Summary of the Special Issue Titled “Applications of Symbolic and Soft Computations in Applied Sciences”

1. Introduction

Recently, symbolic computations and soft computing have played an increasingly important role in various domains of applied sciences. With the growing computational capabilities of modern computers, areas such as soft computing have gained significant relevance in a wide range of applications, from engineering to mathematical modeling in fields like economics, medicine, biology, and beyond.

Computer simulations based on symbolic computation techniques and soft computing approaches empower researchers and engineers to build sophisticated simulation environments that, in turn, drive scientific progress. The development of both classical numerical methods and emerging approaches, particularly those related to artificial intelligence, such as Physics-Informed Neural Networks (PINNs), is gaining momentum and influence year after year.

Given the rapid advancements and broad impact of these computational paradigms, the sharing of knowledge, methodologies, and discoveries in this area is of paramount importance within the scientific community. Not only does it support further innovation, but it also fosters interdisciplinary collaboration, which is key to solving complex real-world problems.

2. Overview of the Special Issue

Five articles have been published in the Special Issue entitled “Applications of Symbolic and Soft Computations in Applied Sciences”. We believe that each of these contributions offers valuable and interesting insights.

Study 1 in

Table 1. Summary of scientific articles in the Special Issue (citations as of 20 October 2025 according to Scopus database).

Authors	Title	Citation
1. Rafał Brociek, Mariusz Pleszczyński	Differential Transform Method and Neural Network for Solving Variational Calculus Problems	4
2. Rana Yousif, Aref Jeribi, Saad Al-Azzawi	Fractional-Order SEIRD Model for Global COVID-19 Outbreak	5
3. Hajid Alsubaie, Amin Yousefpour, Ahmed Alotaibi, Naif D. Alotaibi, Hadi Jahanshahi	Stabilization of Nonlinear Vibration of a Fractional-Order Arch MEMS Resonator Using a New Disturbance-Observer-Based Finite-Time Sliding Mode Control	15
4. Vladimir O. Rybintsev	Estimating the Performance of Computing Clusters without Accelerators Based on TOP500 Results	0
5. Edyta Hetmaniok, Mariusz Pleszczyński	Comparison of the Selected Methods Used for Solving the Ordinary Differential Equations and Their Systems	11

presents methods to solve variational calculus problems. The core of the problem is reduced to solving differential equations with specified boundary conditions. The authors introduce and compare two computational approaches: the Differential Transform Method (DTM) and the Physics-Informed Neural Network (PINN). The first method (DTM) aims to obtain an analytical solution in the form of a functional series. The second approach (PINN) utilizes a neural network to train a model, representing a relatively novel technique within the field of artificial intelligence.

Article 2 in Table 1 presents a model to investigate the spread of COVID-19. In the proposed mathematical framework, a Caputo-type fractional derivative is employed. The use of a fractional derivative instead of an integer-order derivative allows for more accurate modeling of subdiffusive phenomena due to the memory effect inherent in the Caputo derivative, which accounts for the process’s history. The article analyzes the system in terms of the boundedness, existence, uniqueness, and non-negativity of the solutions, as well as the equilibrium points. The comparison of numerical results derived from various types of fractional-order derivatives with empirical data reveals that the Caputo fractional-order derivative provides a closer fit to the real data than the other considered approaches.

Study 3 in Table 1 focuses on a novel disturbance-observer-based terminal sliding-mode control technique to stabilize and control chaos in a fractional-order arch MEMS resonator. Inclusion of control input saturation improves the practicality of the proposed method for real-world applications. A mathematical model of the fractional-order MEMS resonator is developed, followed by a comprehensive analysis of its nonlinear vibrational behavior and chaotic dynamics. The control scheme is applied to the resonator, and its performance is assessed through numerical simulations. The results demonstrate the effectiveness and robustness of the proposed approach in managing uncertain and nonlinear systems.

In study 4 in Table 1, Rybintsev analyzed the TOP500 ranking and defined a functional relationship between the performance of accelerator-free clusters and their system parameters. The analysis was based on the Linpack benchmark. The computed results were then compared with actual test data, showing that the estimation error does not exceed $2\%$ for processors of different generations and manufacturers. The model presented in the study enables accurate prediction of cluster performance when system parameters are modified, eliminating the need for complex real-world testing.

In article 5 in Table 1, Edyta Hetmaniok and Mariusz Pleszczyski focused on presenting and comparing three computational methods dedicated to solving ordinary differential equations and their systems. The comparison emphasized both the accuracy of the methods and their computational efficiency. The methods analyzed in the study include the fourth-order Runge–Kutta method (RK4), the Differential Transformation Method (DTM), and the built-in routine available in Wolfram Mathematica software (version 12).

3. Summary

The Guest Editors are deeply grateful to all the authors who contributed to the Special Issue titled “Applications of Symbolic and Soft Computations in Applied Sciences.” We also extend our sincere thanks to the articles’ reviewers, whose efforts significantly improved the quality of the publications.

We hope that all the articles are of interest to readers as working on this Special Issue has been both a rewarding and motivating experience for us.