Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space.

The main goal of this paper is to investigate the evolution equations for special types of timelike ruled surfaces with significant geometric and physical applications in Lorentz-Minkowski 3-space E³₁ . Using the alternative frame associated with the basic curve of these surfaces, we explored their key geometric properties. Our analysis provided insights into the dynamics of local curvatures during their evolutions, enhancing the understanding of surface behavior. Finally, we present applications of our preliminary findings that contribute to the broader field of differential geometry [ABSTRACT FROM AUTHOR]