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.

The pathogens of dengue and chikungunya are transmitted by Aedes aegypti mosquitoes, presenting a significant likelihood of vectors and hosts being concurrently infected by both viruses. We present biological insights into the co-infection of chikungunya and dengue to elucidate the phenomena clearly. Therefore, we develop a mathematical model to conceptualize the intricate dynamics of chikungunya and dengue co-infection. We investigate the models of dengue fever and chikungunya separately, examining their local and global asymptotic stability. It has been shown that these models have a unique endemic steady state. Moreover, we demonstrate that both diseases are uniformly persistent when the reproduction parameter exceeds one. The endemic indicator, which is the reproduction value of the co-infected model, is computed as the maximum of the reproduction values of dengue and chikungunya. Moreover, the theory of optimal control is implemented using four distinct strategies to reduce infection levels: prevention through insect repellents, control via vaccination, and vector reduction through larvicides and adulticides. Finally, we investigate the system numerically to illustrate the overall dynamics of co-infection under optimal control measures and to assess the effectiveness of these control policies.