Numerical Solution Of Diffusion Equation Research Articles

Numerical Solution of Diffusion Equation with Caputo Time Fractional Derivatives Using Finite-Difference Method with Neumann and Robin Boundary Conditions

Many problems in various branches of science, such as physics, chemistry, and engineering have been recently modeled as fractional ODEs and fractional PDEs. Thus, methods to solve such equations, especially in the nonlinear state, have drawn the attention of many researchers. The most important goal of researchers in solving such equations has been set to provide a solution with the possible minimum error. The fractional PDEs can be generally classified into two main types, spatial-fractional, and time-fractional differential equations. This study was designed to provide a numerical solution for the fractional-time diffusion equation using the finite-difference method with Neumann and Robin boundary conditions. The time fraction derivatives in the concept of Caputo were considered, also the stability and convergence of the proposed numerical scheme have been completely proven a numerical test was also designed and conducted to assess the efficiency and precision of the proposed method. Eventually, it can be said that based on the findings, the present technique can provide accurate results.

Bio‐Imaging quorum sensing signal molecules in a soil‐mimic gel

Rhizosphere, the narrow region surrounding plant roots directly influenced by root exudates and the root associated microbiome, plays an important role in plant productivity and the rhizosphere is well known for stimulating microbial metabolic activities. How microbial communities interact to form stable, metabolically interconnected functional communities is an area of intense interest. The question remains on how microbially produced secreted molecules that function as intercellular communication signals shape the structure and function of microbial communities. Our goal is to detect and quantify signal molecules produced by microbes or plants in the root‐soil environment, spatially and temporarily. As a proof‐of‐concept, we are imaging diffusible extracellular microbial metabolites involved in a bacteria‐bacteria communication process called quorum‐sensing. Quorum‐sensing relies on the accumulation of high concentrations of signal molecules in the environment to control bacterial gene expression, influencing rhizosphere colonization and plant health. Local concentration of quorum sensing molecules are determined using aptamer based sensors. Aptamers that can specifically bind the desired signal molecules are selected through SELEX and immobilized on the surface of nano‐porous membranes. Binding of the signal molecules and aptamer covered surface results in changes in surface charge distribution and steric hinderance and thus modifying the transmembrane ionic transport. Changes in transmembrane impedance can be measured through electrochemical impedance spectroscopy methods to monitor local concentrations of signal molecules. Sensor responses were determined for different concentrations of signal molecules and results showed detection of C4‐HSL in a soil‐mimic solution with KD of 10 nM. We demonstrate that the quorum‐sensing signal molecule C4‐homoserine lactone and (C4‐HSL) can rapidly diffuse in a soil‐mimic gel, providing evidence for our use of a soil‐mimic for further aptasensor development and validation. The sensors were then inserted into soil mimic gel for monitoring C4‐HSL diffusion and the resulted impedance changes were used to determine the C4‐HSL concentration at different positions in the gel. Measurements of local C4‐HSL concentrations variation and numerical solution of diffusion equation were used to create 4D images of C4‐HSL molecule diffusion in the soil‐mimic gel.

Isogeometric analysis of diffusion problems on random surfaces

In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. We describe in detail, how diffusion problems on random surfaces can be modelled and how quantities of interest may be derived. In particular, we employ a low rank approximation algorithm for the high-dimensional space-time correlation of the random solution based on an online singular value decomposition, cp. [7]. Extensive numerical studies are performed to validate the approach. In particular, we consider complex computational geometries originating from surface triangulations. The latter can be recast into the isogeometric context by transforming them into quadrangulations using the procedure from [41] and a subsequent approximation by NURBS surfaces.

Numerical Solution of Diffusion Equation with Caputo Time Fractional Derivatives Using Finite-Difference Method with Neumann and Robin Boundary Conditions

Many problems in various branches of science, such as physics, chemistry, and engineering have been recently modeled as fractional ODEs and fractional PDEs. Thus, methods to solve such equations, especially in the nonlinear state, have drawn the attention of many researchers. The most important goal of researchers in solving such equations has been set to provide a solution with the possible minimum error. The fractional PDEs can be generally classified into two main types, spatial-fractional, and time-fractional differential equations. This study was designed to provide a numerical solution for the fractional-time diffusion equation using the finite-difference method with Neumann and Robin boundary conditions. The time fraction derivatives in the concept of Caputo were considered, also the stability and convergence of the proposed numerical scheme have been completely proven and a numerical test was also designed and conducted to assess the efficiency and precision of the proposed method. Eventually, it can be said that based on findings, the present technique can provide accurate results.

Numerical Solution of Diffusion Equation using a Method of Lines and Generalized Finite Differences

One of the greatest challenges in the area of applied mathematics continues to be the design of numerical methods capable of approximating the solution of partial differential equations quickly and accurately. One of the most important equations, due to the hydraulic and transport applications it has, and the large number of difficulties that it usually presents when solving it numerically is the Diffusion Equation. In the present work, a Method of Lines applied to the numerical solution of the said equation in irregular regions is presented using a scheme of Generalized Finite Differences. The second-order finite difference method uses a central node and 8 neighbor points in order to address the spatial approximation. A series of tests and numerical results are presented, which show the accuracy of the proposed method.

Investigation on the fusion reaction rate of deuterium and tritium under heterogeneous mixing

Mixing between shell material and gas fuel, caused by hydrodynamic instability, isolated defects, or kinetic effects, is the key to understand the degradation of implosion performance in the research of inertial confinement fusion. Understanding the mixing mechanism and reducing its impact is of extreme importance to achieve the ignition and high gain. The impact of mixing morphology on thermonuclear reaction rate in sub grid level has gradually attracted people’s attention in recent years due to its direct influence on burn rate and fusion process, the study on physical model of thermonuclear reaction rate in different mix morphology has important scientific significance and application value. In the paper, the dependence of thermonuclear reaction rate on mass distribution of different fuel concentrations at sub grid scale is derived. Based on thermodynamic equilibrium and ideal gas equation of state, the physical law of the evolution of the thermonuclear reaction rate with mix morphology under the dominance of diffusion mixing is revealed through analytical formula and numerical solution of diffusion equation in one-dimensional spherical geometry. It is convinced that the mixing amount directly affects the thermonuclear reaction rate by mainly affecting the volume fraction of the fuel, and the mixing diffusion time determined by heterogeneous mixing scale and diffusion coefficient directly affects the evolution behavior of the thermonuclear reaction rate. Furthermore, based on mutual diffusion coefficient obtained from direct simulation of diffusion process by Monte Carlo method, the difference of impact to thermonuclear reaction rate for low-<i>Z</i> Carbon and high-<i>Z</i> gold mixing is quantitatively investigated. Heterogeneous mix size with 0.1 μm, 0.01 μm respectively for the low-<i>Z</i> and high-<i>Z</i> mixing can be treated as atomic mix in burn rate aspect, and heterogeneous mix size with 10 μm, 1 μm respectively for the low-<i>Z</i> and high-<i>Z</i> mixing can be treated as ideal chunk mix in burn rate aspect, and heterogeneous mix size in the middle state needs to be evaluated by using the heterogeneous mixing model of thermonuclear reaction rate in the paper. Finally, the physical model is compared with 3D simulation results of the heterogeneous mixing effect experiment called “MARBLE Campaign” carried out on OMEGA laser facility, which is designed as a separated reactant experiments and capsules are filled with deuterated foam and HT gas pores of different size, covering typical mix morphology from atomic mix to chunk mix, which validate the reliability of the theoretical evaluation about the evolution of mixing morphology and its impact to thermonuclear reaction rate. This work is significant for the design and improvement of inertial confinement fusion mixing effect experiment in China.

Numerical Investigation of the Water Vapour Diffusivity inside Homogeneous Porous Media

This work is devoted to numerical investigation of diffusivity function for water vapour interaction with homogeneous porous media. Molecular dynamics simulation is used to determine the diffusivity function. Various approximations of diffusivity are applied for numerical solution of diffusion equation.

New asymptotic expansion for the diffusion problem with high-contrast inclusions: Numerical results

This paper is concerned with the new approach to the numerical solution of diffusion equations with high contrast coefficients. The classical P1 finite element system of equations is replaced by the equivalent system of the saddle point type. For the saddle point problem a new recently invented asymptotic expansion with respect to a small parameter is discussed and the estimates for the convergence factor are given. Numerical results completely confirm the theoretical accuracy and convergence results for the considered method.

Convective drying of osmotically dehydrated apples described by three-dimensional numerical solution of the diffusion equation with analysis of water effective diffusivity spatial distribution

This study presents two liquid diffusion models to represent the convective drying of apple, osmotically dehydrated in sucrose solution, cut into parallelepiped-shaped pieces. Model 1 considered water diffusivity and the volume of the slices with constant values. Model 2 considered water effective diffusivity and the dimensions of the slices as variable. The numerical solution of the three-dimensional diffusion equation in Cartesian coordinates was obtained through the finite volume method, with a fully implicit formulation and boundary condition of the third kind. Process parameters were estimated by an optimizer using experimental data. A spatial distribution analysis was carried out for water effective diffusivity and moisture content in the apple slices. The results showed that the concentration of the osmotic solution used in the pretreatment influenced the drying process and that the mathematical model that considered a variable diffusivity and shrinkage was more suitable to describe the experimental data.

The Green’s function technique for numerical simulations of multichannel electron transfer reactions in electron-donor-acceptor complexes

A method for numerical solution of diffusion equations with delta-localized coupling terms is proposed and applied to the stochastic model of multichannel electron transfer (ET) reaction in the Debye polar solvent. The Green’s function for the model equations is calculated using the Laplace transform technique for the time-dependent reaction fluxes between the reactant and product states. Two computational schemes for ET simulations are suggested: (a) the particle-based Brownian algorithm, (b) the grid scheme, utilizing the probabilities of the diffusion-assisted electronic transitions. The suggested schemes are shown to be more efficient than the previously developed recrossing-algorithm (the RABS method) in the solvent-controlled ET regimes and for slow processes accompanied by rare reactive events. Test simulations with the QM2L code are carried out and demonstrate an accuracy of the method. The results of the simulations agree well with known analytical expressions for the rate of thermal ET and the quantum yield of nonequilibrium multichannel ET in different regimes from the nonadiabatic (the Fermi Golden Rule) limit to the solvent-controlled reactions.

Apple osmotic dehydration described by three-dimensional numerical solution of the diffusion equation

ABSTRACTIn the present study, experimental data of osmotic dehydration kinetics of apple, cut into slices with parallelepiped shape, were simulated using two types of diffusion models. Model 1 considers the constant values of mass diffusivities and volume of the slices. Model 2, on the other hand, considers variable mass diffusivities and also the shrinkage of the product. The numerical solution of the three-dimensional diffusion equation in Cartesian coordinates was obtained through the finite volume method, with a fully implicit formulation and boundary condition of the first kind. Process parameters were determined by optimization using the experimental data sets, through the minimization of an objective function, called χ2. The results of the osmotic dehydration kinetics were compatible with those of other studies found in the literature. Process temperature and osmotic solution concentration had influence on the phenomenon, but temperature was preponderant. A study was conducted on water and sucrose distribution during the osmotic dehydration. The results obtained through the mathematical model that considered the variable diffusivity and shrinkage showed greater adequacy to the experimental data.

Fourth-order numerical solutions of diffusion equation by using SOR method with Crank-Nicolson approach

The aim of this paper is to investigate the effectiveness of the Successive Over-Relaxation (SOR) iterative method by using the fourth-order Crank-Nicolson (CN) discretization scheme to derive a five-point Crank-Nicolson approximation equation in order to solve diffusion equation. From this approximation equation, clearly, it can be shown that corresponding system of five-point approximation equations can be generated and then solved iteratively. In order to access the performance results of the proposed iterative method with the fourth-order CN scheme, another point iterative method which is Gauss-Seidel (GS), also presented as a reference method. Finally the numerical results obtained from the use of the fourth-order CN discretization scheme, it can be pointed out that the SOR iterative method is superior in terms of number of iterations, execution time, and maximum absolute error.

A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017

Numerical solution of diffusion equation to study fast neutrons flux distribution for variant radii of nuclear fuel pin and moderator regions

Abstract In this symbolic investigation, a cylindrical cell in a LWR, which consists of one fuel pin and moderator (water), is considered. The width of this cylindrical cell is divided into 100 equal units. Since the neutron flux in a cylindrical fuel pin is resulting from the diffusion equation: - 1 r d d r φ ( r ) + ∑ a φ ( r ) = S ( r ) $ - \frac{1}{r}\frac{d}{{dr}}\varphi \left( r \right) + \mathop \sum \nolimits_a \varphi \left( r \right) = S\left( r \right)$ , the amount of fast neutron fluxes are obtained on the basis of the numeric solution of this equation, and the applied boundary conditions are considered: φ ′ ( 0 ) = φ ′ ( 1 ) = 0 $\varphi '\left( 0 \right) = \varphi '\left( 1 \right) = 0$ . This differential equation is solved by the tridiagonal method for variant enrichments of uranium. Neutron fluxes are obtained in variant radii of fuel pin and moderator and are finally compared with each other. There are some interesting outcomes resulting from this investigation. It can be inferred that because of the fuel enrichment increment, the fast neutron flux increases significantly at the centre of core, while many of the fast neutrons produced are absorbed after entering the water region, moderation of lots of them causes the reduced neutron flux to get improved in this region.

Three-dimensional numerical analysis of water transfer in wood: determination of an expression for the effective mass diffusivity

The objective of this article is to determine an expression for the effective mass diffusivity related to the description of wood drying using a diffusion model and experimental data. With this objective, a three-dimensional numerical solution of diffusion equation for the parallelepiped, with convective boundary condition, was used to describe the drying process. For the adopted diffusion model, the convective mass transfer coefficient h was considered constant. The effective mass diffusivity D was considered as a function of the local moisture content, and several expressions were tested to describe the process. To this end, the numerical solution was coupled to an optimizer based on the inverse method, which determines the process parameters D and h using an experimental dataset. For both temperatures of the drying air studied in this article (40 and 80 °C), the results obtained by considering the variable effective mass diffusivity are better than those obtained when considering such parameter with a value constant. In addition, with the results for D and h obtained by optimization, the drying kinetics was simulated with success.

A new approach for the numerical solution of diffusion equations with variable and degenerate mobility

We propose a novel approach for the numerical integration of diffusion-type equations with variable and degenerate mobility or diffusion coefficient. Our focus is the Cahn–Hilliard equation which plays a prominent role in phase field models of fluids and soft materials but the methodology has a more general applicability. The central idea is a split method with a linearly implicit component and an analytic step to integrate out the variable mobility. The proposed method is robust, free of high order stability constraints, and its cost is comparable to that of solving the linear Heat Equation with the backward Euler Method. Moreover, by design, the numerical solution is guaranteed to be strictly bounded by the stable, constant states.

Numerical Solution of the Three-Dimensional Diffusion Equation in Solids with Arbitrary Geometry for the Convective Boundary Condition: Application in Drying

In this work, a numerical solution for the diffusion equation applied to solids with arbitrary shape considering convective boundary condition is presented. To this end, the diffusion equation, written in generalized coordinates, was discretized by the finite-volume method with a fully implicit formulation. The transport parameters and the dimensions of the solids are considered constant during all process. For each time step, the system of equations obtained for a given non-orthogonal structured mesh was solved by the Gauss-Seidel method. One computational code was developed in FORTRAN, using the CFV 6.6.0 Studio, in a Windows platform. The proposed solution was validated using analytical and numerical solutions of the diffusion equation for different geometries (parallelepiped and finite cylinder). The analysis and comparison of the results showed that the proposed solution provides correct results for the cases investigated. In order to verify the potential of the proposed numerical solution, we used experimental data of the drying of ceramic roof tiles for the following temperature: T = 55.6 °C. The analysis of the results and the statistical indicators enables to affirm that the developed numerical solution satisfactorily describes the drying processes in this temperature for the convective boundary condition.

Design of Gas-Cooled Fast Reactor 600MWth with Natural Uranium As Fuel Circle Input

This article presents the conceptual design of gas-cooled fast reactor (helium), the small size of the long-lived 600 MWth. Early stages of the design is to determine the geometry of the terrace, the value of the volume fraction and the mass fraction of fuel, cladding and coolant structure to calculate the parameters of reactivity, burnup, power distribution and density changes nuclides U238 and Pu239. The calculation is done using SRAC-CITATION code. SRAC code with JENDL-3.2 Data nuclides produced macroscopic cross section values for the eight energy group. Multi-group numerical solution of diffusion equations for 2-D geometry terrace RZ performed by CITATION code. The study results showed that the scheme Modified CANDLE, thermal power output is 600 MWth, with a fuel cycle for 10 years. This reactor has the advantage of requiring only the input of natural uranium in the fuel cycle, without the need for enrichment processes that affect the economic value. Keywords : Reactor, natural uranium, modified candle, burnup

FRACTAL MODEL OF FISSION PRODUCT RELEASE IN NUCLEAR FUEL

A model of fission gas migration in nuclear fuel pellet is proposed. Diffusion process of fission gas in granular structure of nuclear fuel with presence of inter-granular bubbles in the fuel matrix is simulated by fractional diffusion model. The Grunwald–Letnikov derivative parameter characterizes the influence of porous fuel matrix on the diffusion process of fission gas. A finite-difference method for solving fractional diffusion equations is considered. Numerical solution of diffusion equation shows correlation of fission gas release and Grunwald–Letnikov derivative parameter. Calculated profile of fission gas concentration distribution is similar to that obtained in the experimental studies. Diffusion of fission gas is modeled for real RBMK-1500 fuel operation conditions. A functional dependence of Grunwald–Letnikov derivative parameter with fuel burn-up is established.

Exciton Diffusion in Air-Suspended Single-Walled Carbon Nanotubes

Direct measurements of the diffusion length of excitons in air-suspended single-walled carbon nanotubes are reported. Photoluminescence microscopy is used to identify individual nanotubes and to determine their lengths and chiral indices. Exciton diffusion length is obtained by comparing the dependence of photoluminescence intensity on the nanotube length to numerical solutions of diffusion equations. We find that the diffusion length in these clean, as-grown nanotubes is significantly longer than those reported for micelle-encapsulated nanotubes.