Author(s): Dana Tahseen Abdulrahman

Abstract: The purpose of this study is to compare different numerical, analytical, and modern methods in terms of accuracy, stability, error rate, ease of use, execution time, and rigidity when solving complex differential equations. A thorough literature review methodology and Prisma analysis were used in this process, and both reveal that no single method is appropriate for all cases. The spectral and collocation method provides the highest accuracy and converges fastest when a smooth solution exists. Implicit methods (Implicit RK, BDF, FEM), such as implicit methods, present good stability when facing rigid problems with higher computational cost. Explicit methods such as RK4 and Adams-Bashforth are fast and easy to use but aren't as appropriate when using rigid problems. The PINN and Monte Carlo techniques are both flexible but slightly computationally expensive, and accuracy depends on the model's configuration and sample number. Based on the results of this study, I would recommend choosing the method based on the problem's nature: spectral if accuracy is desired, implicit if stability is desired, and explicit for preliminary results quickly.

DOI: 10.22271/maths.2025.v10.i12a.2207

Pages: 41-51 | Views: 44 | Downloads: 6

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