Parabolic Diffusion Equation Research Articles

Laplacian with singular drift in a critical borderline case

AbstractWe establish well‐posedness and regularity results for parabolic diffusion equation on the torus in the case when the singularities of a general drift reach the critical magnitude. The latter dictates the need to work in an Orlicz space situated between all and .

Wet ball milling activated oyster shells for multifunctional slow-release compound fertilizer production

Slow-release fertilizers have been known to boost crop yields, but their high cost and slow degradation make them less practical for use. In this study, oyster shell (OS) was modified by mechanical ball milling with monosodium glutamate (MSG) wastewater as an activator to prepare activated oyster shell (AOS) with an average particle size of 32.7 nm. Compared to natural oyster shell (NOS), the AOS increased water-soluble Ca by 22 times, expanded the average desorption pore size by 155 %, enhanced the specific surface area by 9 times, and improved the total pore volume by 1587 %. AOS was mixed with traditional fertilizers and extruded to develop a new type of activated oyster-based slow-release compound fertilizer (AOF). Physical effects mainly controlled the slow-release ability of AOF. AOS formed vaterite with a large peak shape, high strength, and well-defined crystal structure, providing more binding sites for NH4+, HPO42−, K+, Ca2+ and Mg2+. In addition, AOS was rich in Ca-OH and Mg-OH, which increased the potential for ligand exchange with fertilizer ions, which further improved the chemical slow-release properties of AOF. The AOF continuously releases N, P, K, Ca, and Mg nutrients over a period exceeding 70 days. The nutrient release behavior in AOF was accurately described by the logistic equation, the Elovich equation, and the parabolic diffusion equation. AOF increased the dry weight of green plum by 190 % and the yield by 61 %. The AOF prepared in this study proved to be an innovative and environmentally sustainable slow-release fertilizer.

Influence of vegetation on heavy metal Cr release process from bottom sediment under unidirectional flows and regular waves

As human activities become more intensive, a substantial number of heavy metals are discharged into estuarine or wetland environments. Due to the poor degradability, heavy metals are prone to adsorption and deposition on suspended particles in bottom sediments. Subsequently, under the influence of disturbances, there is a potential for their re-release, causing secondary pollution. To investigate the release process of the heavy metal Cr from sediment, laboratory experiments were conducted under both unidirectional flow and regular wave conditions. At the initial stage, the temporal trends of particulate (CrP) and dissolved (CrD) Chromium concentrations were both characterized by initial increments followed by stabilization and continuous escalation. Vertically, the stable concentrations of CrP and CrD increased with the presence of vegetation and the enhancement of hydrodynamics. The Elovich equation, pseudo-second-order kinetic equation, Double constant equation (Freundlich model), and parabolic diffusion equation were employed to predict the release process of CrD from bottom sediment. The Elovich equation proved most suitable for describing the release process of CrD, with an R2 exceeding 0.9. In order to assess the influence of vegetation on the Cr release process, the Stem-Reynolds were introduced to modify the Elovich equation. The final maximum error was 12 % (excluding the initial stage), which was much lower than that using the original Elovich equation (maximum error of 32 %). The study findings provide practical support for estuarine and wetland managers to formulate effective heavy metal management measures, which contribute to the conservation and sustainable management of aquatic ecosystems.

Release Kinetics and Plant Availability of Potassium in Soils of Eastern India

ABSTRACT The present experiment was undertaken to characterize the K releasing pattern as well as the bioavailability of K in nine soils under three dominant soil orders of eastern India. Soils were extracted with 0.2 M sodium tetraphenyl boron (NaTPB) and perennial ryegrass (Lolium perenne L.) was grown repeatedly with and without the addition of K fertilizer to study the K release kinetics and bioavailability. The highest K release by 0.2 M NaTPB at 144 h incubation was recorded in Entisols, while the least in Alfisols. Cumulative K released was mathematically best described by a parabolic diffusion equation and the empirical constants up to a 2 h incubation period was significantly related to relative biomass yield and K uptake by ryegrass. Considering relative biomass yield and K uptake as indices of plant K availability, the sequence of K supplying capacity of the soils under different orders was: Entisols>Inceptisols>Alfisols. The K-release rate of 30.8 mg kg-1 h-1 and the ryegrass leaf K concentration of 32.1 g kg-1 were found to be critical for this experiment. Based on the critical K release rate, the maximum amount of K released for plant uptake before showing any K deficiency symptom ranged from 179.3 mg kg-1 to 596 mg kg-1 for soils under the order Entisols, 247.9 mg kg-1 to 687.84 mg kg-1 for soils under Inceptisols and 118.67 mg kg-1 to 184.25 mg kg-1 for soils under Alfisols.

Absolutely stable fitted mesh scheme for singularly perturbed parabolic convection diffusion equations

Absolutely stable fitted mesh scheme for singularly perturbed parabolic convection diffusion equations

Kinetics release of heavy metal Cu from sediment affected by the mimic vegetation under unidirectional flows and regular waves

The process of heavy metal contamination in wetlands is becoming increasingly prevalent due to rapid urbanization. Heavy metals deposited in the sediment can be re-released into the water under disturbances, leading to significant impacts on the water environment and ecosystem. The release of heavy metals in wetlands is a complex phenomenon involving the intricate interplay of physicochemical and biological mechanisms. To gain insights into the adsorption-desorption processes of Cu under the influence of hydro-sediment dynamics, indoor experiments were conducted in this study, considering the presence of mimic vegetation at a short-term scale. The biological processes were not explicitly considered in these experiments. To analyze the kinetic processes of heavy metal migration at the water-sediment interface influenced by mimic vegetation under different hydrodynamic conditions, several equations were utilized, including the pseudo-second-order kinetic equation, double constant rate equation, Elovich equation, and parabolic diffusion equation. The pseudo-second-order kinetic equation can best simulate the Cu release process, with R2 ranging from 0.98 to 0.99. In order to enhance the accuracy of our predictions, the stem-Reynolds number (Red) is introduced into the original pseudo-second-order kinetic equation to account for the influence of mimic vegetation on the release of heavy metals. As a result of this improvement, the release processes of Cu were better predicted using the updated equations. The error between the improved equation and the actual measured value is reduced to less than 10 %. These improved methods are suitable for both unidirectional flow and wave conditions affected by mimic vegetation, and they offer vital parameters for numerical simulations of heavy metal transport in riverine or coastal wetlands. This research contributes to a better understanding of the complex processes governing heavy metal dynamics in wetland environments and provides valuable insights for effective environmental management and protection.

A fully-implicit hybridized discontinuous Galerkin spectral element method for two phase flow in petroleum reservoirs

A hybridized discontinuous Galerkin method with high order spectral element discretization is presented for modeling multiphase immiscible 2D fluid flow in petroleum reservoir simulations. The time dependent nonlinear governing equations for reservoir simulation are presented, and a fully coupled implicit time stepping procedure is implemented using a high order W-Rosenbrock algorithm. For incompressible fluids considered in the present study, the governing PDE's include an elliptic PDE for the pressure field and a parabolic convective diffusion equation for the saturation (water volume fraction) in a two phase oil-water system. The PDE systems are coupled through the highly nonlinear relative permeability and capillary pressure functions relating the fluid velocities of individual species to the individual species pressures and saturations. With HDG/spectral element (HDGSEM) spatial discretization of the coupled governing equations, the system constitutes an index 1 differential algebraic equation. It is shown that the W-Rosenbrock algorithm in combination with the HDGSEM method yields an efficient and effective solution procedure which demonstrates robust, high order temporal accuracy.Numerous physical tests cases are presented demonstrating that the versatility of this approach is sufficient to address the challenges in modeling heterogeneous permeability fields and geologic features of arbitrary shape with discontinuous permeability functions. It is also capable of resolving the sharp time dependent displacement fronts in oil-water systems encountered in secondary oil recovery. It is demonstrated that the high order polynomial basis functions (up to 12th order) used in the HDGSEM algorithm have the ability to resolve complex geometry and achieve accurate results utilizing a simple Cartesian mesh for non-conforming geological features. Eliminating the need to re-grid and conform the mesh to the boundaries of different geological features greatly simplifies the workflow for petroleum engineers.

Suitable Models for Describing Sulphate Desorption Kinetics in Selected Bauchi North Soils of Varying Parent Materials in the Nigerian Sudan Savanna

The modelling of sulphate desorption data is critical for a proper S diagnosis and fertilizer formulation to ensure profitable crop production. Five (5) models such as first-order, second-order, Elovich, fractional power, and parabolic diffusion were used to test the best model describing sulphate desorption kinetics in some soils from Bauchi-north, Sudan Savanna, Nigeria. To achieve this, soil samples were collected from three different parent materials namely Basement complex rock and two sedimentary rocks (Kerri-Kerri Formation and Chad Formation). The study showed that the parabolic diffusion and first-order models were found to describe S desorption data satisfactorily, characterized by relatively high R2 values and lowest S.E values by soil parent materials, respectively. While, the second-order, Elovich and Fractional power equations failed to describe the kinetics desorption of sulphate in all the studied soils, as judged by their high SE values. Therefore, the better fit of S desorption data to the first-order equation is an indication of probable ligand exchange of sulphate ion during the desorption process, and to parabolic diffusion equation suggests that diffusion-controlled phenomena are rate-limiting steps. Based on the findings, we concluded that the rate of sulphate desorption kinetics in these soils is mainly controlled by diffusion-controlled phenomena, which is critical for sulphate-based fertilizer formulations and applications.

Cyanide Release Characteristics of Solid Waste in Gold Smelting Process

To explore the relationship between total cyanide and easily liberatable cyanide in cyanide tailings and its impact on the environment, the leaching and release characteristics of total cyanide and easily liberatable cyanide in a tailings pond were studied using a dynamic leaching experiment, and a dynamic model was established. The results show that the release concentration of total cyanide in cyanide tailings is higher than that of easily liberatable cyanide. With the increase in leaching time, the release of cyanide is more sensitive to leaching intensity. The first-order kinetic, second-order kinetic, modified Elovich, double constant, and parabolic diffusion equations were used to fit the cumulative release of cyanide. According to the kinetic results of cyanide cumulative release, in addition to the first-order kinetic equation, the other four equations revealed good fitting for the cyanide leaching process, which shows that cyanide released under simulated rainwater leaching was not simply via surface diffusion, but controlled by multiple factors.

Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.

The Development of interfaces in a Parabolic p-Laplacian type diffusion equation with weak convection

This work has the objective to analyse the initial growth of interface and structure of nonnegative weak solution for one-dimensional parabolic p-Laplacian type diffusion-convection with non-positive convection coefficient c. In this situation, the interfaces may expand, shrink or remain stationary relying on the competition between these two factors. In this paper, we concentrate on three regions to classify the behavior of local solutions near the asymptotic interface in the irregular domain. In the first and second regions, the slow diffusion dominates over the convection term with expanding interfaces under some restrictions. In the third region, the slow diffusion dominates over the convection, but the interfaces have a waiting time. In our proof, the rescaling method and blow-up techniques are applied.

Решение задачи оптимального управления рассеянием тумана

Solving the problem of optimal control of fog diffusion / Klyomin V.V., Suvorov S.S. // Hydrometeorological Research and Forecasting, 2021, no. 2 (380), pp. 52-65. The paper discusses a possibility of applying one of the fundamental modern optimization methods, namely, the Pontryagin’s method for solving process control problems, whose behavior is described by the diffusion equation. The parabolic diffusion equation is discretized by the method of straight lines and comes to a closed system of ordinary differential equations, which allow finding an optimal control impact in terms of operating speed. The existence of a solution to the problem of optimal control of fog diffusion is proved for the mentioned sampling. The methodology for finding control action switching points is substantiated, the calculations for the revealed two and three switching moments are performed. Keywords: Pontryagin’s method, fog diffusion control, diffusion equation

Recovering the source term for parabolic equation with nonlocal integral condition

The main purpose of this article is to present a Tikhonov method to construct the source function f(x) of the parabolic diffusion equation. This problem is well known to be severely ill‐posed. Therefore, regularization is required. The error estimates between the sought solution and the regularized solution are obtained under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. One numerical test illustrates that the proposed method is feasible and effective.

Migration and Transformation of Heavy Metals in the Soil of the Water-Level Fluctuation Zone in the Three Gorges Reservoir under Simulated Nitrogen Deposition

The accumulation of heavy metals (HMs) in the water-level fluctuation zone (WLFZ) of the Three Gorges Reservoir (TGR) area is potentially harmful to the water environment. In order to reveal whether nitrogen (N) deposition is a potential driving factor for the migration and transformation of HMs (Cd, Cr, Cu, Ni, and Pb), a simulated N deposition experiment was performed on the soil in the WFLZ of the TGR. The results showed that the accumulative release amounts of HMs increased with the increase of N deposition. It was found that the Elovich equation, double-constant equation, and parabolic diffusion equation could well describe the release process of Cu, Cd, Cr, and Ni, while the double-constant equation, parabolic diffusion equation, and first-order equation could be applicable for Pb. The exchangeable fractions of HMs increased to varying degrees after the N deposition treatment, wherein Ni was most significant, indicating that N deposition could increase the ecological risk of HM pollution in the TGR area. The results provide insight into the major factors affecting the release of different HMs under N deposition in this vulnerable region ecologically.

Simulation of Biological Wastewater Treatment in Aerotanks with Moving Biocenosis

Purpose. The main purpose of the article is to develop a numerical model for the analysis of the process of biological wastewater treatment in a reactor with a mobile biocenosis. Methodology. For mathematical modeling of the process of biological wastewater treatment in a reactor with a moving biocenosis, a hydrodynamic model of a non-viscous vortex-free flow is used. We calculated the boundary conditions for the modeling equation on the surfaces of the bioreactor, solid walls, and the upper surface; at the inlet boundary; at the outlet boundary from the building. To calculate the process of movement of activated sludge and substrate in the bioreactor, a mass transfer model is used, which takes into account the convective-diffusion movement of the substrate and activated sludge. The process of biological water purification in that part of the bioreactor where there are no mobile biocenosis carriers will be calculated based on the Monod model. The process of biological water purification in the part of the bioreactor where there are mobile carriers is calculated on the basis of an empirical model in three stages. The first stage is determined using the Harremoes model. At the second stage, the rate of substrate «consumption» in the biofilm is calculated. At the third stage, the change in the substrate concentration in the zone where the biocenosis carriers are located is determined due to convective movement, substrate diffusion in the flow and its destruction in the biofilm on the carriers. The chaotic motion of biocenosis carriers in the reactor is modeled based on the parabolic diffusion equation. Finite-difference schemes are used for numerical integration of modeling equations. Findings. The software implementation of the constructed numerical model is carried out. A computational experiment to determine the efficiency of biological wastewater treatment in different parts of the bioreactor was conducted. Originality. An effective multifactorial numerical model has been created, which allows quick analysis of the efficiency of biological wastewater treatment in an aeration tank with mobile biocenosis carriers. Practical value. The created two-dimensional numerical model can be used for serial calculations at the stage of designing biological wastewater treatment systems and analysis of the efficiency of bioreactors under different operating conditions.

Distribution of phosphorus and the effect of physicochemical properties and clay minerals on phosphorus release in some calcareous soils

Phosphorus (P) is a nutrient essential for plant, which its availability in soils changes with aging process, leaching, precipitation processes, and the landform change. There is limited information available on kinetics release of P in calcareous soils. Therefore, the purpose of this study was to investigate distribution of P and the effect of physicochemical properties and clay minerals on its release in some dominant orders of the calcareous soils in Kohgiluyeh-and-Boyer-Ahmad province, using the 0.01 M calcium chloride ( ) extractant. The results showed that the P released rapidly from the soil and continued slowly. After 72 h, the amount of P released in the studied soils ranged between 3.2 and 25.4 mg kg-1. Evaluation of the fitted different equations on P released in the studied soils revealed that the Simple Elovich and power function equations could well predict the P release process in the studied soils. The results of the correlation between soil properties and released P contents showed that P release coefficients (including the slope and intercept in the Simple Elovich equation) have a significant negative relationship with the amount of smectite and vermiculite minerals. Also, the release coefficient of the parabolic diffusion equation had a significant negative relationship with pH and the amount of illite and palygorskite minerals.

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

In this work, we study the final value problem for a system of parabolic diffusion equations. In which, the final value functions are derived from a random model. This problem is severely ill‐posed in the sense of Hadamard. By nonparametric estimation and truncation methods, we offer a new regularized solution. We also investigate an estimate of the error and a convergence rate between a mild solution and its regularized solutions. Finally, some numerical experiments are constructed to confirm the efficiency of the proposed method.

Numerical Solution of a Two Dimensional Tumour Growth Model with Moving Boundary

We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity--fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, \cred{a viscous fluid model}, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the normal velocity of the outermost layer of cells, and this time--dependence is a challenge in designing and implementing a stable and fast numerical scheme. We recast the model into a form where the hyperbolic equation is defined on a fixed extended domain and retrieve the tumour boundary as the interface at which the cell volume fraction decreases below a threshold value. This procedure eliminates the need to track the tumour boundary explicitly and the computationally expensive re--meshing of the time--dependent domains. A numerical scheme based on finite volume methods for the hyperbolic conservation law, Lagrange $\mathbb{P}_2 - \mathbb{P}_1$ Taylor--Hood finite element method for the \cred{viscous} system, and mass--lumped finite element method for the parabolic equations is implemented in two spatial dimensions, and several cases are studied. We demonstrate the versatility of the numerical scheme in catering for irregular and asymmetric initial tumour geometries. We present numerical simulations for both cases and the results are consistent with theoretical and heuristic expectations such as early linear growth rate and preservation of radial symmetry when the boundary conditions are symmetric.

Regularization and error estimate for an initial inverse nonlocal diffusion problem

We study the backward problem of determining the initial condition for a system of parabolic diffusion equations, which is severely ill-posed in the sense of Hadamard. To stabilize the solution, we develop the quasi-reversibility (QR) and Fourier truncation methods to construct the regularized solutions. We also investigate the error estimates and convergence rates between the regularized solutions and the true solution in L2− and H1−norm. A numerical scheme is presented.

Numerical simulation of mathematical heart model in COMSOL

Electrical activity is essential for the cardiac cell to perform its function. Mathematical modeling of cardiac electrical activity is performed from the cell, tissue and organ levels through to the body surface level. The electrical activity of the cardiac as a whole is thus characterized by a complex multiscale structure. The most complete model of such a complex setting is the anisotropic bidomain model that consists of a system of two degenerate parabolic reaction diffusion equations describing the intra and extracellular potentials in the cardiac muscle, coupled with a system of ordinary deferential equations describing the ionic currents flowing through the cellular membrane. This study describes an anatomically realistic 3D Bidomain model of whole-heart electrical activity. The heart was embedded in a human torso, incorporating spontaneous activation with heterogeneous action potential (AP) morphologies throughout the heart. The aim of this study is the development of a geometrically simple and computationally efficient 3D model of heart. In this paper a finite element formulation, model and simulation of Bidomain equation has been conducted. The FitzHugh-Nagumo (FHN) equations were incorporated into Bidomain model of cardiac electrical activity, which was comprised of a simplified geometry of the whole heart with the torso as an extracellular volume conductor. Laplace equation for the torso also considered. Simplified 3D cardiac model was implemented using COMSOL Multiphysics 5.0 finite element software. Electrical potential at different point on torso is measured. Propagation of electrical excitation on heart surface is also observed. This study represents the first stage toward the development of an accurate computer model of heart activation.  Â