Solution Of Diffusion Equation Research Articles

General rigid bead-rod theory for steady-shear flow

General rigid bead-rod theory yields uniquely the relation between macromolecular architecture and complex viscosity. For this, it relies on the analytical solution of the general diffusion equation for small-amplitude oscillatory shear flow of Bird et al. [Dynamics of Polymeric Liquids, 2nd ed. (Wiley, New York, 1987), Vols. 1–2]. Unfortunately, this general diffusion equation has yet to be solved for any other flow field. In this paper, we do so for steady-shear material functions, namely, viscosity and first normal stress coefficient. We, thus, explain the non-Newtonian behaviors of macromolecular suspensions of any axisymmetric design in steady-shear flow.

An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability

This article proposes a numerical method to find the numerical solutions of the time-fractional diffusion equations involving fractional distributed order operator of Caputo type. Using the finite difference approach, we solve these equations by applying the semi-discrete method regarding the time variable and the fully-discrete method regarding the spatial variable. For the distributed integral part with respect to time, the Gauss–Legendre quadrature formula is applied and to estimate the multi-term time-fractional operator, including the Caputo fractional derivative, the L2 -1 approach is utilized. In addition, the error analysis and stability of the proposed numerical method are studied in this work. Finally, some numerical examples are provided to demonstrate the accuracy and efficiency of the suggested method. These examples are compared to several numerical previous methods stated in the articles, and the results show that the accuracy of our method is superior to these methods.

Solusi Persamaan Difusi Neutron Pada PWR (Pressurized Water Reactor) Berbentuk Heksagonal dengan Bahan Bakar Uranium Daur Ulang

The Research on solution of the neutron diffusion equation with a PWR reactor using recycled uranium fuel at 1⁄6 section of the reactor core with a hexagonal IGT-6 geometry. The purpose of this research is to determine the distribution of the neutron flux in the PWR of recycled uranium fuel. The solution is done by computational simulation using the Dev-C++ programming. The parameters used in this study determine the specifications of the reactor core, determine the volume fraction, determine the atomic density, calculate the macroscopic cross-section with the PIJ module, calculate the neutron diffusion equation, calculate ϕ (x,y) using the Gauss Seidel method. The results obtained in this study are the neutron diffusion equation without a source obtaining the highest relative neutron flux value in group 1 of 4,5729×〖10〗^(-2), with a fission source obtaining the highest relative neutron flux value in group 3 of 7,3327×〖10〗^(-4), with fission and scattering sources obtaining the highest relative neutron flux value found in group 2 of 1,5157×〖10〗^(-3), and 3,200 MW of power is added to the source fission, the value of the neutron flux does not change. This is because the addition of power does not affect the value of the neutron flux.

Spectral Laplace Transform of Signals on Arbitrary Domains

The Laplace transform is a central mathematical tool for analysing 1D/2D signals and for the solution to PDEs; however, its definition and computation on arbitrary data is still an open research problem. We introduce the Laplace transform on arbitrary domains and focus on the spectral Laplace transform, which is defined by applying a 1D filter to the Laplacian spectrum of the input domain. The spectral Laplace transform satisfies standard properties of the 1D and 2D transforms, such as dilation, translation, scaling, derivation, localisation, and relations with the Fourier transform. The spectral Laplace transform is enough general to be applied to signals defined on different discrete data, such as graphs, 3D surface meshes, and point sets. Working in the spectral domain and applying polynomial and rational polynomial approximations, we achieve a stable computation of the spectral Laplace Transform. As main applications, we discuss the computation of the spectral Laplace Transform of functions defined on arbitrary domains (e.g., 2D and 3D surfaces, graphs), the solution of the heat diffusion equation, and graph signal processing.

Robustness of tissue oxygenation estimates by continuous wave space-resolved near infrared spectroscopy.

Continuous wave near infrared spectroscopy (CW-NIRS) is widely exploited in clinics to estimate skeletal muscles and brain cortex oxygenation. Spatially resolved spectroscopy (SRS) is generally implemented in commercial devices. However, SRS suffers from two main limitations: the a priori assumption on the spectral dependence of the reduced scattering coefficient [] and the modeling of tissue as homogeneous. We studied the accuracy and robustness of SRS NIRS. We investigated the errors in retrieving hemodynamic parameters, in particular tissue oxygen saturation (), when was varied from expected values, and when layered tissue was considered. We simulated hemodynamic variations mimicking real-life scenarios for skeletal muscles. Simulations were performed by exploiting the analytical solutions of the photon diffusion equation in different geometries: (1)semi-infinite homogeneous medium and constant ; (2)semi-infinite homogeneous medium and linear changes in ; (3)two-layered media with a superficial thickness , 7.5, 10mm and constant . All simulated data were obtained at source-detector distances , 40, 45mm, and analyzed with the SRS approach to derive hemodynamic parameters (concentration of oxygenated and deoxygenated hemoglobin, total hemoglobin concentration, and tissue oxygen saturation, ) and their relative error. Variations in affect the estimated (up to ), especially if changes are different at the two wavelengths. However, the main limitation of the SRS method is the presence of a superficial layer: errors strongly larger than 20% were retrieved for the estimated when the superficial thickness exceeds 5mm. These results highlight the need for more sophisticated strategies (e.g., the use of multiple short and long distances) to reduce the influence of superficial tissues in retrieving hemodynamic parameters and warn the SRS users to be aware of the intrinsic limitation of this approach, particularly when exploited in the clinical environment.

NONIC B-SPLINE APPROACH FOR ADVECTION DIFFUSION EQUATION

In this paper, a highly accurate method is introduced to achieve the numerical solution of the advection diffusion equation (ADE). This approach contains collocation technique based on nonic B-spline functions in the spatial-domain discretization and Adams Moulton scheme in the temporal-domain discretization. Two test problems are studied to validate effectiveness of the new presented method and efficiency of the approximate results are tested by calculating rate of temporal-convergence and error norm 𝐿∞ for the suggested method. The obtained numerical results are compared in the tables by the other available studies in literature and it is observed that a better approximate solution is provided than the existing methods.

Mathematical Simulation of Honeycomb Weathering via Moisture Transport and Salt Deposition

Honeycomb weathering is a common phenomenon found on various rock surfaces all around the world. However, honeycomb formation mechanisms are still poorly understood. In this study, we propose a model describing moisture transport within the sandstone and erosion resulting from salt deposition during evaporation of moisture off the rock surface. The moisture transport model is based on the non-linear diffusion equation, where the volumetric moisture content is a combined parameter accounting for the moisture and gas (vapor) content. The moisture transport model accounts for the several-orders-of-magnitude decrease in moisture diffusivity, observed during drying. It was assumed that erosion occurs when the evaporation front is located close to the rock surface. The depth of erosion is proportional to the moisture flow rate through the drying surface. The ABAQUS finite-element software suite was used for numerical solution of the non-linear diffusion equation. The iterative scheme of erosion simulation for different drying cycles was implemented using the Python programming language. Computations were conducted in the 2D setting for the square model with dimensions of 50 mm × 50 mm. Simulation results demonstrate the possibility of obtaining various landform shapes (honeycombs, tafoni) by varying only the value of the distribution of moisture content at the bottom side, simulating the rate of internal wetting of rock.

Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon

In the description of transport phenomena, diffusion represents an important aspect. In certain cases, the diffusion may appear together with convection. In this paper, we study the diffusion equation with the self-similar Ansatz. With an appropriate change of variables, we have found an original new type of solution of the diffusion equation for infinite horizon. We derive novel even solutions of diffusion equation for the boundary conditions presented. For completeness, the odd solutions are also mentioned as well, as part of the previous works. We have found a countable set of even and odd solutions, of which linear combinations also fulfill the diffusion equation. Finally, the diffusion equation with a constant source term is discussed, which also has even and odd solutions.

Approximate solution of space fractional order diffusion equations by Gegenbauer collocation and compact finite difference scheme

In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials using zeros of (N - 1) degree shifted Gegenbauer polynomial as collocation points. The important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (N + 1) linear equations. Some theorems are given to establish the convergence and the stability of the proposed method. To validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. The graphical representation are also displayed for various values of \beta Gegenbauer polynomials. It can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature.

Numerical solutions of Atangana-Baleanu time-fractional advection diffusion equation via an extended cubic B-spline technique

The B-spline function is made up of a set of smooth piecewise polynomials that are controlled by a set of control points. A linear combination of B-spline basis of a particular degree can be used to express any spline function of that degree. The spline functions and their derivatives are continuous and depending on the multiplication of knots. In this study, an extended cubic B-spline (ECBS) approximation is used for the numerical solution of time-fractional advection diffusion equation (TFADE) involving Atangana-Baleanu fractional derivative (ABFD). Initially, the non-singular kernel ABFD is discretized using finite difference method (FDM). The spatial derivatives are discretized using ECBS functions. Convergence and stability of the proposed scheme are studied. The results tabulated in tables that show how they accurate and desirable the outcomes are by comparing the obtained approximate results with the available exact solutions. To the authors’ knowledge, this work is the first time to use the ABFD for numerical solution of time-fractional advection diffusion equation using ECBS.

Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method

Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method

Correction: Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion‑Dependent Permeability in Oil Wells Near Sealing Faults

Correction: Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion‑Dependent Permeability in Oil Wells Near Sealing Faults

Global existence for a class of non-equilibrium reaction–diffusion systems with flux limitation

For a class of general reaction–diffusion systems with flux limitation in one dimension, when homogeneous Neumann boundary conditions are considered, we will prove global upper and lower boundness estimates for the positive classical solutions and boundness for their first-order spatial derivatives. In particular, the non-equilibrium radiation flux-limited diffusion coupled with material heat conduction model is included as a special case of the general reaction–diffusion systems. The key point of our method is to at first prove that the second-order spatial derivative of the solution of the second diffusion equation can be linearly controlled by the first-order spatial derivative of the solution of the flux-limited diffusion equation. Then, with this control, we can apply the Bernstein method to the flux-limited diffusion equation, while a stratifying time technique is introduced to get a complete first order spatial derivative estimate. With these global prior estimates, the global existence and uniqueness of positive classical solutions for the general flux-limited diffusion systems are proved.

Modeling of Core-shell Catalytic Pellets of Inert Shell with Various Morphologies for Series Reactions

Modeling of core-shell catalytic pellets with inert shell was performed assuming three different particle shapes such as sphere, cylinder, and slab. For isothermal irreversible first order reaction, A→B, analytical solutions of intra-particle concentration of reactant were derived by solving reaction-diffusion equations, considering thickness of catalytically active core (r2), when the pellets were immersed in infinitely large medium. Unlike conventional pellets or core-shell pellets with inert core, effectiveness factor (η) was affected by r2 as well as ratio of effective diffusivity in inert shell and active core (Υ), and η could be enhanced by increasing Υ. η increased in the order of sphere > cylinder > slab due to surface area per unit volume of medium, and η increased with increasing Biot number (Bi), because of decreased external film resistance. For series reaction, A→B→C, transient change of the concentration of starting reactant (CA) and intermittent product (CB) in bulk fluid of reactor could be predicted by solving material balance equations assuming pseudo-steady state approximation and equal value of Thiele modulus for both reactions in batch reactor and CSTR. When B is desirable product, CB increased by increasing Υ and decreasing r2. Similar to conventional pellets, CB in batch reactor decreased by increase of distribution coefficient of B on surface of pellet, KB, due to increased adsorption, and increased in the order of sphere < cylinder < slab, similar to conventional pellets. In CSTR, CB at steady state could be enhanced by decreasing amount of catalyst and Thiele modulus (Φ), implying that performance of reactor can be controlled by adjusting various reaction parameters. From numerical solutions of reaction diffusion equations for series reactions, the effect of r2 and Υ were also confirmed from transient value of CA and CB in exit stream calculated by finite element method.

Fuzzy Analytical Solution of Horizontal Diffusion Equation into the Vadose Zone

The process of how soil moisture profiles evolve into the soil and reach the root zone could be estimated by solving the appropriate strong nonlinear Richards’ equation. The nonlinearity of the equation occurs because diffusivity D is generally an exponential function of water content. In this work, the boundary conditions of the physical problem are considered fuzzy for various reasons (e.g., machine impression, human errors, etc.), and the overall problem is encountered with a new approximate fuzzy analytical solution, leading to a system of crisp boundary value problems. According to the results, the proposed fuzzy analytical solution is in close agreement with Philip’s semi-analytical method, which is used as a reference solution, after testing 12 different types of soils. Additionally, possibility theory is applied, enabling the decision-makers to take meaningful actions and gain knowledge of various soil and hydraulic properties (e.g., sorptivity, infiltration, etc.) for rational and productive engineering studies (e.g., irrigation systems).

On the Blow-Up of Solutions for a Fractional Diffusion Equation with Nonlinear Memory and Reaction Terms in a Bounded Domain

On the Blow-Up of Solutions for a Fractional Diffusion Equation with Nonlinear Memory and Reaction Terms in a Bounded Domain

Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion-Dependent Permeability in Oil Wells Near Sealing Faults

Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion-Dependent Permeability in Oil Wells Near Sealing Faults

Experimental Study and Modelling of the Sublimation and Desorption Periods for Freeze Drying of Apple, Banana and Strawberry

Slices of fresh apple, banana and strawberry were frozen at -20 oC and freeze-dried using a shelf temperature of 40 oC. Theoretical expressions were proposed to predict vapor transfer kinetics during the primary and secondary drying stages. In the former, a model that predicts the sublimation rate as a function of time, considering the increasing dried layer thickness, was used, which improves greatly the sublimation time equation offered in several textbooks without adding much complexity. In the latter, an analytical solution of the unsteady state diffusion equation was applied. Permeabilities were determined for the primary drying model at an absolute pressure of about 30 Pa, though the relevant kinetic coefficient combines permeability and the mass of ice to sublime relative to the dry matter (sublimation kinetic coefficient). In the secondary drying stage, diffusion coefficients of vapor in the dried layer were in the order of 10−09 m2s−1 for pressures of about 3-5 Pa. In both periods, agreement of predicted and experimental values was more than satisfactory. A minimum freeze-drying time of 12, 6.8 and 8.7 h, considering a final moisture content of 4% w/w, was calculated for apple, banana and strawberry, respectively. Normalized drying curves showed a faster sublimation rate for banana, intermediate for strawberry and slowest for apple. On the other hand, desorption curves showed a faster desorption rate for apple, intermediate for banana and slower for strawberry. In each period, the ordering of the relevant kinetic coefficients (sublimation and diffusion coefficients, respectively) represented the ordering of experimental curves.

Resonance Method for Measuring Parameters of the Spin Transport in a Spin-Valve Structure

The known methods for measuring the spin transport parameters in spin-valve structures are based on the Hanle effect, viz., electron spin precession in an external magnetic field and a decrease in the magnetoresistive signal. These methods make it possible to determine the spin relaxation time in the paramagnetic layer and the relative current polarization constant. We describe an alternative method of measuring in zero external magnetic field, which is based on the resonant increase in the magnetic susceptibility of the paramagnetic layer due to the paramagnetic resonance induced by the nonequilibrium magnetization due to the spin accumulation effect. The proposed method makes it possible to determine the absolute value of spin accumulation in a paramagnet, which can be used as a parameter for numerical solution of three-dimensional diffusion equations of spin transport.

Femtosecond IR and UV laser induced periodic structures on steel and copper surfaces

We analyze in detail laser induced periodic surface structures (LIPSS) on steel and copper surfaces. The chosen metals exhibit extreme differences between their thermal conductivity values, which determines the rate of extinguishing of the laser-induced temperature modulation on the surface. Two wavelengths of femtosecond laser radiation sources in IR and UV spectral range were used to inscribe low spatial frequency LIPSS. Regular LIPSS were successfully formed on steel samples for both wavelengths and it was observed that their regularity decreased with shorter wavelength used to induce the structures. For copper samples, LIPSS of considerably lower regularity were formed and even absent for one of the used scanning method at UV wavelength. Experimental results were analyzed in view of the numerical solution of the 2D heat diffusion equation for induced temperature on the surface. It was found that the temperature modulation decreased considerably faster (lasting tens of picoseconds) in copper, and (additionally) the decrease of the modulation is faster at the higher modulation frequency.