One-dimensional Diffusion Equation Research Articles

Boundary Value Problem of Space-Time Fractional Advection Diffusion Equation

In this article, the analytical and numerical solution of a one-dimensional space-time fractional advection diffusion equation is presented. The separation of variables method is used to carry out the analytical solution, the basis of the system eigenfunction and their corresponding eigenvalue for basic equation is determined, and the numerical solution is based on constructing the Crank-Nicolson finite difference scheme of the equivalent partial integro-differential equations. The convergence and unconditional stability of the solution are investigated. Finally, the numerical and analytical experiments are given to verify the theoretical analysis.

New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient

We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker–Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge–Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Analytical Sensitivity Analysis of a Spent Nuclear Fuel Cask

Abstract Nuclear science and engineering is a field increasingly dominated by computational studies resulting from increasingly powerful computational tools. As a result, analytical studies, which previously pioneered nuclear engineering, are increasingly viewed as secondary or unnecessary. However, analytical solutions to reduced-fidelity models can provide important information concerning the underlying physics of a problem and aid in guiding computational studies. Similarly, there is increased interest in sensitivity analysis studies. These studies commonly use computational tools. However, providing a complementary sensitivity study of relevant analytical models can lead to a deeper analysis of a problem. This work provides the analytical sensitivity analysis of the one-dimensional (1D) cylindrical mono-energetic neutron diffusion equation using the forward sensitivity analysis procedure (FSAP) developed by Cacuci. Further, these results are applied to a reduced-fidelity model of a spent nuclear fuel cask, demonstrating how computational analysis might be improved with a complementary analytic sensitivity analysis.

Analytical Solutions Formulated in the Time Domain for Three-Dimensional Heat Diffusion Equation

Two different strategies are provided to generate solutions to the three-dimensional heat diffusion equation. The first strategy is inspired by the well-known one-dimensional heat polynomial, which consists of an infinite set of polynomials, which are solutions to the one-dimensional heat diffusion equation. The second strategy is based on an exponential type function. None of the solutions presented here can be obtained by the method of separation of variables. The mathematical developments proving that, indeed, the particular solutions generated with both strategies satisfy the three-dimensional heat diffusion equation are presented. The analytical solutions are validated by generating the corresponding numerical solutions with the method of finite differences. When comparing both analytical and numerical solutions, it is found that they are identical. In addition, as part of the results, it is found that there are exponential solutions that reproduce the behavior of polynomial solutions. Finally, an example of the use of heat polynomials in engineering applications is provided.

Neutron Mean Free Path in the Slab Reactor Core using One-Dimensional Multi-group Diffusion Equation

Analysis of the neutron mean free path in the slab reactor core has been carried out using one-dimensional multi-group diffusion equation. This study aims to determine the neutron mean free path in the slab reactor core with the neutron diffusion coefficient calculation using macroscopic cross-section data in the nuclear fuel cell level and the neutron flux distribution. The type of reactor used in this research is a fast reactor with nuclear fuel is uranium-plutonium nitride (U-PuN). The neutron mean free path is calculated for 70 energy groups of neutron by dividing the energy groups, namely the fast energy group, the intermediate energy group and the thermal energy group. The results showed that the neutron mean free path value for U-235 and Pu-239 fuels were obtained almost the same in all energy groups, namely in the fast energy group ranging from 0.11.10-2 to 0.17.10-2 cm, in the intermediate energy group 0.16.10-2 to 1.78.10-2 cm, and in the thermal energy group 0.4.0-2 to 8.04.10-2 cm. The neutron mean free path value for U-238 fuel is much smaller than that for U-235 and Pu-239 fuel, ranging from 0.03.10-2 to 0.36.10-2 cm. These results can be confirmed, because U-238 fuel is a fertile material. Keywords: Neutron mean free path, diffusion equation, neutron flux, slab reactor core

Formation of organic color centers in air-suspended carbon nanotubes using vapor-phase reaction

Organic color centers in single-walled carbon nanotubes have demonstrated exceptional ability to generate single photons at room temperature in the telecom range. Combining the color centers with pristine air-suspended nanotubes would be desirable for improved performance, but all current synthetic methods occur in solution which makes them incompatible. Here we demonstrate the formation of color centers in air-suspended nanotubes using a vapor-phase reaction. Functionalization is directly verified by photoluminescence spectroscopy, with unambiguous statistics from more than a few thousand individual nanotubes. The color centers show strong diameter-dependent emission, which can be explained with a model for chemical reactivity considering strain along the tube curvature. We also estimate the defect density by comparing the experiments with simulations based on a one-dimensional exciton diffusion equation. Our results highlight the influence of the nanotube structure on vapor-phase reactivity and emission properties, providing guidelines for the development of high-performance near-infrared quantum light sources.

Sublimation of a volatile component as a possible mechanism for thermoelectric material degradation

A physical model of the sublimation of a volatile component from a thermoelectric material has been developed. On its basis, two versions of the mathematical description of the degradation process of thermoelectric material are presented. The first of them takes into account only the diffusion of tellurium as a volatile impurity to the evaporation surface, on which the pressure and, consequently, the concentration of atoms of the volatile component are considered to be known. The second explicitly takes into account the volatility of the evaporating component and, hence, the boundary flux on the evaporation surface. In both cases, an analytical solution of the one-dimensional diffusion equation is obtained taking into account the presence of a temperature gradient along the length of the leg. Further, by computer methods in the Mathcad environment, the time dependence of the thickness of the layer with a reduced concentration of the volatile component and the nature of the distribution of the concentration of this component in it was determined. On this basis, the degradation time of thermoelectric material due to the loss of volatile components is estimated and the requirements for the protective coating of thermoelectric branches are established.

Initial-boundary value problem for the time-fractional degenerate diffusion equation

In this paper the initial-boundary value problems for the one-dimensional linear time-fractional diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t and time-degenerate diffusive coefficients t β with β ≥ 1 − α are studied. The solutions of initialboundary value problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t, are shown. The second section present Dirichlet and Neumann boundary value problems, and in the third section has shown the solutions of the Dirichlet and Neumann boundary value problem for the one-dimensional linear time-fractional diffusion equation. The solutions of these fractional diffusive equations are presented using the Kilbas-Saigo function Eα,m,l(z). The solution of the problems is discovered by the method of separation of variables, through finding two problems with one variable. The existence and uniqueness to the solution of the problem are confirmed. In addition, the convergence of the solution has been proven using the estimate for the Kilbas-Saigo function Eα,m,l(z) from [13] and Parseval’s identity

Edge states of a diffusion equationin one dimension: Rapid heat conduction to the heat bath.

We propose a one-dimensional diffusion equation(heat equation) for systems in which the diffusion constant (thermal diffusivity) varies alternately with a spatial period a. We solve the time evolution of the field (temperature) profile from a given initial distribution, by diagonalizing the Hamiltonian, i.e., the Laplacian with alternating diffusion constants, and expanding the temperature profile by its eigenstates. We show that there are basically phases with or without edge states. The edge states affect the heat conduction around heat baths. In particular, rapid heat transfer to heat baths would be observed in a short-time regime, which is estimated to be t<10^{-2}s for the a∼10^{-3}m system and t<1s for the a∼10^{-2}m system composed of two kinds of familiar metals such as titanium, zirconium, and aluminium, gold, etc. We also discuss the effective lattice model which simplifies the calculation of edge states up to high energy. It is suggested that these high-energy edge states also contribute to very rapid heat conduction in a very short-time regime.

Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis

&lt;abstract&gt;&lt;p&gt;In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.&lt;/p&gt;&lt;/abstract&gt;

Physics Informed Neural Networks for Power Transformer Dynamic Thermal Modelling

The emerging methodology of Physics Informed Neural Networks (PINNs) promises to combine available data and physical knowledge to achieve high accuracy and fast evaluation. Dynamic thermal modelling of power transformers is an application specifically set to benefit from these characteristics. Data collected during typical operation is not representative of extreme loading scenarios and the number of thermal sensors is limited. The detailed geometry is often not known by the asset owner which creates high uncertainty for physics-based simulation models. In this study, the transformer is modeled by the one-dimensional heat diffusion equation. PINN is constructed with a loss function including both data-based and physics-based terms. A time-dependent source term from a time series of measurement is also part of the PINN. The result is compared with a finite volume solution demonstrating good agreement. The PINN approach will be useful for further development in thermal modelling for power transformers.

Estimation of Residual Hydrogen in Laser Weld by Two-dimensional Differential Calculation

It has been reported that hydrogen dissolves in the weld bead in laser welding. If the hydrogen concentration in the weld bead is high, there is concern that the weld bead may fracture during press forming of tailored blanks. Therefore, assuming that 1ppm of hydrogen is dissolved in the weld bead at welding, the change in hydrogen concentration over time is estimated by a two-dimensional difference calculation. In the calculations, hydrogen is assumed to follow the Fick’s diffusion equation, and the diffusion coefficient at room temperatures is the value determined by the authors for a 980MPa steel laser welds in the past. The diffusion coefficients at high temperatures were determined from several references. As a result, it was estimated that about 30% of the hydrogen had effused from the surface by the time the weld bead cooled to room temperature. The mechanical properties of the weld bead are considered to be governed by the maximum hydrogen concentration at the center of the weld bead cross-section, but this value cannot be measured directly. Therefore, assuming that all the hydrogen contained in the entire model is in the weld bead and considering the average hydrogen concentration in the weld bead, the ratio of the maximum hydrogen concentration to the average hydrogen concentration is around 1 up to 6h after welding, indicating that the average hydrogen concentration is a good indicator of the maximum hydrogen concentration. It was also confirmed that the logarithm of the average hydrogen concentration is proportional to time as time passes after welding and that the hydrogen concentration distribution in the thickness direction behaves as a solution of a one-dimensional diffusion equation. These results suggest that the hydrogen concentration in the laser weld bead can be considered as an analytical solution of the one-dimensional diffusion equation.

One-dimensional diffusion and stochastic differential equation

In this paper, we study the condition for a one-dimensional diffusion to satisfy a stochastic differential equation. In the case of regular diffusion, we give a sufficient and nearly necessary condition using scale function and speed measure. For general diffusions, we decompose the state space into regular and shunt pieces, and give a condition for X satisfying a stochastic differential equation on any shunt pieces.

Fully discretized methods based on boundary value methods for solving diffusion equations

Based on boundary value methods, we establish a kind of new fully discretized methods for solving one-dimensional diffusion equations. The proposed methods are composed of a series of full discretizations with multi-time-level and multi-space-level. For the full discretizations, we give the local truncation error. Moreover, we analyze the stability of the proposed methods and obtain the corresponding error estimate. Meanwhile, we make some numerical experiments to show that the proposed methods are stable and own high accuracy.

Enhanced water uptake in the longitudinal direction by shiitake mycelium in shiitake cultivation logs: water content distribution in logs measured by magnetic resonance imaging.

In the cultivation of shiitake mushrooms (Lentinula edodes), the farmer needs to know the time needed to water in order to adjust the water content of the logs. To study the enhanced water uptake in the longitudinal direction by shiitake mycelium in shiitake cultivation logs, six dried test logs (Quercus serrata, diameter of 38 to 48 mm, length of 110 to 118 mm) were used. Three test logs had shiitake mycelium grown on them, and the remaining three test logs had mold generated on them. Liquid water was supplied to the bottom surface of the test log which had its longitudinal direction along the line of gravity. Water content distribution in the logs was measured in chronological order using magnetic resonance imaging (MRI) with 1 Tesla. The calibration curve for converting the signal intensity of the MR image into the water content in the test log was determined by cutting the test log at 5-mm intervals and measuring the water content distribution using the mass method. Spatial distribution of the water content of the test log without shiitake mycelium depending on the cumulative water supply time was obtained, and the distribution shape was always concave corresponding to the exact solution of an unsteady one-dimensional diffusion equation with one diffusion coefficient. In the case of the test log in which shiitake mycelium grew, within a few hours after liquid water supply the water content increased in the whole region where shiitake mycelium grew, and the shape of the water content distribution in the longitudinal direction became convex. Based on observation of water penetration into logs by MRI and an optical microscope, it is believed that the driving force behind increased rise in liquid water in the longitudinal direction in the test log is the capillary force acting in vessels.

A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

Modeling of zircon nucleation and growth rates using crystal size distributions in a cooling magmatic intrusion

With a rapid increase in the use of zircons for geochronology and in situ isotopic and chemical analyses it is important to quantify growth rates and nucleation rates of zircons in natural magmatic systems. Here we present a mathematical model of the nucleation and growth of a population of zircon crystals in a cooling magmatic intrusion and apply nucleation laws to derive zircon crystal size distributions (CSD) and particle number density (n per cm3 of melt) from first principles. The model is based on a numerical solution of the one-dimensional diffusion equation that accounts for the nucleation and growth of individual crystals and the dependence of the equilibrium concentration of Zr on temperature. As experimental studies of zircon nucleation at natural concentration levels of Zr are nearly impossible, we rely on measured CSDs of age-invariant (within error of the method <±5ky) zircon populations in rapidly quenched eruptive products of known crystallization duration to calibrate our model. The CSDs of zircon crystals are calculated at different rates of cooling of the magma chamber and are compared to measurements. The combination of the zircon growth software with measurements of zircon CSDs and particle number density permits direct estimation of the zircon nucleation rates and their evolution with time in cooling magmatic systems. We observe a delayed zircon nucleation of 500–2000 years after the melt becomes zircon saturated. Zr supersaturations that drive grown in simulations range from 1 to ∼30ppm and vary non-monotonically as crystallization proceeds. Initially, the nucleation rates increase exponentially from 10−3 to ∼1crystals/cm3/yr but then decrease by an order of magnitude as the distance between zircons decreases. Growth rates stay in 10−15cm/s range for the most duration of zircon growth. The ratio of growth to nucleation rates in our model also varies insignificantly and is used to estimate crystallization time based on the CSD slope, which decreases with the increased duration of cooling. Flatter slopes correspond to longer crystallization durations as is also noted for major phases. Due to non-monotonic variations in growth and nucleation rates with time, the observed deficiency of small zircons resulting in concave up CSD shapes can be reproduced from first principles assuming monotonic cooling of the magmatic system. Variations in the mode of cooling from linear to monotonic during conductive heat loss of a cooling intrusion, and minor coeval precipitation (<10vol%) of major phases has little impact on zircon CSD and its evolution. This study has implication to individual zircons IDTIMS dates and demonstrates that most zircons crystallize much later than magma becomes saturated with Zr, despite sometimes having nominally long tail crystallization rate in the inherited cores. For natural CSDs from eight different eruptions randomly selected zircons predate eruption age only by ∼300–1000years.

All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations

We focus on a time-dependent one-dimensional space-fractional diffusion equation with constant diffusion coefficients. An all-at-once rephrasing of the discretized problem, obtained by considering the time as an additional dimension, yields a large block linear system and paves the way for parallelization. In particular, in case of uniform space–time meshes, the coefficient matrix shows a two-level Toeplitz structure, and such structure can be leveraged to build ad-hoc iterative solvers that aim at ensuring an overall computational cost independent of time. In this direction, we study the behavior of certain multigrid strategies with both semi- and full-coarsening that properly take into account the sources of anisotropy of the problem caused by the grid choice and the diffusion coefficients. The performances of the aforementioned multigrid methods reveal sensitive to the choice of the time discretization scheme. Many tests show that Crank–Nicolson prevents the multigrid to yield good convergence results, while second-order backward-difference scheme is shown to be unconditionally stable and that it allows good convergence under certain conditions on the grid and the diffusion coefficients. The effectiveness of our proposal is numerically confirmed in the case of variable coefficients too and a two-dimensional example is given.

Analytical Solution for Boltzmann Collision Operator for the1-D Diffusion equation

Abstract: In this paper, we have presented the analytical solution of the collision operator for the Boltzmann equation of onedimensional diffusion equation using the analytical solution of the one-dimensional Navier Stoke diffusion equation. Keywords: Boltzmann equation; analytical collision operator; one-dimensional diffusion equation.

Fate of turbid glacial inflows in a hydroelectric reservoir

Turbidity from glacial meltwater limits light penetration with potential ecological consequences. Using profiles of temperature, conductivity, and turbidity, we examine the physical processes driving changes in the epilimnetic turbidity of Carpenter Reservoir, a long and narrow, glacier-fed reservoir in southwest British Columbia, Canada. Following the onset of permanent summer stratification, the relatively dense inflows plunged into the hypolimnion, and despite the high glacial load entering the reservoir, the epilimnion cleared due to particle settling. Using a one-dimensional (longitudinal) diffusion equation for a decaying substance to describe the variation in epilimnetic turbidity, we obtain two nondimensional parameters: the epilimnetic inflow parameter, mathcal {I}, a measure of the turbidity flux into the epilimnion; and the dispersion parameter, {mathcal {D}}, a measure of longitudinal dispersion. In the case of Carpenter Reservoir: mathcal {I}ll 1, indicating that turbidity declines over the summer; and {mathcal {D}}ll 1, indicating a strong gradient in turbidity along the epilimnion. Using our theoretical formulation of epilimnetic turbidity variations in conjunction with monthly field surveys, we compute the particle settling velocity ({sim}{0.25},{hbox {m},hbox {d}^{-1}}), the longitudinal dispersion coefficient (50–70 {hbox {m}^{2},hbox {s}^{-1}}), and the flux of turbid water into the epilimnion ({sim }1{%} of the total inflow). Our approach is applicable to other reservoirs and can be used to investigate changes in turbidity in response to changes in mathcal {I} and {mathcal {D}}.