Diffusion Equation Research Articles

Initial Value and Terminal Value Problems for Distributed Order Fractional Diffusion Equations

Initial Value and Terminal Value Problems for Distributed Order Fractional Diffusion Equations

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.

Development of HLB3-002 (recombinant human PH20 stand-alone) and its applicability as an anticancer diffusion agent.

e15195 Background: HLB3-002 is a standalone of Halozyme's rHuPH20 (Original) and has the same amino acid sequence. HLB3-002 is being developed as a drug diffusion agent that decomposes subcutaneous hyaluronan, allowing substances such as anticancer drugs to be administered subcutaneously. This study was conducted to compare the in vivo efficacy and PK of original and HLB3-002 to confirm equivalence and thereby prove its applicability as a drug diffusion agent. Methods: In the in vivo efficacy test, Trypan blue was mixed with each original and HLB3-002 and administered subcutaneously to confirm diffusion and reconstruction equivalence in the nude mouse. The PK test was conducted using Rituximab, prepared in the same formulation as Rituximab SC by adding HLB3-002. A single dose was administered subcutaneously to the nape of the rat, and 12 points were collected for 28 days. The PK samples were measured by ELISA. When the Cmax and AUClast ratios of Rituximab between 2 groups were within 80.00 to 125.00%, they were evaluated as biologically equivalent. Results: As a result of efficacy test, the diffusion area of HLB3-002 was 104.7% compared to the original, which was confirmed to have an equivalent diffusion efficacy. In the skin reconstruction test, it was confirmed that both original and HLB3-002 rebuilt the skin layer to a similar to the negative control after 48 hours. As a result of the PK, Compared to Rituximab SC, the Cmax of Rituximab+HLB3-002 was similar at 98.53% and AUClast at 95.50%. Conclusions: Through an in vivo efficacy and PK test, we confirmed an equivalent drug diffusion effect, confirming that HLB3-002 applicability as an anticancer drug diffusion agent.[Table: see text]

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times t=T1 and t=T2. The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an a priori and an a posterior parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.

A Conservative and Positivity-Preserving Method for Solving Anisotropic Diffusion Equations with Deep Learning

A Conservative and Positivity-Preserving Method for Solving Anisotropic Diffusion Equations with Deep Learning

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

A Direct Discontinuous Galerkin Method for Time Fractional Diffusion Equations with Fractional Dynamic Boundary Conditions

A Direct Discontinuous Galerkin Method for Time Fractional Diffusion Equations with Fractional Dynamic Boundary Conditions

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

The role of adhesion on soft lubrication: A new theory

Recent experiments reveal that adhesive interactions can play a key role in causing surface instability in soft lubrication. Instances of instability include fluid entrapment in isolated pockets upon a soft sphere's normal contact with a hard substrate and surface wrinkling of a soft substrate as a hard sphere slides across it. These phenomena underscore a substantial distinction between hard and soft lubrication. They are of paramount importance from a fundamental standpoint, providing an entirely new explanation for the transition mechanism from elasto-hydrodynamic to the mixed lubrication regimes. Here, we introduce a new theory to elucidate these observations. Our theory modifies the Reynolds elasto-hydrodynamic equation by incorporating adhesive interaction across the fluid layer, investigating the interplay between adhesion, fluid flow and elastic instability. Our analysis proposes the addition of a new dimensionless parameter in lubrication theory, that compares the stiffness of the adhesive interaction to that of the substrate. When this parameter exceeds unity, the soft solid surface exhibits instability to small perturbations in its shape. In mathematical terms, the Reynolds equation undergoes a transition from a nonlinear diffusion equation to a nonlinear wave equation at this critical point. Post-transition, the diffusivity of the nonlinear diffusion equation turns negative, rendering the problem ill-posed. We investigate the transition using the method of characteristics and present an exact analytic solution. This solution offers insights into the occurrence of a vanishing liquid film thickness at specific locations, resulting in dry contact—initiating transition to mixed lubrication.

Nonlinear two-component system of time-fractional PDEs in [formula omitted]-dimensions: Invariant subspace method combined with variable transformation

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction–diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

A fast solvable operator-splitting scheme for time-dependent advection diffusion equation

It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.

Emergence of non-trivial solutions from trivial solutions in reaction–diffusion equations for pattern formation

Reaction–diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction–diffusion models: the Schnakenberg model and the Gray–Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models.

Time-fractional fabric to quantify non-Fickian diffusion in porous media: New vision from previous studies

The diffusion kinetics of different substances in porous media is a priori described by the second Fick's law of diffusion. Herein, we demonstrate that such a description sometimes yields erratic results. In contrast, the time-fractional diffusion equation is found to reflect the diffusion kinetics in the case of Fick's law failure. The analytic solutions of the time-fractional diffusion equation are based on the Mittag–Leffler function. At large times, it is shown that the utilization of the approximation of the Mittag–Leffler function for the large values of its argument for the description of the experimental data gives almost the same results as those obtained using the Mittag–Leffler function itself. The time-fractional diffusion is applied on a phenomenological basis. Nevertheless, we speculate that the deviations from Fick's law may be governed by strong adsorption of the diffusing species inside the small micropores of a porous material.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra–Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schläfli integral of the Volterra–Bessel function. In this respect, we study the asymptotic analysis of the Volterra–Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

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.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Research on multi‐mechanism synergistic effects of gas cross‐scale percolation in tight gas reservoirs of Yanchang oilfield

AbstractRevealing the gas transport patterns at the microscale and nanoscale is of great scientific and engineering significance for unconventional oil and gas exploration and development. This paper first systematically analyzes the properties of gas transport mechanism under different Knudsen number conditions. Second, the coupling modes of the cross‐scale multi‐drive mechanism are systematically categorized, and three typical coupling calculation methods are pointed out. Again, the rationality of the cross‐scale multi‐agency coupling is quantitatively calculated. The results show that (a) Under the influence of Fick's diffusion equation, when Knr > 0, the coupled ‘slip flow–Fick diffusion–Knudsen diffusion’ equation does not converge to the Knudson diffusion equation. (b) The ‘Fick diffusion–Knudsen diffusion’ coupled equation is calculated to be more than 102 times as large as the Knudsen diffusion equation.

Drying Kinetics and Thermodynamic Properties of Ultrasound Pretreatment Bitter Melon Dried by Infrared

Determination of drying characteristics of ripe and unripe bitter melon in an infrared dryer at 50, 60, and 70°C with airborne ultrasonic pretreatment with a power of 20 kW for 0, 10, and 20 minutes and a frequency of 20 kHz is the purpose of this study. At the end of drying, among the five used models that fit the moisture ratio data, the Midilli and logarithmic models were selected to properly characterize the drying behavior in the infrared dryer with ultrasonic pretreatment of ripe and unripe bitter melon samples. Moisture transfer from bitter melon samples’ ripeness and early maturity was defined using Fick’s diffusion equation. Then, the Arrhenius equation was utilized to determine the effective moisture diffusivity. Also, the activation energy of unripe and ripe bitter melon slices was reduced with the enhancement in the ultrasound pretreatment time. Water activity (aw) amounts of ripe and unripe bitter melon were achieved at various drying temperatures and ultrasound pretreatment. The trends of the experiment indicate a decrease in enthalpy (ΔH) and entropy (ΔS) amounts of bitter melon with enhancing temperature and ultrasonic. The Gibbs free energy (ΔG) increases with enhancing drying temperature and ultrasonic pretreatment. Specific energy consumption decreased with enhancing drying temperature and ultrasound pretreatment duration for both ripe and unripe bitter melon samples.

Fixed Point Results with Applications to Fractional Differential Equations of Anomalous Diffusion

The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and (α,η)-Reich type interpolative contraction in the framework of orthogonal F-metric space and prove some fixed point results. Our primary result serves as a cornerstone, from which established findings in the literature emerge as natural consequences. To enhance the clarity of our novel contributions, we furnish a significant example that not only strengthens the innovative findings but also facilitates a deeper understanding of the established theory. The concluding section of our work is dedicated to the application of these results in establishing the existence and uniqueness of a solution for a fractional differential equation of anomalous diffusion.

Study of the Accuracy of the Ronen Method at Material Interfaces

The Ronen Method (RM) is a nonlinear iterative scheme that demands successive resolutions of the diffusion equation, where local diffusion constants are modified to reproduce more accurate estimates of the neutron currents by a transport operator. The methodology is currently formulated using the formalism of the collision probability method for evaluation of the current. The RM was recently tested on a complete suite of one-dimensional (1-D) multigroup benchmark problems. Small differences in the flux (less than 2%) were reported at material interfaces and close to the vacuum boundary with respect to the reference solution from transport. This work investigates first a possible numerical equivalence between transport and diffusion in some representative 1-D problems from the same benchmark test suite. The equivalence is sought with optimal diffusion coefficients computed using reference transport solutions that allow for adjusting the diffusion model. The RM, which attempts to obtain such equivalent diffusion coefficients without knowing the reference solution, is then compared to the optimal coefficients. The accuracy of the flux distribution at material interfaces is investigated for different approximations of the vacuum boundary and by decreasing progressively the RM convergence tolerance set in the iterative scheme. Using tighter convergence criteria, the RM calculates more accurate flux distributions at all material interfaces, regardless of the value of the diffusion coefficient and the extrapolated distance set at the beginning of the iterative scheme. Maximum flux deviations are remarkably reduced when the RM convergence tolerance is set to eight or more significant digits, leading to improvements in the flux deviation of two orders in magnitude and providing numerical proof for equivalence with transport in the tested configurations.