This study explores the analytical solution for magneto-hydrodynamic (MHD) free convection flow in an unsteady state occurring between two inclined plates. It takes into account factors such as buoyancy forces, an aligned magnetic field, thermal radiation, chemical reactions, and radiation absorption. By implementing the Analytical Perturbation Method, the analysis yields expressions for velocity, temperature, and concentration profiles, highlighting the interactions among magnetic fields, gravitational forces, and thermal radiation. The findings illustrate notable impacts on flow characteristics and thermal behavior, which {is} subsequently examined through graphical representations. Additionally, calculations yield tabulated data on heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number).

This study investigates the properties, interactions, and attractors of irregular iterated function systems (IFS). Unlike traditional IFS, which rely on uniform contraction functions and symmetrical attractors, irregular IFS capture complex structures characterized by non-uniform scaling and unpredictable patterns. We show that the Hausdorff distance between the attractors of the product of irregular IFS can be bounded by the maximum contraction factors of the coordinate irregular IFSs. This extends classical IFS results to irregular settings. The Hutchinson operator for the product of irregular IFS uniquely fixes a point, corresponding to the product of the coordinate attractors, preserving the structure of attractors even in irregular cases. A modified Collage Theorem for product irregular IFSs estimates the Hausdorff distance between an arbitrary set and the attractor, providing tools for approximating fractal sets in higher dimensions. These findings provide new insights into the behavior of irregular IFS and their products, enhancing the understanding of composite fractal structures.

To understand the impacts of awareness program run by social media on the dynamics of wildlife species, a non-linear mathematical model is proposed and analysed. The mathematical model involves three dynamical variables namely: the density of human population, the density of wildlife species and awareness program run by social media. Existence and stability of equilibrium points are discussed. We further extend our mathematical model by introducing diffusion and analyse the reaction-diffusion model. Numerical simulations are performed to verify and validate our analytical results. By this study, we observe that the density of wildlife species increases as implementation rate of awareness program increases. It is also found that the introduction of diffusion destabilises the system which was stable in the absence of diffusion.

A relaxation equation refers to a mathematical model that describes how a system approaches equilibrium or steady-state over time, often after being disturbed. It essentially involves an exponential decay towards a steady state, with a characteristic time that governs how quickly the system relaxes to equilibrium. Relaxation equations are common in many fields like physics, chemistry, engineering, and economics. Motivated by these facts, in this article, we consider the discrete fractional relaxation equation and establish sufficient conditions on the existence, uniqueness and stability of its periodic solutions using suitable fixed point theorems. We also demonstrate the applicability of established results through an example.

This paper investigates the effect of temperature on drug distribution between the stomach and bloodstream compartments of the human body, incorporating the influence of time-dependent transfer and elimination rates. The study reveals that drug distribution behaviour is significantly influenced by variations in surrounding temperatures and the dynamic nature of transfer and elimination parameters. A theoretical analysis is performed to establish the existence, uniqueness, and stability of solutions for the proposed models using Caputo fractional derivatives. Numerical simulations are conducted using the predictor-corrector method to validate the theoretical findings and examine the impact of different fractional orders (\(\kappa\)) on drug kinetics. Graphical representations of the numerical results highlight the combined influence of memory effects, time-dependent parameters, and temperature variations on drug dynamics, providing valuable insights into the pharmacokinetics of drugs modeled in this work.

The nonlocal Volterra-Fredholm system for a class of nonlinear fractional $\psi$-Caputo integro-differential equations of Sobolev type in Banach spaces is examined in this study. The $\psi$-Caputo fractional derivative generalizes the classical Caputo derivative by incorporating a function $\psi$, allowing greater flexibility in modeling memory effects. We first establish novel Darbo-type fixed point theorems. These theorems are then applied to derive a solvability theorem for the nonlocal Volterra-Fredholm problem associated with nonlinear fractional equations of Sobolev type. Furthermore, we provide an example to show how our theoretical findings might be applied. These results have potential applications in physics and engineering, particularly in modeling complex systems with memory and hereditary properties.

The delicate interplay among agricultural ecosystems, insect pest populations, and natural predators is a complex phenomenon that significantly influences sustainable crop management. This work uses a mathematical model to investigate the intricate dynamics governing the interrelationship that involves population densities of bats, crop biomass, and insect pests. The model is grounded in a nonlinear system of ordinary differential equations to unveil the nuanced mechanism underpinning the complex nexus that controls the coexistence of bats, crop biomass, and insect pests. The mathematical model analyses the dynamics and identifies conditions for system stability. The numerical simulation results serve as evidence for the conclusions drawn. The outcomes of the study indicate that when bats cohabited with crops and insect pests at their maximum density, the agricultural system stabilizes. It underscores the pivotal role assumed by bats as natural predators, wielding a substantial influence in the regulation of insect pest populations and, consequently, in the preservation of crop biomass density.

This study examines the dynamics of a smoking model using the Caputo ($\mathcal{C}$) fractional derivative, which effectively captures memory effects inherent in complex systems. We conduct a mathematical analysis of the fractional model, ensuring the positivity of solutions, invariant region and demonstrating the existence and uniqueness of solutions through fixed-point theory. For numerical simulations, we employ a generalized predictor-corrector method tailored for the $\mathcal{C}$ derivative. The model is computationally solved, and results are graphically illustrated across various fractional-order values.