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This document discusses recent trends in mathematical models and solitons used to analyze real-world problems. The focus is on the application of fractional calculus in modeling phenomena such as anomalous diffusion, non-Markovian processes, and random walk. The document provides summaries of 15 papers accepted for publication in a special issue, covering topics such as digestive system modeling, COVID-19 pandemic modeling, control systems for motors, symmetry in pine wilt disease, soliton solutions for various equations, and epidemic modeling. The papers employ various mathematical techniques and provide insights into different aspects of real-world problems. [Extracted from the article]

In this paper, we introduce a numerical and analytical study of the HIV‐1 infection of CD4+ T‐cells conformable fractional mathematical model. This model is studied analytically by Kudryashov and modified Kudryashov methods and numerically by finite difference method. A comparison between the results of the analytical and numerical methods is investigated. We also give some figures that show how accurate the solutions will be obtained from analytical and numerical methods. [ABSTRACT FROM AUTHOR]

This article developed a functional integration matrix via the Hermite wavelets and proposed a novel technique called the Hermite wavelet collocation method (HWM). Here, we studied two models: the coupled system of an ordinary differential equation (ODE) is modeled on the digestive system by considering different parameters such as sleep factor, tension, food rate, death rate, and medicine. Here, we discussed how these parameters influence the digestive system and showed them through figures and tables. Another fractional model is used on the COVID-19 pandemic. This model is defined by a system of fractional-ODEs including five variables, called S (susceptible), E (exposed), I (infected), Q (quarantined), and R (recovered). The proposed wavelet technique investigates these two models. Here, we express the modeled equation in terms of the Hermite wavelets along with the collocation scheme. Then, using the properties of wavelets, we convert the modeled equation into a system of algebraic equations. We use the Newton–Raphson method to solve these nonlinear algebraic equations. The obtained results are compared with numerical solutions and the Runge–Kutta method (R–K method), which is expressed through tables and graphs. The HWM computational time (consumes less time) is better than that of the R–K method. [ABSTRACT FROM AUTHOR]

This paper presents a newly developed method, namely, the rational sine-Gordon expansion method to find novel exact solutions to nonlinear differential equations. This method is based on the sine-Gordon expansion method. To generalize the approach, we utilize the ansatz, which is a rational function as different to a polynomial function. In this way, we have more general wave solutions for nonlinear dynamic systems. We apply this method to the (2 + 1)-dimensional conformable Zakharov–Kuznetsov modified equal width (ZK-MEW) equation and the modified regularized long wave (MRLW) equation. Some new solutions are reported. Consequently, we submit the new soliton solutions to the literature. [ABSTRACT FROM AUTHOR]

In this article, a fractional model of the Capsicum annuum (C. annuum) affected by the yellow virus through whiteflies (Bemisia tabaci) is examined. We analyzed the model by equilibrium points, reproductive number, and local and global stability. The optimal control methods are discussed to decrease the infectious B. tabaci and C. annuum by applying the Verticillium lecanii (V. lecanii) with the Atangana–Baleanu derivative. Numerical results described the population of plants and comparison values of using V. lecanni. The results show that using 60% of V. lecanni will control the spread of the yellow virus in infected B. tabaci and C. annuum in 10 days, which helps farmers to afford the costs of cultivating chili plants. [ABSTRACT FROM AUTHOR]