We analyze a Dirichlet $\left(p\right(x),\mu q(x\left)\right)$-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter $\mu$ varies on the non-negative real axis. The results are obtained by applying the properties of pseudomonotone operators, jointly with certain a priori estimates.

In this article, we investigate the solution of the fractional multidimensional Navier–Stokes equation based on the Caputo fractional derivative operator. The behavior of the solution regarding the Navier–Stokes equation system using the Sumudu transform approach is discussed analytically and further discussed graphically.

The integral-order derivative is not suitable where infinite variances are expected, and the fractional derivative manages to consider effects with more precision; therefore, we considered timefractional Emden–Fowler-type equations and solved them using the rational homotopy perturbation method (RHPM). The RHPM method is based on two power series in rational form. The existence and uniqueness of the equation are proved using the Banach fixed-point theorem. Furthermore, we approximate the term h(z) with a polynomial of a suitable degree and then solve the system using the proposed method and obtain an approximate symmetric solution. Two numerical examples are investigated using this proposed approach. The effectiveness of the proposed approach is checked by representing the graphs of exact and approximate solutions. The table of absolute error is also presented to understand the method′s accuracy.

A closed conformal vector field in de Sitter space $S_1^{n+1}\left(\overline{c}\right)$ induces a vector field on a spacelike hypersurface M of $S_1^{n+1}\left(\overline{c}\right)$, referred to as the induced vector field on M. This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming the constancy of the mean curvature. Specifically, we establish that under certain conditions, a compact spacelike hypersurface in $S_1^{n+1}\left(\overline{c}\right)$ is a sphere, that is, a totally umbilical hypersurface with constant mean curvature. We also present a different characterization of compact spacelike hypersurfaces in de Sitter space as spheres by using a lower bound on the integral of the Ricci curvature of the compact hypersurface in the direction of the induced vector field. We also consider de Sitter space as a Robertson–Walker space and provide several characterizations of spheres within its spacelike hypersurfaces.

This research work is dedicated to solving the n-generalized Korteweg–de Vries (KdV) equation in a fractional sense. The method is a combination of the Sumudu transform and the Adomian decomposition method. This method has significant advantages for solving differential equations that are both linear and nonlinear. It is easy to find the solutions to fractional-order PDEs with less computing labor.

Unbalanced lifestyles and other underlying medical conditions are responsible for the worrying pace at which diabetes mellitus is becoming a global health crisis. Recent studies suggest that placing a diabetic patient into remission through a rigorous lifestyle change program can normalize blood glucose levels. This research focuses on fractional order derivative-based mathematical modeling and analysis of the diabetes mellitus model with remission parameters. Firstly, the existence and uniqueness of the solution of the diabetes mellitus model are discussed. Non-negativity and boundedness are also examined. Afterward, the concept of the Jacobian matrix is used to investigate the stability of the model’s equilibrium points. The Daftardar-Gejji and Jafari Method has finally been applied to approximate the solutions. The conclusions drawn from numerical simulations of the diabetic model with fractional-order derivatives show a clear dependence on the remission parameters and fractional-order derivative.