In this paper, we study warped product semi-slant submanifolds of locally metallic Riemannian manifolds. A Chen-type inequality for such submanifolds is derived. We construct a non trivial example of such classes of submanifolds. We also provide several applications of the obtained inequality.

Within this paper, we have formulated the first Chen inequality for bi-warped product sub-manifolds within Riemannian space forms. This inequality intricately involved extrinsic invariants such as mean curvature and the lengths of the warping functions, while also incorporating intrinsic invariants like sectional curvature and $\delta$-invariants. Furthermore, we extensively explored and analyzed the scenarios where equality conditions were met within the context of this inequality.

We began by considering invariant, anti-invariant, proper slant, and pointwise slant submanifolds of a Lorentzian concircular structure manifold. Subsequently, we explored two distinct categories of warped product submanifolds. The first category encompassed the fiber submanifold as an anti-invariant submanifold, while the second category included the fiber submanifold as a pointwise slant submanifold. We established several fundamental results concerning these submanifold classes. Additionally, we demonstrated the existence of such submanifold classes through specific examples. Moreover, we derived inequalities for the squared norm of the second fundamental form.

The objective of the present study is to illustrate the significance and applicability of fractional-order derivatives in resolving mathematical models. In this study, we investigated two mathematical models, including the nonlinear time-fractional water pollution model and the time-fractional Bloch model, which arises in bioengineering. These models were successfully solved by using a unique technique known as the $q$-homotopy analysis Yang transform method, and approximate solutions were obtained. Using the Caputo operator, which accounts for memory effects, the model was examined in fractional order. The results clearly indicate that fractional-order derivatives give a better match for the behavior of the systems under consideration. This method not only allows for improved comprehension of complex phenomena but also provides an opportunity for further research in diverse areas, such as environmental science and bio engineering. The results obtained by the considered method are in the form of a series that converges swiftly and also we proposed convergence analysis for the considered method. This technique helps us modify the convergence area of the series solution by utilizing an auxiliary parameter called $\hslash$, also referred to as the convergence control parameter. The outcomes were analyzed using 3D plots and graphs, and also we conducted numerical simulations. The obtained results demonstrate that the proposed method is highly accurate and successful in examining nonlinear issues that arise in various scientific and technological domains.