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In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the L 1 scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification. [ABSTRACT FROM AUTHOR]

Due to its remarkable thermodynamic development in various engineering sciences, the dispersion of nanostructures in classical base liquids is receiving greater attention from researchers and scientists. In light of the aforementioned inspirations, the suggested research is associated with using a water-based ternary-hybrid nanofluid that contains three unique nanostructures, silicon dioxide, titanium dioxide and aluminum oxide to optimize the heating process in science and engineering. In this situation, the influence of a chemically reactive magneto-hydrodynamics (MHD) Darcy–Forchheimer ternary hybrid composites transport involving nonlinear radiation across an exponentially permeable stretched surface is highlighted with the model's physical initial and boundary constraints. The simulation of the heat expression also includes the impacts of nonlinear thermal radiations, energy supply, energy dissipation and magnetic force. Furthermore, the interpretation of Buongiorno's theory involves spontaneous and thermophoresis diffusions. Entropy analysis is explained using the second law of thermodynamics. We converted the model framework for fluid movement, energy and mass transfer (together with ternary hybrid nanofluid characteristics) into self-similar nondimensional differential equations, that were then further analytically evaluated by using the homotopy asymptotic method (HAM) technique via Mathematica software. The acquired results are illustrated numerically and graphically to study the behavior of fluid flow, heat, concentration distribution, drag force, rate of heat and mass transfer for various emergent components to gain scientific comprehension. With the growing values of magnetic, porosity and Forchheimer number, the nature of the velocity profile goes down. [ABSTRACT FROM AUTHOR]