In this paper, connections between various subclasses of harmonic univalent functions by using a convolution operator involving the Pascal distribution series are investigated. Furthermore, an example is provided, illustrating graphically with the help of Maple, to illuminate the convolution operator. .................................................................................................................................

We consider classes of harmonic univalent functions $f_k=h_k+\overline{g_k}$, $(k=1,2)$ that are shears of the analytic map $h_k-g_k=1/2\log \left[\left(1+z\right)/\left(1-z\right)\right]$ with dilatation ${\omega }_k={\mathrm{e}}^{\mathrm{i}{\theta }_k}{z}^k$. We prove that if the convolution $f_1\ast f_2$ is locally one-to-one and sense-preserving, then $f_1\ast f_2$ is convex in the direction of the real axis.

This study deals with the third-order Boussinesq type system, an essential model for coastal and civil engineering to perform the nonlinear water wave model in a harbour and coastal design. The generalized exponential function method and the invariant subspace method are utilized to exhibit exact solutions. The analysis also involves examining the time derivative through the Riemann-Liouville fractional derivative operator and employing the invariant subspace method. Conservation laws are also investigated by exploiting Lie point symmetries, the nonlocal conservation method, and the self-adjointness concept. The present work's outputs contribute to increasing our understanding of the dynamics governed by the system in question.