Search Results (269)

Partial Differential Equation (PDE)-based hydrologic models demand extensive preprocessing, creating a bottleneck and slowing down the model setup process. Mesh generation typically lacks integration with hydrological features like river networks. We present GHOST Mesh (GMesh), an automated, watershed-oriented mesh generator built within the Watershed Modeling Framework (WMF), to address this. While primarily designed for the GHOST hydrological model, GMesh’s functionalities can be adapted for other models. GMesh enables rapid mesh generation in Python by incorporating Digital Elevation Models (DEMs), flow direction maps, network topology, and online services. The software creates Voronoi polygons that maintain connectivity between river segments and surrounding hillslopes, ensuring accurate surface–subsurface interaction representation. Key features include customizable mesh generation and variable refinement to target specific watershed areas. We applied GMesh to Iowa’s Bear Creek watershed, generating meshes from 10,000 to 30,000 elements and analyzing their effects on simulated stream flows. Results show that higher mesh resolutions enhance peak flow predictions and reduce response time discrepancies, while local refinements improve model performance with minimal additional computation. GMesh’s open-source nature streamlines mesh generation, offering researchers an efficient solution for hydrological analysis and model configuration testing. Full article

A new class of poroelastic dynamic contact problems stemming from hydraulic fracture theory is introduced and studied. The two-phase medium consists of a solid phase and pores which are saturated with a Newtonian fluid. The porous body contains a fluid-driven crack endowed with non-penetration conditions for the opposite crack surfaces. The poroelastic model is described by a coupled system of hyperbolic–parabolic partial differential equations under the unilateral constraint imposed on displacement. After full discretization using finite-element and Hilber–Hughes–Taylor methods, the well-posedness of the resulting variational inequality is established. Formulation of the complementarity conditions with the help of a minimum-based merit function is used for the semi-smooth Newton method of solution presented in the form of a primal–dual active set algorithm which is tested numerically. Full article

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article

In this paper, we propose a fuzzy formulation of the classic Frankot–Chellappa algorithm by which surfaces can be reconstructed using normal vectors. In the fuzzy formulation, the surface normal vectors may be uncertain or ambiguous, yielding a fuzzy Poisson partial differential equation that requires appropriate definitions of fuzzy derivatives. The solution of the resulting fuzzy model is approached by adopting a fuzzy variant of the discrete sine transform, which results in a fast and robust algorithm for surface reconstruction. An adaptive defuzzification strategy is also introduced to improve noise handling in highly uncertain regions. In experiments, we demonstrate that our fuzzy Frankot–Chellappa algorithm achieves accuracy on par with the classic approach for smooth surfaces and offers improved robustness in the presence of noisy normal data. We also show that it can naturally handle missing data (such as gaps) in the normal field by filling them using neighboring information. Full article

In this article, we propose a novel nonlinear cross-diffusion framework to model the distribution of susceptible and infected individuals within their habitat using a reduced SIR model that incorporates saturated incidence and treatment rates. The study investigates solution boundedness through the theory of parabolic partial differential equations, thereby validating the proposed spatio-temporal model. Through the implementation of the suggested cross-diffusion mechanism, the model reveals at least one non-constant positive equilibrium state within the susceptible–infected (SI) system. This work demonstrates the potential coexistence of susceptible and infected populations through cross-diffusion and unveils Turing instability within the system. By analyzing codimension-2 Turing–Hopf bifurcation, the study identifies the Turing space within the spatial context. In addition, we explore the results for Turing–Bogdanov–Takens bifurcation. To account for seasonal disease variations, novel perturbations are introduced. Comprehensive numerical simulations illustrate diverse emerging patterns in the Turing space, including holes, strips, and their mixtures. Additionally, the study identifies non-Turing and Turing–Bogdanov–Takens patterns for specific parameter selections. Spatial series and surfaces are graphed to enhance the clarity of the pattern results. This research provides theoretical insights into the implications of cross-diffusion in epidemic modeling, particularly in contexts characterized by localized mobility, clinically evident infections, and community-driven isolation behaviors. Full article

In this paper, we discuss the prediction of the delivery efficiency of magnetic carriers based on their properties and field parameters. We developed a theory describing the behavior of magnetic capsules in the capillaries of living systems. A partial differential equation for the spatial distribution of magnetic capsules has been obtained. We propose to characterize the interaction between the magnetic field and the capsules using a single vector, which we call “specific magnetic force”. To test our theory, we performed experiments on a model of a capillary bed and on a living organism with two types of magnetic capsules that differ in size and amount of magnetic material. The experimental results show that the distribution of the capsules in the field correlated with the theory, but there were fewer actually accumulated capsules than predicted by the theory. In the weaker fields, the difference was more significant than in stronger ones. We proposed an explanation for this phenomenon based on the assumption that a certain level of magnetic force is needed to keep the capsules close to the capillary wall. We also suggested a formula for the relationship between the probability of capsule precipitation and the magnetic force. We found the effective value of a specific magnetic force at which all the capsules attracted by the magnet reach the capillary wall. This value can be considered as the minimum level for the field at which it is, in principle, possible to achieve a significant magnetic control effect. We demonstrated that for each type of capsule, there is a specific radius of magnet for which the effective magnetic force is achieved at the largest possible distance from the magnet’s surface. For the capsules examined in this study, the maximum distance where the effective field can be achieved does not exceed 1.5 cm. The results of the study contribute to our understanding of the behavior of magnetic particles in the capillaries of living organisms when exposed to a magnetic field. Full article

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by the geodetic latitude φ and geodetic longitude λ, and eccentricities e_e, e_x, e_y. On its surface, the value of gravity potential has a constant value, defining a level triaxial ellipsoid. In addition, the gravity intensity is known on the surface, which allows us to formulate a Dirichlet boundary value problem for determining the gravity intensity as a series of spherical harmonics. This expression for gravity intensity is presented here for the first time, filling a gap in the study of triaxial ellipsoids and spheroids. Given that the triaxial ellipsoid has very small eccentricities, a first order approximation can be made by retaining only the terms containing e_e² and e_x². The resulting expression in spherical harmonics contains even degree and even order harmonic coefficients, along with the associated Legendre functions. The maximum degree and order that occurs is four. Finally, as a special case, we present the geometrical degeneration of an oblate spheroid. Full article

Stream–aquifer interactions, as well as surface water/groundwater interactions within wetlands, require a solution of complex partial differential equations of flow and contaminant transport, namely a deterministic approach. Groundwater level (GWL) responses to precipitation, particularly for extreme value events such as annual maxima, require a probabilistic approach to evaluate potential climate trends. It is commonly assumed that the distribution of annual maxima series (AMS) precipitation follows the generalized extreme value distribution (GEV). If the extremes of the data are nonstationary, it is possible to incorporate this knowledge into the parameters of the GEV. This approach is also applied to the computed annual maxima of daily groundwater level data. Nonstationary versus stationary time series for both groundwater level and AMS 24-h duration precipitation are compared for National Oceanic and Atmospheric Administration (NOAA) stations with nearby wells. Predicted extreme value analysis (EVA) climate trends for wells penetrating limestone aquifers directly beneath rainfall monitoring stations at major airports indicate similar GWL response. Groundwater levels at wells located near coastlines are partially impacted by sea level rise. An extreme value analysis of the GWL is shown to be a useful tool to confirm hydrologic connections and long-term climate trends. Full article

A variational formulation and variationally consistent boundary conditions were derived for a coupled system of two boron nitride nanotubes (BNNTs), with the piezoelectric and surface effects taken into account in the formulation. The coupling between the nanotubes was defined in terms of Winkler and Pasternak interlayers. The equations governing the vibrations of the coupled system were expressed as a system of four partial differential equations based on nonlocal elastic theory. After deriving the variational principle for the double BNNT system, Hamilton’s principle was expressed in terms of potential and kinetic energies. Next, the differential equations for the free vibration case were presented and the variational form for this case was derived. The Rayleigh quotient was formulated for the vibration frequency, which indicated that piezoelectric and surface effects led to higher vibration frequencies. Next, the variationally consistent boundary conditions were formulated in terms of moment and shear force expressions. It was observed that the presence of the Pasternak interlayer between the nanotubes led to coupled boundary conditions when a shear force and/or a moment was specified at the boundaries. Full article

Concrete dams are prone to various hidden dangers after long-term operation and may lead to significant risk if failed to be detected in time. However, the existing hollowing detection techniques are few as well as inefficient when facing the demands of comprehensive coverage and intelligent management for regular inspections. Hence, we proposed an innovative, non-destructive infrared inspection method via constructed dataset and proposed deep learning algorithms. We first modeled the surface temperature field variation of concrete dams as a one-dimensional, non-stationary partial differential equation with Robin boundary. We also designed physics-informed neural networks (PINNs) with multi-subnets to compute the temperature value automatically. Secondly, we obtained the time-domain features in one-dimensional space and used the diffusion techniques to obtain the synthetic infrared images with dam hollowing by converting the one-dimensional temperatures into two-dimensional ones. Finally, we employed adaptive joint learning to obtain the spatio-temporal features. We designed the experiments on the dataset we constructed, and we demonstrated that the method proposed in this paper can handle the low-data (few shots real images) issue. Our method achieved 94.7% of recognition accuracy based on few shots real images, which is 17.9% and 5.8% higher than maximum entropy and classical OTSU methods, respectively. Furthermore, it attained a sub-10% cross-sectional calculation error for hollowing dimensions, outperforming maximum entropy (70.5% error reduction) and OTSU (7.4% error reduction) methods, which shows our method being one novel method for automated intelligent hollowing detection. Full article

In the current paper, an analysis of magnetohydrodynamic Williamson nanofluid boundary layer flow is presented, with multiple slips in a porous medium, using a newly designed human-brain-inspired Ricker wavelet neural network solver. The solver employs a hybrid approach that combines genetic algorithms, serving as a global search method, with sequential quadratic programming, which functions as a local optimization technique. The heat and mass transportation effects are examined through a stretchable surface with radiation, thermal, and velocity slip effects. The primary flow equations, originally expressed as partial differential equations (PDEs), are changed into a dimensionless nonlinear system of ordinary differential equations (ODEs) via similarity transformations. These ODEs are then numerically solved with the proposed computational approach. The current study has significant applications in a variety of practical engineering and industrial scenarios, including thermal energy systems, biomedical cooling devices, and enhanced oil recovery techniques, where the control and optimization of heat and mass transport in complex fluid environments are essential. The numerical outcomes gathered through the designed scheme are compared with reference results acquired through Adam’s numerical method in terms of graphs and tables of absolute errors. The rapid convergence, effectiveness, and stability of the suggested solver are analyzed using various statistical and performance operators. Full article

This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function method with the complete discriminant polynomial system. This method proves to be effective in discovering precise solutions for stochastic fractional partial differential equations (SFPDEs), including special cases. Applying this method to the SFRLWE yields new exact solutions, offering fresh insights. We investigated how noise affects stochastic solutions and discovered that more intense noise can result in flatter surfaces. We note that multiplicative noise can stabilize the solution, and we show how fractional derivatives influence the dynamics of noise. We found that the noise strength and fractional derivative affect the width, amplitude, and smoothness of the obtained solutions. Additionally, we conclude that multiplicative noise impacts and stabilizes the behavior of SFRLWE solutions. Full article

This work aims to explore the magnetohydrodynamic mixed convection boundary layer flow (MHD-MCBLF) on a slanted extending cylinder using Eyring–Powell fluid in combination with Levenberg–Marquardt algorithm–artificial neural networks (LMA-ANNs). The thermal properties include thermal stratification, which has a higher temperature surface on the cylinder than on the surrounding fluid. The mathematical model incorporates essential factors involving mixed conventions, thermal layers, heat absorption/generation, geometry curvature, fluid properties, magnetic field intensity, and Prandtl number. Partial differential equations govern the process and are transformed into coupled nonlinear ordinary differential equations with proper changes of variables. Datasets are generated for two cases: a flat plate (zero curving) and a cylinder (non-zero curving). The applicability of the LMA-ANN solver is presented by solving the MHD-MCBLF problem using regression analysis, mean squared error evaluation, histograms, and gradient analysis. It presents an affordable computational tool for predicting multicomponent reactive and non-reactive thermofluid phase interactions. This study introduces an application of Levenberg–Marquardt algorithm-based artificial neural networks (LMA-ANNs) to solve complex magnetohydrodynamic mixed convection boundary layer flows of Eyring–Powell fluids over inclined stretching cylinders. This approach efficiently approximates solutions to the transformed nonlinear differential equations, demonstrating high accuracy and reduced computational effort. Such advancements are particularly beneficial in industries like polymer processing, biomedical engineering, and thermal management systems, where modeling non-Newtonian fluid behaviors is crucial. Full article

This study examines the flow dynamics of synovial fluid within a lubricated knee joint during movement, incorporating the effect of inertia and linear re-absorption at the synovial membrane. The fluid behavior is modeled using a couple-stress fluid framework, which accounts for mechanical phenomena and employs a lubricated membrane. synovial membrane plays a crucial role in reducing drag and enhancing joint lubrication for the formation of a uniform lubrication layer over the cartilage surfaces. The mathematical model of synovial fluid flow through the knee joint presents a set of non-linear partial differential equations solved by a recursive approach and inverse method through the software Mathematica 11. The results indicate that synovial fluid flow generates high pressure and shear stress away from the entry point due to the combined effects of inertial forces, linear re-absorption, and micro-rotation within the couple-stress fluid. Axial flow intensifies at the center of the knee joint during activity in the presence of linear re-absorption and molecular rotation, while transverse flow increases away from the center and near to synovium due to its permeability. These findings provide critical insights for biomedical engineers to quantify pressure and stress distributions in synovial fluid to design artificial joints. Full article

This paper investigates the frost heave and thaw settlement characteristics of the pipe–soil system during the freeze–thaw cycle, along with the underlying mechanisms. A numerical simulation platform for the complex pipe–soil system was developed using the heat conduction equation, moisture migration equation, and stress–strain equation, all of which account for the ice–water phase change process. The simulations were performed with the coefficient-type partial differential equation (PDE) module in COMSOL Multiphysics. By employing coupled thermal–hydraulic–mechanical (THM) simulation methods, the study analyzed the changes in volumetric water content, volumetric ice content, moisture migration patterns, and temperature field distribution of a water pipeline after three years of service under real engineering conditions in the cold region of northern Xinjiang, China. The study also examined the effects of parameters such as pipeline burial depth, specific heat capacity, thermal conductivity, permeability of saturated soil, and initial saturation on the displacement field. The results show that selecting soil layers with high specific heat capacity (e.g., 1.68 kJ/kg·°C) and materials with high thermal conductivity (e.g., 2.25 W/m·°C) can reduce surface frost heave displacement by up to 40.8% compared to low-conductivity conditions. The maximum freezing depth near the pipeline is limited to 0.87 m due to the thermal buffering effect of water flow. This research provides a scientific reference and theoretical foundation for the design of frost heave resistance in water pipelines in seasonally frozen regions. Full article