Mathematica software. This includes numerical analysis with analytical solutions, convergence study, and evaluation of computational complexity. Special emphasis is placed on the expected contributions in terms of optimization of the method parameters and development of practical guidelines for solving PDEs in real applications. A detailed overview of PDJs is given, starting with their mathematical classification according to characteristics (elliptic, parabolic, and hyperbolic). Classical problems with fixed initial and boundary conditions and problems with variable boundaries are considered. The main part of this paper consists of the implementation of the finite difference method in Mathematica. Finite difference schemes for each type of PDJ are analyzed in detail, including explicit and implicit approximations. The influence of network parameters on the stability and accuracy of solutions is examined. The final part of this paper emphasizes the advantages of Mathematica software in increasing researcher productivity through the integration of computational, symbolic, and visualization components. The limitations of the method and software environment are discussed, indicating that a combination with other numerical methods can result in more accurate and faster solutions.linear partial differential equations; finite difference method; one-dimensional wave equation; Mathematica software;

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