Digital Twin (DT) technology models and synchronizes physical systems through virtual counterparts. Existing mathematical formulations of DTs concentrate on physical-virtual synchronization and say little about how representations of the same system at different levels of detail should be related. This paper builds a mathematical foundation for Granular Digital Twins (GDTs) by combining Granular Computing with DT theory. We introduce information granules, the Granular Representation Mapping, the Granular Twin Space, and the Granular Digital Twin, and then construct a Granular Twin Algebra whose aggregation, refinement, and adaptive transformation operators act on an ordered family of granula... More >

Compartmental and tracer-kinetic models are widely used to describe radiopharmaceutical uptake in brain PET and SPECT, but in simple analytical models the fraction of transferred activity that reaches the brain is seldom isolated as a parameter of its own. This study defines a dimensionless Brain Uptake Constant, $\psi$, as the uptake fraction in a two-stage first-order model of cerebral radiopharmaceutical activity and examines its behaviour in a purely synthetic setting. Activity in the systemic pool, $A(t)$, is lost at a first-order rate $k$; cerebral activity, $A_c(t)$, receives a source term $\psi k A(t)$ and is lost at an effective decay rate $\lambda$. The closed-form solution was eva... More >

Glioblastoma (GBM) is a highly infiltrative primary brain tumor, and reaction-diffusion models are widely used to describe how its cells spread and proliferate. We derive the classical Fisher-KPP equation from mass conservation, Fickian diffusion, and logistic growth, and extend it to fractionated radiotherapy through a linear-quadratic survival update applied to the tumor-density field at each fraction. The Fisher-KPP equation is solved by finite differences on a three-dimensional voxel grid with 0.5-mm spacing; Geant4 is used only for Monte Carlo photon transport and dose calculation. The schedule of 60~Gy in 30 fractions from the clinical case reported by Roda et al. is reproduced, and th... More >

In this paper, we study the unsteady magnetohydrodynamic (MHD) flow and the heat and mass transfer of a Williamson tri-hybrid nanofluid (THNF) composed of water and Al$_2$O$_3$, TiO$_2$ and Cu nanoparticles over a stretching sheet. The model incorporates thermal radiation, activation energy, chemical reaction, a heat source and the bioconvection of motile gyrotactic microorganisms. An artificial neural network (ANN) is employed for the predictive modeling of such THNF flows under these combined effects. Similarity transformations reduce the governing nonlinear partial differential equations (PDEs) to coupled ordinary differential equations (ODEs), which are solved numerically with a boundary... More >

This study investigates the two-dimensional magnetohydrodynamic (MHD) slip flow of a tangent-hyperbolic (TGH) aluminum-alloy hybrid nanofluid over a stretching surface in a porous medium. The model incorporates magnetic effects, velocity and thermal slip, thermal radiation, heat generation, viscous dissipation, chemical reaction, and activation energy. The governing nonlinear partial differential equations are transformed into ordinary differential equations using similarity transformations and solved numerically in MATLAB using the bvp4c solver. The effects of key parameters on the velocity, temperature, and concentration profiles are analyzed, together with the skin-friction coefficient, N... More >

This paper proposes a modified two-point block backward differentiation formula (MBBDF) for the numerical solution of highly stiff and oscillatory ordinary differential equations. The method achieves third-order accuracy with a reduced error constant and computes two solution values simultaneously at each step. For the selected parameter value $\rho=-\dfrac{7}{8}$, stability analysis shows that the resulting scheme is A-stable, making it suitable for the numerical integration of stiff initial value problems. The method is also consistent and zero-stable, and hence convergent. Numerical experiments on benchmark stiff problems show that the relative accuracy of the considered methods depends o... More >

This study investigates the order and convergence properties of an extended two-point superclass Block Backward Differentiation Formula (BBDF) incorporating two intermediate (off-step) points for the numerical solution of stiff initial value problems (IVPs). Stiff systems frequently arise in scientific and engineering applications and require numerical methods that combine stability, accuracy, and computational efficiency. While classical Backward Differentiation Formula (BDF) methods are well known for their stability, their sequential nature limits efficiency, particularly for large-scale problems. To address these limitations, the proposed framework extends the BBDF approach by introducin... More >

Daily mean PM\(_{2.5}\) concentrations in Hyderabad routinely exceed both the 24-hour (\(60\,\mu\text{g/m}^3\)) and the annual (\(40\,\mu\text{g/m}^3\)) National Ambient Air Quality Standards, motivating the development of transparent and inexpensive descriptive models of their seasonal behaviour. To this end, we develop and calibrate a parsimonious first-order linear ordinary differential equation (ODE) that describes the daily evolution of ambient PM\(_{2.5}\) as the balance between a seasonally varying emission source \(E(t)=E_{0}+A\cos(2\pi(t-\varphi)/365)\) and a wind-driven first-order removal \(\lambda(t)=\alpha v(t)+\lambda_{0}\). The five parameters are estimated by Differential Evo... More >