This study employs the modified extended direct algebraic method (MEDAM) to investigate the generalized nonlinear fractional $(3+1)$-dimensional wave equation with gas bubbles. This advanced analytical framework is used to construct a comprehensive class of exact wave solutions and explore the associated dynamical characteristics of diverse wave structures. The analysis yields several categories of soliton solutions, including rational, hyperbolic (sech, tanh), and trigonometric (sec, tan) function forms. To the best of our knowledge, these soliton solutions have not been previously documented in the existing literature. By selecting appropriate standards for the permitted constraints, the qualitative behaviors of the derived solutions are illustrated using polar, contour, and two- and three-dimensional surface graphs. Furthermore, a stability analysis is performed on the obtained soliton solutions to ascertain their robustness and dynamical stability. The suggested analytical approach not only deepens the theoretical understanding of nonlinear wave phenomena but also demonstrates substantial applicability in various fields of applied sciences, particularly in engineering systems, mathematical physics, and fluid mechanics, including complex gas–liquid interactions.

In this paper, we define and study the notion of hereditary class on nearly μ-Lindelöf space. Moreover, we study the effects of some types of continuity of hereditary class on nearly μ-Lindelöf space by properties of the function. Also, more variations between these spaces and some known spaces are investigated.

In this paper, we examine the 𝑤𝑡−distance characteristics over 𝑏−metric space and the conditions required to ensure the presence of the fixed point by letting 𝛽−function appropriately.In addition, we prove some fixed point theorems.