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Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes significantly to the discussion of inequalities. The concepts of symmetry and convexity are related and we can apply this because of the close link that has grown between the two in recent years. In this study, harmonic convexity, also known as harmonic s-convexity for fuzzy number valued functions (F-NV-Fs), is defined in a more thorough manner. In this paper, we extend harmonically convex F-NV-Fs and demonstrate Hermite–Hadamard (H.H) and Hermite–Hadamard Fejér (H.H. Fejér) inequalities. The findings presented here are summaries of a variety of previously published studies.

The main objective of this study is to introduce new versions of fractional integral inequalities in fuzzy fractional calculus utilizing the introduced preinvexity. Due to the behavior of its definition, the idea of preinvexity plays a significant role in the subject of inequalities. The concepts of preinvexity and symmetry have a tight connection thanks to the significant correlation that has developed between both in recent years. In this study, we attain the Hermite-Hadamard (H·H) and Hermite-Hadamard-Fejér (H·H Fejér) type inequalities for preinvex fuzzy-interval-valued functions (preinvex F·I·V·Fs) via Condition C and fuzzy Riemann–Liouville fractional integrals. Furthermore, we establish some refinements of fuzzy fractional H·H type inequality. There are also some specific examples of the reported results for various preinvex functions deduced. To support the newly introduced ideal, we have provided some nontrivial and logical examples. The results presented in this research are a significant improvement over earlier results. This paper’s awe-inspiring notions and formidable tools may energize and revitalize future research on this worthwhile and fascinating topic.

In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I∙V-F), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove Hermite–Hadamard type (𝓗–𝓗 type) inequality. Furthermore, 𝓗–𝓗 type inequalities for the product of two left and right convex I∙V-Fs have been established. Finally, for left and right convex I∙V-Fs, we found the Riemann–Liouville fractional integral Hermite–Hadamard type inequality (𝓗–𝓗 Fejér type inequality). The findings of this research show that this methodology may be applied directly and is computationally simple and precise.

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

In this study, we provide an efficient simulation to investigate the behavior of the solution to the Brusselator system (a biodynamic system) with the Rabotnov fractional-exponential (RFE) kernel fractional derivative. A system of fractional differential equations can be used to represent this model. The fractional-order derivative of a polynomial function tp is approximated in terms of the RFE kernel. In this work, we employ shifted Vieta–Lucas polynomials in the spectral collocation technique. This process transforms the mathematical model into a set of algebraic equations. By assessing the residual error function, we can confirm that the provided approach is accurate and efficient. The outcomes demonstrate the effectiveness and simplicity of the technique for accurately simulating such models.

In this study, an integer-order rabies model is converted into the fractional-order epidemic model. To this end, the Caputo fractional-order derivatives are plugged in place of the classical derivatives. The positivity and boundedness of the fractional-order mathematical model is investigated by applying Laplace transformation and its inversion. To study the qualitative behavior of the non-integer rabies model, two steady states and the basic reproductive number of the underlying model are worked out. The local and global stability is investigated at both the steady states of the fractional-order epidemic model. After analytic treatment, a structure-preserving numerical template is constructed to numerically solve the fractional-order epidemic model. Moreover, the positivity, boundedness and symmetry of the numerical scheme are examined. Lastly, numerical experiment and simulations are accomplished to substantiate the significant traits of the projected numerical design. Consequences of the study are highlighted in the closing section.