In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold $(M,g)$ with potential function f, there exists a unique smooth function $\rho$ on M that connects u to the gradient of potential function $\nabla f$ . We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere ${\mathbf{S}}^n\left(c\right)$ and the Euclidean space ${\mathbf{E}}^n$ . We also show that the connecting function influences on a topology of the Riemannian manifold.

In this research paper, we discussed some geometric axioms of a relativistic string cloud spacetime attached with strange quark matter. We determined the conformal $ \eta $-Ricci soliton on a relativistic string cloud spacetime attached with strange quark matter with a $ \varphi(\mathcal{R}ic) $-vector field. In addition, we illustrated some physical significance of conformal pressure $ P $ in terms of conformal $ \eta $-Ricci soliton with the same vector field. Besides this, we deduced a generalized Liouville equation from the conformal $ \eta $-Ricci soliton. Furthermore, we examine the harmonic relevance of conformal $ \eta $-Ricci soliton on string cloud spacetime attached with strange quark matter with a potential function $ \psi $. Finally, we turned up a necessary and sufficient condition for the 1-form $ \eta $, which is the $ {g} $-dual of the vector field $ \gamma $ on a string cloud spacetime attached with strange quark matter to be a solution for the Schrödinger-Ricci equation.

The main target of this work is presenting two efficient accurate algorithms for solving numerically one of the most important models in physics and engineering mathematics, Fisher–Kolmogorov–Petrovsky–Piskunov’s equation (Fisher‐KPP) with fractional order, where the derivative operator is defined and studied by the fractional derivative in the sense of Liouville–Caputo (LC). There are two main processes; in the first one, we use the compact finite difference technique (CFDT) to discretize the derivative operator and generate a semidiscrete time derivative and then implement the Vieta–Lucas spectral collocation method (VLSCM) to discretize the spatial fractional derivative. The presented approach helps us to transform the studied problem into a simple system of algebraic equations that can be easily resolved. Some theoretical studies are provided with their evidence to analyze the convergence and stability analysis of the presented algorithm. To test the accuracy and applicability of our presented algorithm a numerical simulation is given.

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples.

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

A torqued vector field $\xi$ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative.