One-dimensional Diffusion Equation Research Articles

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

AbstractThe aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

The surface temperatures of flat areas on the Moon

Lunar surface temperature (LST) is a quantity of special interest for interpreting the thermal character of the regolith. It is affected by the solar irradiance, earthshine, and heat flow for the flat areas on the Moon. We present an improved transient temperature model to calculate temperatures of lunar flat surfaces. The model consists of one-dimensional thermal diffusion equation and two boundary conditions. The improved lunar surface boundary condition allows one to precisely determine the effective solar irradiance (ESI) and earthshine. The simulated surface temperatures suggest consistency with the measured temperatures from the thermocouples of Apollo 15 and 17 heat flow experiments. From the LST simulated with the improved model, it is found the annual-seasonal variations present obvious latitude characters, with the highest surface temperature occurring in late October, November, and December separately at high (>∼72°) latitudes, middle latitudes, and low (<∼20°) latitudes, respectively; the lowest surface temperatures occur in late July. Furthermore, the discrepancies between the maximum and minimum temperatures decrease as latitude increases, as do the maximum and minimum lunar surface temperatures. The surface daytime temperature would change by 179.4 K with a 1322.5 W/m2 change in the ESI. A 0.12 W/m2 and 0.02 W/m2 change of the earthshine and heat flow would lead to 0.5 and 0.09 K in surface nighttime temperature, respectively.

Solution of Fishers equation using homotopy perturbation Aboodh transform method

In this study, a nonlinear one-dimensional reaction Diffusion Equation called Fisher’s equation is solved using Aboodh Transform coupled with HPM. This method is applied to the general one-dimensional Fishers equation. This technique is demonstrated with three unique examples with different initial conditions solved with the values of β=1 and a=0. This simplicity of Aboodh transform makes the rapid rate of convergence from approximate to exact solution answers makes it easy and dependable.

Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory

When applying the finite-differences method to numerically solve the one-dimensional diffusion equation, one must choose discretization steps Δx, Δt in space and time, respectively. By applying large-deviation theory on the discretized dynamics, we analyze the numerical errors due to the discretization, and find that the (relative) errors are especially large in regions of space where the concentration of particles is very small. We find that the choice Δt=Δx2/(6D) , where D is the diffusion coefficient, gives optimal accuracy compared to any other choice (including, in particular, the limit Δt→0 ), thus reproducing the known result that may be obtained using truncation error analysis. In addition, we give quantitative estimates for the dynamical lengthscale that describes the size of the spatial region in which the numerical solution is accurate, and study its dependence on the discretization parameters. We then turn to study the advection–diffusion equation, and obtain explicit expressions for the optimal Δt and other parameters of the finite-differences scheme, in terms of Δx, D and the advection velocity. We apply these results to study large deviations of the area swept by a diffusing particle in one dimension, trapped by an external potential ∼|x| . We extend our analysis to higher dimensions by combining our results from the one dimensional case with the locally one-dimension method.

Elucidation of Mass Transport Phenomena in Highly Concentrated Electrolytes during Current Cycling Using In-Situ Interferometry and Finite Difference Method

Understanding electrolyte mass transfer during charge–discharge reactions is essential for developing next-generation storage batteries with high energy densities. In this study, we investigated Li+ transport in a highly concentrated electrolyte (HCE) consisting of an equimolar mixture of lithium bis(fluorosulfonyl)amide (LiFSA) and tetraglyme (G4) under current reversal and re-reversal. Concentration profiles of the electrolyte at a distance of 0–600 μm from the Li electrodes were obtained using in situ laser interferometry. The Li+ transference numbers and LiFSA diffusion coefficients were calculated from these profiles. Raman spectroscopy suggested that the coordination structure surrounding Li+ ions in the electrolytes mainly contributed to the transference number. A one-dimensional unsteady diffusion equation and the finite difference method were employed to simulate the concentration profiles. The maximum error percentage between the measured and simulated values was only 3%, confirming the accuracy and validity of the interferometric measurements. Our findings on Li-ion transfer in HCEs could promote the rational design of high-energy-density Li-ion batteries with higher cation transference numbers of electrolytes and charge–discharge rates.

Application of data-driven model reduction techniques in reactor neutron field calculations

High-order harmonic techniques can be used to recreate neutron flux distributions in reactor cores using the neutron diffusion equation. However, traditional source iteration and source correction iteration techniques have sluggish convergence rates and protracted calculation periods. The correctness of the implicitly restarted Arnoldi method (IRAM) in resolving the eigenvalue problems of the one-dimensional and two-dimensional neutron diffusion equations was confirmed by computing the benchmark problems SLAB_1D_1G and two-dimensional steady-state TWIGL using IRAM. By integrating Galerkin projection with Proper Orthogonal Decomposition (POD) techniques, a POD-Galerkin reduced-order model was developed and the IRAM model was used as the full-order model. For 14 macroscopic cross-section values, the TWIGL benchmark problem was perturbed within a 20% range. We extracted 100 sample points using the Latin hypercube sampling method, and 70% of the samples were used as the testing set to assess the performance of the reduced-order model The remaining 30% were utilized as the training set to develop the reduced-order model, which was employed to rebuild the TWIGL benchmark problem. The reduced-order model demonstrates good flexibility and can efficiently and accurately forecast the effective multiplication factor and neutron flux distribution in the core. The reduced-order model predicts keff and neutron flux distribution with a high degree of agreement compared to the full-order model. Additionally, the reduced-order model's computation time is only 10.18% of that required by the full-order model.The neutron flux distribution of the steady-state TWIGL benchmark was recreated using the reduced-order model. The obtained results indicate that the reduced-order model can accurately predict the keff and neutron flux distribution of the steady-state TWIGL benchmark.Overall, the proposed technique not only has the potential to accurately project neutron flux distributions in transient settings, but is also relevant for reconstructing neutron flux distributions in steady-state conditions; thus, its applicability is bound to increase in the future.

Phosphorus Sources and Transport Pathways in the North Chaohu Lake Catchment of China

To understand the cause of the high content of phosphorus in the North Chaohu Lake Catchment of China, the distribution of the total phosphorus in groundwater and the dissolution of rock phosphorus near Chaohu Lake were investigated, and a few interesting findings have been revealed. Firstly, four main processes affecting the groundwater quality in North Chaohu Lake Catchment were identified with the results of factor analysis, including anthropogenic activities and the dissolution of carbonate rocks, phosphorus, and fluorine-bearing minerals. Secondly, the dissolution of rock phosphorus can be well described using the one-dimensional diffusion equation, with a semi-infinite boundary condition, and the Langmuir kinetic equation. A relatively low ambient temperature (less than 25 °C) was probably responsible for the small diffusion coefficient compared to that of the previous studies. Thirdly, the high-potential maximum dissolution of rock phosphorus (Cmax) indicated that the dissolution of rock phosphorus could pose a risk to the ambient water body, and Cmax was found to be independent of the total rock phosphorus content.

An inverse source Cauchy-weighted time-fractional diffusion problem

In the present paper, we are concerned with an inverse source Cauchy weighted problem involving a one-dimensional diffusion equation involving a time-fractional Riemann-Liouville derivative with 0

ANOMALOUS DIFFUSION MODELS AND MANDELBROT SCALING-LAW SOLUTIONS

In this paper, the anomalous diffusion models are studied in the framework of the scaling-law calculus with the Mandelbrot scaling law. A analytical technology analogous to the Fourier transform is proposed to deal with the one-dimensional scaling-law diffusion equation. The scaling-law series formula via Kohlrausch–Williams–Watts function is efficient and accurate for finding exact solutions for the scaling-law PDEs arising in the Mandelbrot scaling-law phenomena.

Modeling large-scale bioreactors with diffusion equations. Part II: Characterizing substrate, oxygen, temperature, carbon dioxide, and pH profiles.

Large-scale fermentation processes involve complex dynamic interactions between mixing, reaction, mass transfer, and the suspended biomass. Empirical correlations or case-specific computational simulations are usually used to predict and estimate the performance of large-scale bioreactors based on data acquired at bench scale. In this two-part-study, one-dimensional axial diffusion equations were studied as a general and predictive model of large-scale bioreactors. This second part focused on typical fed-batch operations where substrate gradients are known to occur, and characterized the profiles of substrate, pH, oxygen, carbon dioxide, and temperature. The physically grounded steady-state axial diffusion equations with first- and zeroth-order kinetics yielded analytical solutions to the relevant variables. The results were compared with large-scale Escherichia coli and Saccharomyces cerevisiae experiments and simulations from the literature, and good agreement was found in substrate profiles. The analytical profiles obtained for dissolved oxygen, temperature, pH, and were also consistent with the available data. Distribution functions for the substrate were defined, and efficiency factors for biomass growth and oxygen uptake rate were derived. In conclusion, this study demonstrated that axial diffusion equations can be used to model the effects of mixing and reaction on the relevant variables of typical large-scale fed-batch fermentations.

Inverse problem of determining the kernel of integro-differential fractional diffusion equation in bounded domain

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.

Droplet absorption and spreading into thin layers of polymer hydrogels

From biological tissues to microactuators and absorption of solvents into layers of paint, macroscopically non-porous materials with the capacity to swell when in contact with a solvent are ubiquitous. In these systems, owing to strong solid–fluid interactions, chemically driven flows can yield large geometric changes. We study experimentally and theoretically the canonical problem of the swelling of a thin hydrogel layer by a single water drop. Using a bespoke experimental set-up, we observe fast absorption leading to a radially spreading axisymmetric blister. We use a fully three-dimensional linear poroelastic framework with nonlinear kinematic equations to obtain governing equations, which we then reduce with thin-layer scalings to a one-dimensional nonlinear diffusion equation for the evolution of the blister geometry. In the limits of large and small deformations, the evolution of the blister characteristic height and radius are self-similar, following power laws in time. Our experimental measurements show that the evolution of the blister is broadly within the theoretical predictions in the large and small deformation regimes. In the general intermediate deformation regime, the measurements are well captured by our reduced one-dimensional diffusion model, which does not require the sophisticated and computationally expensive numerical approaches necessary for the original two-dimensional nonlinear coupled transport problem. By adapting modelling techniques from the fluid dynamics of thin porous elastic layers to a polymer swelling problem, our modelling framework extends the range of polymer swelling problems that can be treated with semi-analytical methods. Moreover, our detailed experimental data can serve as a test case for future nonlinear poroelastic frameworks of swelling polymer materials.

Inverse coefficient problem for a fractional-diffusion equation with a Bessel operator

The second initial-boundary value problem in a bounded domain for a fractional-diffusion equation with the Bessel operator and the Gerasimov-Caputo derivative is investigated. Theorems of existence and uniqueness of the solution of the inverse problem of determining the lowest coefficient in a one-dimensional fractional diffusion equation under the condition of integral observation are obtained. The Schauder principle was used to prove the existence of the solution.

An inverse source time-fractional diffusion problem via an input-output mapping

In this paper, we investigate an inverse source problem involving a one-dimensional diffusion equation of a time-fractional RiemannLiouville derivative with 0 < α < 1. First, results on the existence and regularity of the weak solution of the direct problem are obtained. For the determination of the unknown time-dependent source term, we use a monotone and distinguishable input-output mapping defined by the additional over-determination integral data for the considered sub-diffusion problem. Finally, the uniqueness of the solution of the inverse problem is proved.

Complex Three-Dimensional Mathematical Model of the Ignition of a Coniferous Tree via a Cloud-to-Ground Lightning Discharge: Electrophysical, Thermophysical and Physico-Chemical Processes

Thunderstorms are the main natural source of forest fires. The ignition mechanism of trees begins with the impact of cloud-to-ground lightning discharge. A common drawback of all predicting systems is that they ignore the physical mechanism of forest fire as a result of thunderstorm activity. The purpose of this article is to develop a physically based mathematical model for the ignition of a coniferous tree via cloud-to-ground lightning discharge, taking into account thermophysical, electrophysical, and physicochemical processes. The novelty of the article is explained by the development of an improved mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account the processes of soot formation caused by the thermal decomposition phase of dry organic matter. Mathematically, the process of tree ignition is described by a system of non-stationary nonlinear differential equations of heat conduction and diffusion. In this research, a locally one-dimensional method is used to solve three-dimensional partial differential equations. The finite difference method is used to solve one-dimensional heat conduction and diffusion equations. Difference analogues of the equations are solved using the marching method. To resolve nonlinearity, a simple iteration method is used. Temperature distributions in a structurally inhomogeneous trunk of a coniferous tree, as well as distributions of volume fractions of phases and concentrations of gas mixture components, are obtained. The conditions for tree trunk ignition under conditions of thunderstorm activity are determined. As a result, a complex three-dimensional mathematical model is developed, which makes it possible to identify the conditions for the ignition of a coniferous tree trunk via cloud-to-ground lightning discharge.

On the Inverse Problem of Time-Dependent Coefficient in a Time Fractional Diffusion Problem by Newly Defined Monic Laquerre Wavelets

Abstract The primary aim of this research is to establish the time-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation in Caputo sense by means of newly defined Monic Laquerre wavelets (MLW) and collocation points. We first give the definition of MLW by taking Monic Laquerre's polynomials into account. Later, time fractional diffusion problem is reduced into a system of ordinary fractional and algebraic equations by utilizing MLW. The residual power series method (RPSM) and the overdetermined data are applied to this system to determine the solution and the unknown time-dependent coefficient together in series form. In the end, illustrative examples are presented to show the stability and accuracy of the proposed wavelet method for the inverse problem of determining unknown time-dependent coefficient in fractional diffusion problems. The reliability of the proposed algorithm for the inverse problems is supported by high degree of accuracy in the given examples.

(Invited) 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 [1]. 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.This work is supported in part by MIC (SCOPE JP191503001 to Y.K.K.), JSPS (JP18H05329, JP20H00220 to S.M., JP20H02558 to Y.K.K., JP20K15112, JP22H01893 to D.K., JP20K15137 to K.O.), MEXT (Nanotechnology Platform JPMXP09F19UT0077), NSF (RAISE-TAQS PHY-1839165 to Y.H.W.), and JST (CREST JPMJCR20B5 to S.M.). D.K. acknowledges support from RIKEN Special Postdoctoral Researcher Program. K.O. is supported by JSPS Research Fellowship. We thank the Advanced Manufacturing Support Team at RIKEN for technical assistance.[1] D. Kozawa, X. Wu, A. Ishii, J. Fortner, K. Otsuka, R. Xiang, T. Inoue, S. Maruyama, Y. Wang, and Y. K. Kato, Nature Commun. 13, 2814 (2022).

Modeling the Anisotropic Transport Behavior for Axisymmetric 2D Electrode Particles Using a Modified Pseudo-2D Model Framework

This presentation extends widely used pseudo-2D (P2D) Li-ion battery models to include electrode particles that have strong crystal-scale anisotropies and possibly non-spherical shapes. Typical P2D models predict Li transport with representative spherical electrode particles by solving a one-dimensional diffusion equation in spherical coordinates. However, battery electrodes (graphite, NMC, etc.) may have strong crystalline anisotropies, leading to transport and mechanical properties that vary by orders of magnitude depending on crystallographic orientations. The anisotropic transport can cause large concentration gradients, possibly leading to particle fracture associated with concentration-induced stress. The remainder of the P2D model, such as Li-ion transport within the electrolyte solvent, remains essentially unchanged. The charge-transfer boundary conditions at the particle surfaces do need to accommodate the crystallographic anisotropy.The present model discretizes representative electrode particles in a two-dimensional finite-volume axisymmetric geometry, accounting for the anisotropic behaviors. The computational burden is greater than it is for the one-dimensional spherical solution. However, the P2D model with two-dimensional axisymmetric particle discretization still runs in only a few minutes on a typical personal computer.The model shows how electrode-particle anisotropy affects the battery performance, such as charge-discharge characteristics and electrochemical impedance spectroscopy. The model also predicts the concentration gradients and correlated stress for various axisymmetric, non-spherical particles.

EFFICIENT EXPLICIT NUMERICAL TECHNIQUE FOR MODELING ADVECTION DIFFUSION REACTION FOR A WATER QUALITY MODEL IN AN OPENED UNIFORM FLOW – A 1D PERSPECTIVE

This research paper presents a novel and efficient explicit numerical technique for modeling advection diffusion reactions in an opened uniform flow from a one-dimensional perspective. The proposed hybrid scheme combines the benefits of explicit finite difference schemes, resulting in an accurate and fast solution for the advection-diffusion equation in water stream problems. The effectiveness of the scheme is demonstrated through its successful implementation in the solution of the water quality problems, where the advection-diffusion equation plays a crucial role. The results obtained using this technique show improved accuracy and computational efficiency. Overall, this research offers a valuable contribution to the field of numerical modeling in water quality and provides a useful tool for researchers and practitioners working in the area of approximating the one-dimensional diffusion equation for the measurement of pollutant concentration.

Numerical Simulation of One-Dimensional Advection Diffusion Equation by New Hybrid Explicit Finite Difference Schemes

In this paper, the aim is to develop two Hybrid Explicit Schemes based on the Finite Difference method for a one- dimensional Advection Diffusion Equation. Moreover, the study considered the advection-diffusion equation as an initial boundary value problem (IBVP) for numerical solutions obtained from various second-order explicit methods along with the solution by proposed methods. Von-Neumann stability analysis is used to analyze the stability of the developed schemes graphically. In the numerical analysis of errors, the L 2 has been computed to compare proposed methods with existing methods in literature and has been carried out in different conditions and step sizes. Proposed methods are robust explicit methods for the purpose of solution of the 1-D advection-diffusion equation.