Class Of Diffusion Equations Research Articles

A Direct Method to Approximate Solution of the Space-fractional Diffusion Equation

A class of partial differential equations with fractional derivatives in the spatial variables are called space-fractional diffusion equations. They can be applied to simulate anomalous diffusion, in which the classical diffusion equation does not accurately describe how a plume of particles disperses. Analytically solving fractional diffusion equations can be problematic due to the typically complex structures of fractional derivative models. Hence, this study proposes the utilisation of a satisfier function in combination with the Ritz method to effectively address fractional diffusion equations in the Caputo sense. By employing this approach, the equations are transformed into an algebraic system, so facilitating their solution and providing a numerical result. This method can achieve a high level of accuracy in solving the Caputo fractional diffusion equations by utilising only a small number of terms from the shifted Legendre polynomials in two variables. The precision and effectiveness of our approach may be evaluated, as it yielded dependable approximations of the solutions.

DECAY RATES AND INITIAL VALUES FOR TIME-FRACTIONAL DIFFUSION-WAVE EQUATIONS

We consider a solution u(·,t) to an initial boundary value problem for time-fractional diffusion-wave equation with the order α ∈ (0,2)\ {1} where t is a time variable. We first prove that a suitable norm of u(·,t) is bounded by the rate t −α for 0 α 1 and t 1−α for 1 α 2 for all large t > 0. Second, we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation α = 1, the decay rate can keep some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.

Galerkin-finite difference method for fractional parabolic partial differential equations

The fractional form of the classical diffusion equation embodies the super-diffusive and sub-diffusive characteristics of any flow, depending on the fractional order. This study aims to approximate the solution of parabolic partial differential equations of fractional order in time and space. For this, firstly, we briefly discussed on some existing methods to solve Partial Differential Equations (PDEs) and fractional Differential Equations (DEs), then introduce a combined technique such as the Galerkin weighted residual method for the space fractional term with modified Bernoulli polynomials as basis functions, and the finite difference approximation for the time fractional term, respectively. The mathematical formulation of the proposed method is explained elaborately. Then we describe the order of convergence for the time fractional term only, as the convergence of the Galerkin method is obvious. We impose this technique on the fractional Black-Scholes model subsequently. Finally, we experimented with our proposed technique on some numerical problems. All the results are depicted in both tabular and 3D visualizations as well. We compare our results with the available methods in the literature, and our accuracy is considerable. To summarize: •The paper introduces an approach that integrates the Galerkin weighted residual method with modified Bernoulli polynomials to handle space fractional terms, alongside employing a finite difference approximation for time fractional terms.•The convergence analysis is focused.•The technique is implemented on the fractional Black-Scholes model and other numerical problems, with outcomes depicted through tables and 3D visualizations.

Numerical approximation of a two-dimensional fractional neutron diffusion model describing dynamics of neutron flux in a nuclear reactor

The goal of this work is to develop a high-order numerical technique to solve a two-dimensional fractional neutron diffusion (FND) model describing dynamical behavior of a lead-cooled fast reactor. This method is based on L1 technique for temporal discretization and a high-order compact Alternating Direction Implicit (ADI) finite difference scheme for the spatial discretization. Numerical simulations are conducted to demonstrate the applicability of the method. We calculate the neutronic flux in the core and examine how the order of the temporal-fractional derivative influences the neutronic flux. The results obtained with FND model for different values of anomalous diffusion coefficient are compared against those obtained with P1 model and classical neutron diffusion equation. In addition, the computational time (CPU time) is provided in order to justify the computational efficiency of the proposed method.

Reinvestigating the Kinetic Model for the Suspended Sediment Concentration in an Open Channel Flow

The prediction of sediment transport, related to different environmental and engineering problems, requires accurate mathematical models. Most available mathematical models for the concentrations of suspended sediments are based on the classical advection diffusion equation, which remains not efficient enough to describe the complete behavior related to sediment–water two-phase flows and the feedback between the turbulent unsteady flow and suspended sediments. The aim of this paper is to reinvestigate the kinetic model for turbulent two-phase flows, which accounts for both sediment–turbulence interactions and sediment–sediment collisions. The present study provides a detailed and rigorous derivation of the kinetic model equations, clarifications about the mathematical approach and more details about the main assumptions. An explicit link between the kinetic model and the classical advection diffusion equation is provided. Concentration profiles for suspended sediments in open channel flows show that the kinetic model is able to describe the near-bed behavior for coarse sediments.

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

A Differential Diffusion Theory for Participating Media

AbstractWe present a novel approach to differentiable rendering for participating media, addressing the challenge of computing scene parameter derivatives. While existing methods focus on derivative computation within volumetric path tracing, they fail to significantly improve computational performance due to the expensive computation of multiply‐scattered light. To overcome this limitation, we propose a differential diffusion theory inspired by the classical diffusion equation. Our theory enables real‐time computation of arbitrary derivatives such as optical absorption, scattering coefficients, and anisotropic parameters of phase functions. By solving derivatives through the differential form of the diffusion equation, our approach achieves remarkable speed gains compared to Monte Carlo methods. This marks the first differentiable rendering framework to compute scene parameter derivatives based on diffusion approximation. Additionally, we derive the discrete form of diffusion equation derivatives, facilitating efficient numerical solutions. Our experimental results using synthetic and realistic images demonstrate the accurate and efficient estimation of arbitrary scene parameter derivatives. Our work represents a significant advancement in differentiable rendering for participating media, offering a practical and efficient solution to compute derivatives while addressing the limitations of existing approaches.

Analytic solutions to the two-dimensional solute advection-dispersion equation coupled with heat diffusion equation in a vertical aquifer section

Analytical and semi-analytical solutions to solute transport equations are not only used to predict the fate and transport of contaminants in soil or groundwater systems, but also to provide reference standards for validation of numerical models. However, when the solute transport equation is coupled with heat, the corresponding analytical solutions are rarely reported, perhaps due to the complexity of the coupled models. The main purpose of this paper is to present an analytical solution to the solute transport equation coupled with heat for the case of modeling solute transport in a vertical section of a homogenous infinite aquifer with steady uniform groundwater flow under constant-flux solute source. In earlier work (Shan and Javandel, 1997), analytical solutions to the solute transport equation in both finite and infinite aquifer thickness were presented, influence of heat (Soret effect) was generally not considered. Heat can, however, have a significant impact on movement of concentration front. In the present work, we first couple the solute equation with a generalized classic heat diffusion equation. To do this, we assume that the heat is not influenced by the solute (Dufour effect) such that only a one-way coupled model is considered. The repeated integral transformation method (RITM) is utilized to obtain the analytical solutions to the proposed coupled model. As a limiting case, a comparison with analytical solution (Shan and Javandel, 1997), equation (21) and other literature findings is done verify the new proposed solutions and validate our results. The influences of the thermopheresis effect and other model parameters are pictured graphically by the use of MATLAB R2015a. In comparison with the earlier work, the present results show that movement of contaminant in the medium is affected by the presence of heat, in particular along the direction of flow, consistent with the literature. The analysis of longitudinal and transverse concentration profiles suggest that the proposed solutions are applicable for monitoring of groundwater contaminants and risk assessment. Also, the solutions should be useful for comparison with numerical models.

Implementation, Validation and Study Case of a Landscape Evolution Model Algorithm

This paper presents a landscape evolution model based on physical processes – hillslope processes and fluvial erosion, transport, and deposition – solved by numerical methodology. That is, through the solution of differential equations approximated by numerical methods. In this case, hillslope processes are modeled through the classical diffusion equation, discretized by the finite volume method. Fluvial erosion, transport, and deposition are modeled by the fluvial potential equations (stream power law). For this, the approximation is performed by the finite difference method. The topography – initial condition – is set by digital elevation models, obtained from satellite images. These are Raster datasets, that each cell contains a representative elevation value. The drainage is determined through the classical algorithm D8, which performs a scan on the digital elevation model, tracing routes of greater slopes between the cells. The algorithm execution flowchart is presented, and the model is validated. Finally, a geomorphological study is presented in the Piratini river basin, showing thar developed model mimics largescale natural phenomena of watershed processes.

Inverse problems for fractional Schrödinger and subdiffusion equations

The inverse problems of determining the right-hand side of the Schrödinger and the sub-diffusion equations with the fractional derivative is considered. In the problem 1, the time-dependent source identification problem for the Schrödinger equation , in a Hilbert space is investigated. To solve this inverse problem, we take the additional condition with an arbitrary bounded linear functional . In the problem 2, we consider the subdiffusion equation with a fractional derivative of order , and take the аbstract operator as the elliptic part. The right-hand side of the equation has the form , where is a given function and the inverse problem of determining element is considered. The condition is taken as the over-determination condition, where is some interior point of the considering domain and is a given element. Obtained results are new even for classical diffusion equations. Existence and uniqueness theorems for the solutions to the problems under consideration are proved.

Rapid Multicomponent Alloy Solidification with Allowance for the Local Nonequilibrium and Cross-Diffusion Effects.

Motivated by the fast development of various additive manufacturing technologies, we consider a mathematical model of re-solidification of multicomponent metal alloys, which takes place after ultrashort (femtosecond) pulse laser melting of a metal surface. The re-solidification occurs under highly nonequilibrium conditions when solutes diffusion in the bulk liquid cannot be described by the classical diffusion equation of parabolic type (Fick law) but is governed by diffusion equation of hyperbolic type. In addition, the model takes into account diffusive interaction between different solutes (nonzero off-diagonal terms of the diffusion matrix). Numerical simulations demonstrate that there are three main re-solidification regimes, namely, purely diffusion-controlled with solute partition at the interface, partly diffusion-controlled with weak partition, and purely diffusionless and partitionless. The type of the regime governs the final composition of the re-solidified material, and, hence, may serve as one of the main tools to design materials with desirable properties. This implies that the model is expected to be useful in evaluating the most effective re-solidification regime to guide the optimization of additive manufacturing processing parameters and alloys design.

Renewal equationsfor single-particle diffusion through a semipermeable interface.

Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional diffusion through a barrier with constant permeability κ_{0} can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflected BMs that are restricted to either the left or right of the barrier. Each round is killed (absorbed) at the barrier when its Brownian local time exceeds an exponential random variable parameterized by κ_{0}. A new round is then immediately started in either direction with equal probability. It has recently been shown that the probability density for snapping out BM satisfies a renewal equationthat relates the full density to the probability densities of partially reflected BM on either side of the barrier. Moreover, generalized versions of the renewal equationcan be constructed that incorporate non-Markovian, encounter-based models of absorption. In this paper we extend the renewal theory of snapping out BM to single-particle diffusion in bounded domains and higher spatial dimensions. In each case we show how the solution of the renewal equationsatisfies the classical diffusion equationwith a permeable boundary condition at the interface. That is, the probability flux across the interface is continuous and proportional to the difference in densities on either side of the interface. We also consider an example of an asymmetric interface in which the directional switching after each absorption event is biased. Finally, we show how to incorporate an encounter-based model of absorption for single-particle diffusion through a spherically symmetric interface. We find that, even when the same non-Markovian model of absorption applies on either side of the interface, the resulting permeability is an asymmetric time-dependent function with memory. Moreover, the permeability functions tend to be heavy tailed.

Fractional derivatives for the core losses prediction: State of the art and beyond

The core losses prediction in laminated magnetic circuits has generated relentless scientific research efforts. In this domain, the almost forty years old Statistical Theory of Losses (STL) is still prevailing. Modern electrical energy conversion exposes laminated magnetic circuits to higher frequencies and larger excitation fields. STL, which requires a full flux penetration, poorly considered these intense stimuli. Recent studies proposed upgrading STL by modifying the classical eddy current loss term using fractional derivative operators. Excellent predictions were observed, but the lumped aspect inherent to STL means working with averaged quantities which constitutes a limitation. Alternatively, local predictions and good simulation trajectories were obtained for the frequency dependence of hysteresis by replacing the classic magnetic diffusion equation with a fractional one. Still, the physical origin of the fractional diffusion equation remains questionable. In the light of these results, we propose in this new study to review the use of the fractional derivative operators for the core losses prediction. We eventually revealed an alternative way to combine the classic diffusion equation to a fractional material law, leading to the ultimate fractional method, with physical fundaments, few parameters, and accurate results on significant frequency bandwidths.

The gas permeability properties of poly(vinyltrimethylsilane) treated by low‐temperature plasma

AbstractThe results of one‐sided treatment of polyvinyltrimethylsilane (PVTMS) films using low‐temperature air plasma are presented. The treatment was carried out at 25°C in air at a pressure in the reaction chamber of 10–25 Pa and a processing time of 10–60 s. The surface of the plasma‐treated PVTMS films was analyzed by X‐ray photoelectron spectroscopy and atomic force microscopy and the contact angle values determined. The effect of the plasma conditions, such as the treatment time and the pressure in the reactor chamber, on the transport properties of the modified membrane for He, O2, N2, CO2, and CH4 were investigated in detail. The permeability of all plasma‐treated PVTMS films for the studied gases except for He was found to decrease depending on the modification conditions, leading to an increase in selectivity for such pairs as He/CH4, CO2/CH4, and O2/N2. The selectivity of the modified PVTMS achieved a maximum value when the films were treated with plasma for 20–40 s at a chamber pressure of 15–20 Pa. The effective diffusion coefficients of the studied gases were obtained experimentally. It was shown that the volumetric diffusion of gases, particularly O2 and N2, in PVTMS before as after plasma treatment is described by the classical diffusion equations. The theoretical approaches to the influence of boundary conditions on selective gas transfer through polymeric membrane are considered. To estimate the prospects for the long‐term use of membranes based on modified PVTMS, the stability of the parameters gas permeability and selectivity of the modified samples was investigated over 9 months. It was shown that the modification of PVTMS by the method of low‐temperature plasma in air is a promising method for expanding the spectrum of membrane processes based on existing commercial membranes.

Data smoothing with applications to edge detection

Abstract The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} with N ≥ 1 N\ge 1 . The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.

Recovery of the Order of Derivation for Fractional Diffusion Equations in an Unknown Medium

In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion-type equations, including both classical and time-fractional diffusion equations, from the flux measurement observed at one point on the boundary. The mathematical model for time-fractional diffusion equations involves a Djrbashian--Caputo fractional derivative in time. We prove a uniqueness result in an unknown medium (e.g., diffusion coefficients, obstacle, initial condition, and source), i.e., the recovery of the order of derivation in a diffusion process having several pieces of unknown information. The proof relies on the analyticity of the solution at large time, asymptotic decay behavior, strong maximum principle of the elliptic problem, and suitable application of the Hopf lemma. Further we provide an easy-to-implement reconstruction algorithm based on a nonlinear least-squares formulation, and several numerical experiments are presented to complement the theoretical analysis.

Theoretical Analysis on the Stability of Single Bulk Nanobubble

As predicted by classical macroscopic theory, the lifetime for nanoscale gas bubbles is extremely short, which causes conflict when detecting stable bulk nanobubbles experimentally in recent years. In fact, the stability of bulk nanobubbles depends on the surrounding liquid environment. Also, the dynamic process of gas in water involves the dissolution, diffusion, release, and transportation of gas as well as the properties of nanobubbles inside. Here, based on previous reports, we introduce the gas transport parameter ℓ in the classical diffusion equation by considering the gas diffusion near the bulk nanobubble at different locations in a container and consider the MacLeod-Sugden relationship between the surface tension and densities of liquid and gas for computing the lifetime of single bulk nanobubbles in an open system. The results show that the single nanobubble lifetime depends on the inner density and gas transport length. It could reach the order of 10–100 s for a single nanobubble with an initial radius of 200 nm, and provides a new idea to prolong the lifetime of the single bulk nanobubble. Meanwhile, compared with the continuous influence of the inner density on the gas diffusion flux near the nanobubble, the range of the gas transport near the nanobubble on the gas diffusion flux is limited, which is affected by the dissolution time of the nanobubble. Our findings would be helpful to explore the storage conditions of nanobubbles and the mechanism of mass transfer at the gas-liquid interface at the micro-scale.

Optimal convergence rate of the explicit Euler method for convection–diffusion equations

Revisiting the explicit Euler method of the classical diffusion equation, a new difference scheme with the optimal convergence rate four is achieved under the condition of the specific step-ratio r=1/6. Applying the corrected idea to the convection–diffusion equation, a new corrected numerical scheme is obtained which owns a similar fourth-order optimal convergence rate. Rigorous numerical analysis is carried out by the maximum principle. Compared with the standard difference schemes, the new proposed difference schemes have obvious advantage in accuracy. Extensive numerical examples with and without exact solutions confirm our theoretical results. Moreover, extending our technique to nonlinear problems such as the Fisher equation and viscous Burgers’ equation is available.

Multiplicity of nodal solutions in classical non-degenerate logistic equations

<abstract><p>This paper provides a multiplicity result of solutions with one node for a class of (non-degenerate) classical diffusive logistic equations. Although reminiscent of the multiplicity theorem of López-Gómez and Rabinowitz [<xref ref-type="bibr" rid="b1">1</xref>, Cor. 4.1] for the degenerate model, it inherits a completely different nature; among other conceptual differences, it deals with a different range of values of the main parameter of the problem. Actually, it is the first existing multiplicity result for nodal solutions of the classical diffusive logistic equation. To complement our analysis, we have implemented a series of, very illustrative, numerical experiments to show that actually our multiplicity result goes much beyond our analytical predictions. Astonishingly, though the model with a constant weight function can only admit one solution with one interior node, our numerical experiments suggest the existence of non-constant perturbations, arbitrarily close to a constant, with an arbitrarily large number of solutions with one interior node.</p></abstract>

Convolution integral for fractional diffusion equation

A modified version of the classical diffusion equation, called the fractional diffusion equation (FDE), has been developed to describe anomalous fluid flow through porous media. Since the FDE does not contain the rate as the input of the well-reservoir system, in this paper, we re-develop the FDE containing both rate (input) and pressure (output). Based on the new FDE, we show that the conventional convolution integral can still be used to relate the rate and pressure in reservoirs with anomalous behavior. A practical application of the convolution integral and how to use it in practice is presented using a synthetic example.