Application of Fractional Calculus as an Interdisciplinary Modeling Framework

From a mathematical fantasy to a complex and rigorous mathematical theory, the subject of fractional calculus has applications in diverse and widespread fields of engineering and science, having a rapid growth of its applications. One of the greatest ways to make discoveries in math and science is finding answers to many new questions and interesting results. Even if fractional calculus has found an important place in science and engineering as a powerful tool for modeling complex phenomena with many excellent results, there are still some unresolved challenges. The purpose of this Special Issue was to bring together researchers from various fields of Physics, Medicine, Biology, Biosciences, Engineering, Robotics, and Signal Processing, including Applied Mathematics and to create an international and interdisciplinary framework for sharing innovative research work related to fractional calculus. This Special Issue aims to cover all theoretical and applied aspects of fractional calculus and related approaches.

This Special Issue contains nine published papers. The sequential minimal optimization (SMO) method is an algorithm for solving optimization problems that arise from the training process of support vector machines. In paper 1, the fractional-order sequential minimal optimization (FOSMO) method is proposed, based on the SMO method and fractional order calculus for classification. Results show that the FOSMO method can obtain better accuracy than the normal SMO method.

Paper 2 applied a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional-order-governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for the solution. The effectiveness and accuracy of the algorithm are verified by the numerical example.

The Wheeler–DeWitt equation for a flat and compact Friedmann–Lemaitre–Robertson–Walker cosmology at the pre-inflation epoch is studied in the contexts of the standard and fractional quantum cosmology 3. Working within the semi-classical regime and applying the Wentzel–Kramers–Brillouin (WKB) approximation, we show that some fascinating consequences are obtained for our simple fractional scenario that are completely different from their corresponding standard counterparts.

Paper 4 aims to present a general identification procedure for fractional first-order plus dead-time models. This identification method is general for processes having S-shaped step responses, where process information is collected from an open-loop step-test experiment, and has been conducted by fitting three arbitrary points on the process reaction curve. Some numerical examples are provided to show the simplicity and effectiveness of the proposed procedure.

A second-order sliding-mode control with a fractional module using an adaptive fuzzy controller is developed for an active power filter in paper 5. A second-order sliding surface using a fractional module, which can decrease the discontinuities and chattering, is designed to make the system work stably and simplify the design process. A simulation and experimental discussion illustrated good robustness and stability compared with an integer order of one.

The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated in paper 6. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The research results will provide a valuable theoretical basis for mechanical manufacturing and engineering practice.

In paper 7, the authors solved Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials, obtained by replacing x with $x^{\alpha }$, with positive $\alpha$. Fractional derivatives are in the Caputo sense. The simplicity of the method recommends it for applications in engineering and nature.

In paper 8, the q-homotopy analysis transform method was used to generate an analytical solution for the moisture content distribution in a one-dimensional vertical groundwater recharge problem. The acquired results demonstrate the efficiency and reliability of the projected scheme and are also suitable to carry out the highly nonlinear complex problems in a real-world scenario.

The existence of landscape constraints in the home range of living organisms that adopt Lévy-flight movement patterns prevents them from making arbitrarily large displacements. In work 9, the authors investigated the influence of the $\lambda$-truncated fractional-order diffusion operator on the spatial propagation of the epidemics caused by infectious diseases, where $\lambda$ is the truncation parameter.