In this article, we use the prominent Karmarkar condition to investigate some novel features of astronomical objects in the f(R, ϕ) gravity; R and ϕ represent the Ricci curvature and the scalar field, respectively. It is worth noting that we classify the exclusive set of modified field equations using the exponential type model of the f(R, ϕ) theory of gravity f(R, ϕ) = ϕ(R + α(e ^{β R − 1})). We show the embedded class-I approach via a static, spherically symmetric spacetime with an anisotropic distribution. To accomplish our objective, we use a particular interpretation of metric potential (g _rr ) that has already been given in the literature and then presume the Karmarkar condition to derive the second metric potential. We employ distinct compact stars to determine the values of unknown parameters emerging in metric potentials. To ensure the viability and consistency of our exponential model, we execute distinct physical evolutions, i.e. the graphical structure of energy density and pressure evolution, mass function, adiabatic index, stability, equilibrium, and energy conditions. Our investigation reveals that the observed anisotropic findings are physically appropriate and have the highest level of precision.

This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.

The present paper is focused on time-like circular surfaces and singularities in Minkowski 3-space. The timelike circular surface with a constant radius could be swept out by moving a Lorentzian circle with its center while following a non-lightlike curve called the spine curve. In the present study, we have parameterized timelike circular surfaces and examined their geometric properties, such as singularities and striction curves, corresponding with those of ruled surfaces. After that, a different kind of timelike circular surface was determined and named the timelike roller coaster surface. Meanwhile, we support the results of this work with some examples.

We studied the asymptotic behavior of fourth-order advanced differential equations of the form ${\left(a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+q\left(\upsilon \right)g\left(w\left(\delta \left(\upsilon \right)\right)\right)=0$. New results are presented for the oscillatory behavior of these equations in the form of Philos-type and Hille–Nehari oscillation criteria. Some illustrative examples are presented.

In this article, we derived an equality for $\mathcal{CR}$-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a $\mathcal{CR}$-warped product submanifolds in K$\ddot{a}$hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of $\mathcal{CR}$-warped product submanifolds in the K$\ddot{a}$hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.