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

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  • Ana Laura Mendonça Almeida Magalhães

    (Programa de Pós-graduação em Engenharia Mecânica Pontifícia Universidade Católica de Minas Gerais (PUCMINAS), Av. Dom José Gaspar, 500 Prédio 10 Coração Eucarístico, Belo Horizonte 30535-901, MG, Brazil)

  • Pedro Paiva Brito

    (Programa de Pós-graduação em Engenharia Mecânica Pontifícia Universidade Católica de Minas Gerais (PUCMINAS), Av. Dom José Gaspar, 500 Prédio 10 Coração Eucarístico, Belo Horizonte 30535-901, MG, Brazil)

  • Geraldo Pedro da Silva Lamon

    (Programa de Pós-graduação em Engenharia Mecânica Pontifícia Universidade Católica de Minas Gerais (PUCMINAS), Av. Dom José Gaspar, 500 Prédio 10 Coração Eucarístico, Belo Horizonte 30535-901, MG, Brazil)

  • Pedro Américo Almeida Magalhães Júnior

    (Programa de Pós-graduação em Engenharia Mecânica Pontifícia Universidade Católica de Minas Gerais (PUCMINAS), Av. Dom José Gaspar, 500 Prédio 10 Coração Eucarístico, Belo Horizonte 30535-901, MG, Brazil)

  • Cristina Almeida Magalhães

    (Departamento de Engenharia Mecânica, Centro Federal de Educação Tecnológica de Minas Gerais (Cefet-MG), Av. Amazonas 7675, Nova Gameleira, Belo Horizonte 30510-000, MG, Brazil)

  • Pedro Henrique Mendonça Almeida Magalhães

    (Departamento de Engenharia Elétrica, Universidade Federal de Minas Gerais (UFMG), Av. Pres. Antônio Carlos, 6627 Pampulha, Belo Horizonte 31270-901, MG, Brazil)

  • Pedro Américo Almeida Magalhães

    (Programa de Pós-graduação em Engenharia Mecânica Pontifícia Universidade Católica de Minas Gerais (PUCMINAS), Av. Dom José Gaspar, 500 Prédio 10 Coração Eucarístico, Belo Horizonte 30535-901, MG, Brazil)

Abstract

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.

Suggested Citation

  • Ana Laura Mendonça Almeida Magalhães & Pedro Paiva Brito & Geraldo Pedro da Silva Lamon & Pedro Américo Almeida Magalhães Júnior & Cristina Almeida Magalhães & Pedro Henrique Mendonça Almeida Magalhãe, 2024. "Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain," Mathematics, MDPI, vol. 12(12), pages 1-39, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1870-:d:1415514
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    References listed on IDEAS

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    1. Abdumauvlen Berdyshev & Dossan Baigereyev & Kulzhamila Boranbek, 2023. "Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model," Mathematics, MDPI, vol. 11(17), pages 1-27, September.
    2. Mary C. Seiler & Fritz A. Seiler, 1989. "Numerical Recipes in C: The Art of Scientific Computing," Risk Analysis, John Wiley & Sons, vol. 9(3), pages 415-416, September.
    3. Qian Chen & Peng Wang & Detong Zhu & Nan-Jing Huang, 2024. "A Second-Order Finite-Difference Method for Derivative-Free Optimization," Journal of Mathematics, Hindawi, vol. 2024, pages 1-12, March.
    4. Jian Sun & Ling Wang & Dianxuan Gong, 2023. "A Joint Optimization Algorithm Based on the Optimal Shape Parameter–Gaussian Radial Basis Function Surrogate Model and Its Application," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
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