In this paper, by characterizing a weak contractive condition based on using $C-$ functions and $\alpha -$ admissible multi-valued mapping of type S, we present some fixed point results for $(\alpha ,F)-$ admissible multi-valued mappings in the setting of $b-$ metric spaces. Some examples and an application are added in order to show the reliability of our obtained results. Our results amend, unify, and generalize some existing results in the literature. The scientific novelty of our main results is to take new contraction self-mappings in $b-$ metric spaces for multi-valued mappings.

This research investigates the impact of virtual laboratories (VLabs) based on artificial intelligence (AI) on developing students’ motivation toward learning mathematics. A semiexperimental approach is used to achieve the research objectives. The research sample, consisting of 80students from the seventh grade, is selected by the purposeful sampling approach. The research sample is randomly distributed into three groups: two experimental groups and one control group. The first group of 26 students is taught using the AI-based VLabs, while the second group of 27 students is taught using a VLab based on 3D visual imaging, and the third group, a control group consisting of 27 students, is taught using the traditional approach. The research instrument, a questionnaire for learning motivation, was designed after ensuring its validity and reliability. The findings of the motivation questionnaire indicate that students in the first experimental group have more motivation to learn mathematics than students in the second experimental group and the control group. The results also show that the students in the second experimental group have more motivation to learn mathematics than the students in the control group. Given the said findings, the research recommends using virtual laboratories based on artificial intelligence and all its applications in the learning process due to their impact on students’ mathematics learning.

The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der Waals equation explains the phase separation phenomenon. Noncovalent Van der Waals or dispersion forces usually have an effect on the structure, dynamics, stability, and function of molecules and materials in different branches of science, including biology, chemistry, materials science, and physics. Solutions are obtained, including dark, dark-singular, periodic wave, singular wave, and many more exact wave solutions by using the modified extended tanh function method. Using the fractional derivatives makes different solutions different from the existing solutions. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other applications of the Van der Waals equation. The solutions may be useful in distinct fields of science and civil engineering, as well as some basic physical ones like those studied in geophysics. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs by using Mathematica software. The obtained results are newer than the existing results. Stability analysis is also performed to check the stability of the concerned model. Furthermore, modulation instability is studied to study the stationary solutions of the concerned model. The results will be helpful in future studies of the concerned system. In the end, we can say that the method used is straightforward and dynamic, and it will be a useful tool for debating tough issues in a wide range of fields.

This paper presents the mathematical and physical analysis, as well as distinct types of exact wave solutions, of an important fluid flow dynamics model called the truncated M-fractional (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. This model is used to explain waves in shallow water, fluid dynamics, and other areas. We obtain kink, bright, singular, and other types of exact wave solutions using the modified extended direct algebraic method and the improved $(G^{\prime }/G)$-expansion method. Some solutions do not exist. These solutions may be useful in different areas of science and engineering. The results are represented as three-dimensional, contour, and two-dimensional graphs. Stability analysis is also performed to check the stability of the corresponding model. Furthermore, modulation instability analysis is performed to study the stationary solutions of the corresponding model. The results will be helpful for future studies of the corresponding system. The methods used are easy and useful.

In this paper, we establish the existence of some fixed point results for generalized(Alpha,Beta,F)-Geraghty contraction in metric-like spaces. We provide an example in order to support our results where some consequence applications of such result will be considered in this article. The obtained results improve and extend some well-known common fixed point results in the literature.

The purpose of this paper is to consider various results in the context of G ${}_b$ -metric spaces that have been recently published after the paper (Aghajani, A.; Abbas, M.; Roshan, J.R. Common fixed point of generalized weak contractive mappings in partially ordered G ${}_b$ -metric spaces. Filomat 2014, 28, 1087–1101). Our new results improve, complement, unify, enrich and generalize already well known results on G ${}_b$ -metric spaces. Moreover, some coupled and tripled coincidence point results have been provided.