This study presents a novel framework for uncertainty propagation in power-law, Bingham, and Casson fluids through rectangular ducts under stochastic viscosity (Case I) and pressure gradient conditions (Case II). Using the computationally efficient Stochastic Finite Difference Method with Homogeneous Chaos (SFDHC), validated via comparison with quasi-Monte Carlo simulations, we demonstrate significantly lower computational costs across varying Coefficients of Variation (COVs). For viscosity uncertainty (Case I), results show a 0.54–2.8% increase in mean maximum velocity with standard deviations reaching 75.3–82.5% of the COV, where the power-law model exhibits the greatest sensitivity (velocity variations spanning 71.2–177.3% of the mean at COV = 20%). Pressure gradient uncertainty (Case II) preserves mean velocities but produces narrower and symmetric distributions. We systematically evaluate the effects of aspect ratio, yield stress, and flow behavior index on the stochastic velocity response of each fluid. Moreover, our analysis pioneers a performance hierarchy: Herschel–Bulkley fluids show the highest mean and standard deviation of maximum velocity, followed by power-law, Robertson–Stiff, Bingham, and Casson models. A key finding is the extreme fluctuation of the Robertson–Stiff model, which exhibits the most drastic deviations, reaching up to 177% of the average velocity. The significance of fluid-specific stochastic analysis in duct system design is underscored by these results. This is especially critical for non-Newtonian flows, where system performance and reliability are greatly impacted by uncertainties in viscosity and pressure gradient, which reflect actual operational variations.

This study presents an innovative approach to quantifying uncertainty in Herschel–Bulkley (H-B) fluid flow through rectangular ducts, analyzing four scenarios: uncertain apparent viscosity (Case I), uncertain pressure gradient (Case II), uncertain boundary conditions (Case III) and uncertain apparent viscosity and pressure gradient (Case IV). Using the stochastic finite difference with homogeneous chaos (SFDHC) method, we produce probability density functions (PDFs) of fluid velocity with exceptional computational efficiency (243 times faster), matching the accuracy of Monte Carlo simulation (MCS). Key statistics and maximum velocity PDFs are tabulated and visualized for each case. Mean velocity shows minimal variation in Cases I, III, and IV, but maximum velocity fluctuates significantly in Case I (63.95–187.45% of mean), Case II (50.15–156.68%), and Case IV (63.70–185.53% of mean), vital for duct design and analysis. Examining the effects of different parameters, the SFDHC method’s rapid convergence reveals the fluid behavior index as the primary driver of maximum stochastic velocity, followed by aspect ratio and yield stress. These findings enhance applications in drilling fluid management, biomedical modeling (e.g., blood flow in vascular networks), and industrial processes involving non-Newtonian fluids, such as paints and slurries, providing a robust tool for advancing understanding and managing uncertainty in complex fluid dynamics.

In the present work, we aim to study the inverse problem of recovering an unknown spatial source term in a multi-term time-fractional diffusion equation involving the fractional Laplacian. The forward problem is first analyzed in appropriate fractional Sobolev spaces, establishing the existence, uniqueness, and regularity of solutions. Exploiting the spectral representation of the solution and properties of multinomial Mittag–Leffler functions, we prove uniqueness and derive a stability estimate for the spatial source term from finaltime observations. The inverse problem is then formulated as a Tikhonov regularized optimization problem, for which existence, uniqueness, and strong convergence of the regularized minimizer are rigorously established. On the computational side, we propose an efficient reconstruction algorithm based on the conjugate gradient method, with temporal discretization via an L¹-type scheme for Caputo derivatives and spatial discretization using a Galerkin approach adapted to the nonlocal fractional Laplacian. Numerical experiments confirm the accuracy and robustness of the proposed method in reconstructing the unknown source term.