This work purposes to establish two small numerical modifications for the Fractional Euler method (FEM) and the Modified Fractional Euler Method (MFEM) to deal with fractional initial value problems. Two such modifications, which are named Improved Modified Fractional Euler Method 1 (IMFEM 1) and Improved Modified Fractional Euler Method 2 (IMFEM 2), endeavor to further enhance FEM and MFEM in terms of attaining more accuracy. By utilizing certain theoretical results, the resultant error bounds of the proposed methods are analyzed and estimated. Several numerical comparisons are carried out to validate the efficiency of our proposed methods.

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

This paper establishes a novel fractional-order version of a recently expanded form of the Susceptible-Exposed-Infectious-Recovery (SEIR) Mpox model. This model is investigated by means of demonstrating some significant findings connected with the stability analysis and the vaccination impact, as well. In particular, we analyze the fractional-order Mpox model in terms of its invariant region, boundedness of solution, equilibria, basic reproductive number, and its elasticity. In accordance with an effective vaccine, we study the progression and dynamics of the Mpox disease in compliance with various scenarios of the vaccination ratio through the proposed fractional-order Mpox model. Accordingly, several numerical findings of the proposed model are depicted with the use of two numerical methods; the Fractional Euler Method (FEM) and Modified Fractional Euler Method (MFEM). Such findings demonstrate the influence of the fractional-order values coupled with the vaccination rate on the dynamics of the established disease model.

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator ($D{R}_{\lambda ,q}^{m,n}$) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator ($D{R}_{\lambda ,q}^{m,n}$), we establish the q-analogues of two new integral operators ($F_{\lambda ,{\gamma }_{1,}{\gamma }_{2,\dots }{\gamma }_l}^{m,n,q}$ and $G_{\lambda ,{\gamma }_{1,}{\gamma }_{2,\dots }{\gamma }_l}^{m,n,q}$), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator ($D{R}_{\lambda ,q}^{m,n}$) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature.

American scientific publishing group

We rigorously examine the concept of bipolar vague soft expert sets (BPVSESs) and their defining characteristics. Fundamental operations such as complement, union, and intersection are firmly established as foundational elements of the framework. Additionally, the notion of bipolar vague soft expert topology (BPVSET) is introduced, along with eight innovative definitions. Among these, the definition of the bipolar vague soft expert pre-open set, often abbreviated as the p-open set, is particularly significant for constructing diverse structures. This study also provides a strong and healthy articulation of the concepts of interior and closure, offering a detailed exploration of their interactions. Furthermore, it develops foundational topological concepts in bipolar vague soft expert topology by introducing and analyzing bases, sub-bases, and local bases. The notions of first and second countability in the bipolar vague soft expert topology context are formally defined, while separability is explored via countable dense sets. These results enhance the theoretical framework of bipolar vague soft expert topological space, supporting soft topological modeling under uncertainty and parameterization. A comprehensive investigation into these foundational concepts culminates in a series of compelling results concerning the basis of bipolar vague soft expert topological spaces. Finally, it introduces a decision-making framework based on bi-polar vague soft expert sets to support cancer diagnosis.