Obálka

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

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Vydáno v:Mathematics vol. 12, no. 12 (2024), p. 1870
Hlavní autor: Ana Laura Mendonça Almeida Magalhães
Další autoři: Pedro Paiva Brito, Geraldo Pedro da Silva Lamon, Pedro Américo Almeida Magalhães Júnior, Cristina Almeida Magalhães, Pedro Henrique Mendonça Almeida Magalhães, Pedro Américo Almeida Magalhães
Vydáno:
MDPI AG
Témata:
Boundary conditions
Variables
Approximation
Numerical analysis
Methods
Partial differential equations
Difference equations
Mathematical analysis
Differential equations
Finite difference method
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Abstrakt:The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain.
ISSN:2227-7390
DOI:10.3390/math12121870
Zdroj:Engineering Database