The conformal fractional derivative (CFD) has become a hot research topic since it has a physical interpretation and is easier to grasp and apply to problems compared with other fractional derivatives. Its application to heat transfer, diffusion, diffusion-advection, and wave propagation problems can be found in the literature. Fractional diffusion equations have received great attention recently due to their applicability in physical, chemical, and biological processes and engineering. The diffusion of the pollutants within the ground, which is an important environmental problem, can be modeled with a diffusion equation. Diffusion in some porous materials or soil can be modeled more accurately with fractional derivatives or the conformal fractional derivative. In this study, the diffusion problem of a spilled pollutant leaking into the ground modeled with the conformal fractional time derivative in spherical coordinates has been solved analytically using the Fourier series for a constant mass flow rate and complete symmetry under the assumptions of homogeneous and isotropic soil, constant soil temperature, and constant permeability. The solutions have been simulated spatially and in time. A parametric analysis of the problem has been performed for several values of the CFD order. The simulation results are interpreted. It has also been suggested how to find the parameters of the soil to see whether the CFD can be used to model the soil or not. The approach described here can also be used for modeling pollution problems involving different boundary conditions.

This study investigates the application of conformable fractional derivatives (CFD) for modeling groundwater flow dynamics. Fractional derivatives offer a powerful alternative to overcome the limitations of traditional models in groundwater management, a critical challenge due to increasing population demands and diminishing water resources. In this context, the conformable fractional time derivative was integrated into the groundwater flow equation to enable more precise and accurate calculations of complex flow dynamics in aquifer systems. The analytical solution of the developed CFD-based groundwater flow equation was established and compared with both the classical Theis solution and experimental data. The proposed solution incorporated the well-known Theis equation, modified to include the fractional-order derivative parameter α, which allowed for greater flexibility in representing the temporal evolution of drawdown. When α = 1, the solution reverted to the classical Theis equation. Especially, the CFD-based solution with α = 0.9 exhibited superior alignment with experimental results, outperforming the classical Theis model. Performance evaluation metrics confirmed the enhanced predictive capabilities of the CFD-based model, with the coefficient of determination value of 0.9806 compared to 0.8781 for the classical model, and reductions in mean absolute error, mean squared error, and root mean squared error by 37%, 45%, and 48%, respectively. These findings highlight the potential of the proposed approach in improving groundwater management and aquifer characterization by accurately modeling storage effects and non-local behaviors in groundwater systems.

In this article, a uniform Haar wavelet approach is devised to numerically solve the differential equations. The uniform Haar wavelet coefficients are generated by employing collocation points. The generalized approach for function and matrix approximation using Haar wavelets is proposed. This study aims to decide which method is more useful by reflecting on the differences between the two methods. Also, the application of Haar wavelets to the solution of a first and second-order ODE is described in this research. To assess its applicability and efficiency, two test problems are used. The findings obtained are compared to those obtained using the function and matrix approximation methods. For numerically solving first and second-order ODEs, the Haar wavelet methodology gives a more reliable and exact method. By estimating error norms for various problems, the performance and accuracy of the method have been shown.

Scalar centrality measures assign one score to each sector of a production network and cannot compare its exposure to two shock sources. We develop a matrix-algebraic framework for this comparison. The Asymmetry Multiplier ${\gamma }_j(\alpha ,\beta )=L_{j\alpha }^{\top }/L_{j\beta }^{\top }$, the ratio of the price responses of a target sector to equal increments in unit value added at two sources, factors into a direct ratio and a network amplification ratio. Using sub-eigenvectors of the technical coefficient matrix, we derive lower and upper bounds that hold for every productive network and reduce to Perron–Frobenius bounds under irreducibility. We also prove that linear price responses bound those of competitive constant-returns economies from above and that no labeling-independent node-level measure determines the multiplier. A symmetric structural decomposition separates technological and final demand effects. As a proof of concept, the framework is applied to electricity and mining cost shocks in Türkiye. In 2022, the multiplier ranges from 0.20 to 1.86 across eleven sectors, indirect paths attenuate the direct ratio by a factor of 8 to 22 in six sectors, and Basic Metals is more exposed to mining than to electricity cost shocks, through imported mining inputs. For most sectors, the dominant channel is robust to disaggregation, coefficient perturbation and the annual tables of 2012–2022.