Nonlinear Diffusion Model Research Articles

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Random walk model for dual cascades in wave turbulence.

Dual cascades in turbulent systems with two conserved quadratic quantities famously arise in both two-dimensional hydrodynamic turbulence and also in wave turbulence based on four-wave interactions. Examples for the latter include surface waves and nonlinear Schrödinger equationswith cubic nonlinearity. However, numerical simulations in forced-dissipative equilibrium of two-dimensional turbulence and of a one-dimensional wave system reveal that the physical nature of their cascades is starkly different. This is demonstrated by comparing their spectra in a finite inertial range and by comparing the temporal fluctuations of their spectral fluxes. In particular, the flux fluctuations are much larger in the wave case and frequently lead to instantaneous flux values that have the opposite sign of the mean flux, a phenomenon that is completely absent in the hydrodynamic case. A simple random walk model for the dual cascade in wave turbulence is then formulated that is very successful in explaining these effects. In particular, the model is able to replicate the detailed shape of the observed turbulent spectrum in a finite inertial range, and it also offers a ready explanation for the large flux fluctuations. It is also shown that a nonlinear diffusion model for the wave system cannot explain the observed spectral shapes. Overall, this suggests that in wave turbulence the systematic spectral fluxes observed in a dual cascade do not require an irreversible dynamical mechanism, rather, they arise as the inevitable outcome of blind chance.

Analysis on a High Accuracy Fully Implicit Solution for Strong Nonlinear Diffusion Problem - Convergence, Stability, and Uniqueness

Some fundamental properties are analyzed for a fully implicit finite difference (FIFD) solution of conservative strong nonlinear diffusion problem. The scheme is constructed by combining a second-order backward difference temporal discretization and a central finite difference spatial discretization, and therefore highly nonlinear. Theoretical analysis is carried out under a coercive condition according with the diffusion feature of the strong nonlinear diffusion model. Benefiting from the boundedness estimates of the FIFD solution itself and its first- and second-order spatial difference quotients, some novel argument techniques are developed to overcome the difficulties coming from the nonlinear approximation for the strong nonlinear conservative diffusion operator. Consequently, it is proved rigorously that the FIFD scheme is unconditionally stable, its solution is unique and convergent to the exact solution of the original problem with second-order space-time accuracy. Numerical examples are provided to confirm its advantages on precision and efficiency over its first-order time accurate counterpoint.

On fluorophore imaging by nonlinear diffusion model with dynamical iterative scheme

Fluorescence imaging aims at recovering the absorption coefficient of fluorophore in biological tissues. Due to the nonlinear dependence of excitation and emission fields on the unknown coefficient, such an inverse problem with the boundary measurement as inversion input is nonlinear. We reformulate this inverse problem as an optimization problem for a non-convex cost functional consisting of the unknown absorption coefficient depending both on the excitation field and the emission one with a penalty term. The existence of minimizer of the cost functional is proved rigorously, with explicit expression for the Fréchet derivative of the cost functional. For seeking the local minimizer considered as the approximate solution to the inverse problem efficiently, we develop a dynamical process which makes the non-convex cost functional quadratic at each iteration step, by freezing the unknown coefficient for the excitation field. This new dynamical quadratic optimization process is proven convergent and decreases the amount of computations for the fluorescence imaging, while the reconstruction accuracy keeps almost unchanged. Such advantages are verified by several numerical examples for different configurations of the absorption coefficient to be identified, comparing our proposed scheme with the algorithm for the original non-convex cost functional.

A variational PDNet network using a learning reaction–diffusion equation

Due to their high performance in modeling and forecasting a large amount of real-world complex phenomena, deep convolutional neural networks have received a great deal of attention over the past ten years. It has been extensively utilized in several recent vision problems, such as image denoising and deblurring. By incorporating recent developments in learning-based nonlinear diffusion models and highlighting the connection between partial differential equations (PDEs) and networks, especially residual networks, we propose a flexible learning reaction–diffusion system with spatially computed convolution kernels. This trained PDE is employed to prevent the low-level features from deteriorating, which is needed for tasks involving denoising. In fact, we introduce an improved network architecture for a denoising network entitled VPDNet through an optimal control problem. More precisely, the network architecture is governed by an explicit Euler discretization of a reaction–diffusion system. We give the main algorithm to identify the kernel parameters using the Alternating Direction Method of Multipliers (ADMM) combined with a Primal–Dual algorithm. Finally, numerical simulations are provided with appropriate comparisons to demonstrate the efficiency of the elaborated network.

Image reconstruction based on nonlinear diffusion model for limited-angle computed tomography

The problem of limited-angle computed tomography (CT) imaging reconstruction has a wide range of practical applications. Due to various factors such as high x-ray absorption, structural characteristics of the scanned object, and equipment limitations, it is often impractical to obtain a complete angular scan, resulting in limited-angle scan data. In this paper, we propose an iterative image reconstruction algorithm for limited-angle CT. The algorithm carries out a traditional CT reconstruction and a nonlinear diffusion process alternatively. Specifically, a subtle partial differential equation is constructed to guide the nonlinear diffusion process to eliminate limited-angle artifacts in the reconstructed image. Numerical experiments on both analytic data and real data validate the efficacy of the proposed nonlinear diffusion reconstruction algorithm. Furthermore, a linear diffusion reconstruction algorithm which combines a traditional CT reconstruction algorithm and a linear diffusion process is also presented in the paper.

Partial Lyα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} thermalization in an analytic nonlinear diffusion model

During recombination, the cosmic background radiation is disturbed, in particular, by Lyman-alpha emissions from neutral hydrogen. It is proposed to account for the subsequent time-dependent partial thermalization of the Lyα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} energy content in an analytically solvable nonlinear diffusion model. The amplitude of the partially thermalized and redshifted Lyα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} line is found to be too low to be visible in the cosmic microwave spectrum, in accordance with previous numerical models and Planck observations.

EWRD: Entropy‐Weighted Low‐Light Image Enhancement via Reverse Diffusion Model

Low‐light enhancement significantly aids vision tasks under poor illumination conditions. Existing methods primarily focus on enhancing severely degraded low‐light areas, improving illumination accuracy, or noise suppression, yet they often overlook the loss of additional details and color distortion during calculation. In this paper, we propose an innovative entropy‐weighted low‐light image enhancement method via the reverse diffusion model, aiming at addressing the limitations of the traditional Retinex decomposition model in preserving local pixel details and handling excessive smoothing issues. This method integrates an entropy‐weighting mechanism for improved image quality and entropy, along with a reverse diffusion model to address the detail loss in total variation regularization and refine the enhancement process. Furthermore, we utilize long short‐term memory networks for the learning reverse process and the simulation of image degradation, based on a thermodynamics‐based nonlinear anisotropic diffusion model. Comparative experiments reveal the superiority of our method over conventional Retinex‐based approaches in terms of detail preservation and visual quality. Extensive tests across diverse datasets demonstrate the exceptional performance of our method, evidencing its potential as a robust solution for low‐light image enhancement.

Comprehensive analysis of cationic dye removal from synthetic and industrial wastewater using a semi-natural curcumin grafted biochar/poly acrylic acid composite hydrogel.

Polymer composites offer a tailored framework as an exceptional candidate for water treatment due to their tunable chemical structure, porous 3D architecture, physiochemical stability, accessibility, pH-sensitivity and ease of use. In this study, curcumin-engineered biochar is embedded into a cross-linked polyacrylic acid hydrogel matrix using in situ polymerization for developing a semi-natural adsorbent for the removal of cationic dye from an aqueous solution. The physicochemical features of the generated composite hydrogel are significantly influenced by the implementation of curcumin-grafted biochar into the polyacrylic acid substrate. Comprehensive characteristic approaches were employed to explore all aspects of the adsorbent's properties, especially its removal efficacy. The methodical adsorption study was accomplished by monitoring dynamic factors such as pH, adsorbent content, time frame, and initial dye concentration. The presence of the porous aromatized structure of biochar, active oxygen-enrich functional groups (carboxyl, hydroxyl, keto, enol, ether) coupled with the conjugated curcumin structure facilitate the effective establishment of hydrogen bonds, electrostatic interactions, π-π interactions, electron donor-acceptor and charge-assisted H-bonding with the malachite green (MG) and rhodamine B (Rho) molecules. The highest adsorption capacities of MG and Rho reached 521 mg g-1 and 741 mg g-1 respectively, in the range of neutral pH, considering their molecular nature, functionalities, and unique adsorption mechanisms. The isothermal modeling was carried out with Henry, Langmuir, Jovanovic, Freundlich, Temkin, and Koble-Corrigan models to determine the adsorption system. Additionally, the kinetic data were assessed with Bangham, pseudo-first-order, pseudo-second-order, intra-particle, and liquid film diffusion models to ascertain the rate-limiting phase. The Koble-Corrigan and Langmuir isotherm models (R2 > 0.997) as well as pseudo-second-order (R2 > 0.998) and Elovich (R2 = 0.983 and 0.995) kinetics models provide a substantial level of concordance with empirical findings. The analysis of non-linear diffusion models revealed that the Bangham (R2 > 0.995) pore and liquid film diffusion (R2 > 0.960) models has major influence on the rate of the adsorption procedure. The binary adsorption test demonstrated higher efficacy of the synthesized adsorbent in the removal of malachite as compared to rhodamine. This study sheds light on the design of a cost-effective semi-natural polymeric composite for treating dye-polluted wastewaters, a major milestone toward environmental and ecological sustainability.

Direct Visualization of Dark Interlayer Exciton Transport in Moiré Superlattices.

Moiré superlattices have emerged as an unprecedented manipulation tool for engineering correlated quantum phenomena in van der Waals heterostructures. With moiré potentials as a naturally configurable solid-state that sustains high exciton density, interlayer excitons in transition metal dichalcogenide heterostructures are expected to achieve high-temperature exciton condensation. However, the exciton degeneracy state is usually optically inactive due to the finite momentum of interlayer excitons. Experimental observation of dark interlayer excitons in moiré potentials remains challenging. Here we directly visualize the dark interlayer exciton transport in WS2/h-BN/WSe2 heterostructures using femtosecond transient absorption microscopy. We observe a transition from classical free exciton gas to quantum degeneracy by imaging temperature-dependent exciton transport. Below a critical degeneracy temperature, exciton diffusion rates exhibit an accelerating downward trend, which can be explained well by a nonlinear quantum diffusion model. These results open the door to quantum information processing and high-precision metrology in moiré superlattices.

Exact sharp-fronted solutions for nonlinear diffusion on evolving domains

Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either report numerical solutions or require an assumption of linear diffusion to determine exact solutions. Unfortunately, numerical solutions do not reveal the relationship between the model parameters and the solution features. Additionally, experimental observations typically report the presence of sharp fronts, which are not captured by linear diffusion. Here we address both limitations by presenting exact sharp-fronted solutions to a model of degenerate nonlinear diffusion on a growing domain. We obtain the solution by identifying a series of transformations that converts the model of a nonlinear diffusive process on an evolving domain to a nonlinear diffusion equation on a fixed domain, which admits known exact solutions for certain choices of diffusivity functions. We determine expressions for critical time scales and domain growth rates such that the diffusive population never reaches the domain boundaries and hence the solution remains valid.

Comparison of several numerical solvers for a discretized nonlinear diffusion model with source terms

Abstract The numerical solution of the nonlinear system of equations resulting from a real engineering problem is discussed. We use the approximate solution of a system of two nonlinear integrodifferential equations to build the nonlinear system of equations. This system can be solved by Newton’s method if the solution is differentiable, or using some derivative-free methods, such as Steffensen’s method. Here we show that Steffensen’s method does not always converge and secant method requires more iterations than Traub’s method and Newton’s method. We recommend Traub’s method in case the solution is not differentiable.

Nanoscale effect on the formation of the amorphous Ni silicide by rapid thermal annealing from crystalline and pre-amorphized silicon

The Ni monosilicide alloyed with Pt is widely used as contact material in advanced microelectronics devices and a good knowledge of silicide formation kinetics is required for the process control. In this work, the nature, and the growth kinetics of the first silicide obtained during the solid-state reaction between the Ni0.9Pt0.1 and the Si are studied for different pre amorphization implant (PAI) conditions as well as for a reference sample without PAI. Reactions between a 10 nm thick Ni0.9Pt0.1 film and Si (100) substrate are analyzed after several rapid thermal anneals (RTA). The nature of the first silicide is determined by Fourier Transform of TEM images and by chemical TEM-EDX analyses. The silicide growth behavior is determined by measuring the silicide thickness by X ray reflectivity (XRR) after the partial reaction induced by RTAs at different temperatures and times. To determine the growth law, the linear parabolic model is first considered but a nonlinear reactive diffusion model must be developed to accurately reproduce the experimental results. From this model, the effective diffusion coefficient as well as its activation energy were determined for the three samples with PAI and the reference sample without PAI. The influence of the driving force on the nonlinear diffusion for thin films is proved, and the impact of the amorphous substrate on the kinetics parameters is quantified and compared to the literature.

Hygromechanical Behavior of Polyamide 6.6: Experiments and Modeling.

This paper investigates water absorption in polyamide 6.6 and the resulting hygroscopic swelling and changes in mechanical properties. First, sorption and swelling experiments on specimens from injection molded plates are presented. The observed swelling behavior is dependent on the melt flow direction of the injection molding process. Additionally, thermal analysis and mechanical tensile tests were performed for different conditioning states. The water sorption is accompanied by a decrease in the glass transition temperature and a significant reduction in stiffness and strength. Next, a sequentially coupled modeling approach is presented. A nonlinear diffusion model is followed by mechanical simulations accounting for swelling and concentration-dependent properties. For the mechanical properties, the notion of a "gap" temperature caused by the shift of the glass transition range due to water-induced plasticization is employed. This model enables the computation of local moisture concentration fields and the resultant swelling and changes in stress-strain behavior.

Field-Theoretic Renormalization Group in Models of Growth Processes, Surface Roughening and Non-Linear Diffusion in Random Environment: Mobilis in Mobili

This paper is concerned with intriguing possibilities for non-conventional critical behavior that arise when a nearly critical strongly non-equilibrium system is subjected to chaotic or turbulent motion of the environment. We briefly explain the connection between the critical behavior theory and the quantum field theory that allows the application of the powerful methods of the latter to the study of stochastic systems. Then, we use the results of our recent research to illustrate several interesting effects of turbulent environment on the non-equilibrium critical behavior. Specifically, we couple the Kazantsev–Kraichnan “rapid-change” velocity ensemble that describes the environment to the three different stochastic models: the Kardar–Parisi–Zhang equation with time-independent random noise for randomly growing surface, the Hwa–Kardar model of a “running sandpile” and the generalized Pavlik model of non-linear diffusion with infinite number of coupling constants. Using field-theoretic renormalization group analysis, we show that the effect can be quite significant leading to the emergence of induced non-linearity or making the original anisotropic scaling appear only through certain “dimensional transmutation”.

Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration–consumption interaction

A no-flux initial-boundary value problem for ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) is considered in smoothly bounded subdomains of with and suitably regular initial data, where φ is assumed to reflect algebraic type cross-degeneracies by sharing essential features with for some . Based on the discovery of a gradient structure acting at regularity levels mild enough to be consistent with degeneracy-driven limitations of smoothness information, in this general setting it is shown that with some measurable limit profile and some null set , a corresponding global generalized solution, known to exist according to recent literature, satisfies ρ(u(⋅,t))⇀⋆ρ(u∞)in L∞(Ω) and v(⋅,t)→0in Lp(Ω)for all p⩾1 as , where , . In the particular case when either and is arbitrary, or and , additional quantitative information on the deviation of trajectories from the initial data is derived. This is found to imply a lower estimate for the spatial oscillation of the respective first components throughout evolution, and moreover this is seen to entail that each of the uncountably many steady states of () is stable with respect to a suitably chosen norm topology.

Reversible Navier-Stokes equationon logarithmic lattices.

The three-dimensional reversible Navier-Stokes (RNS) equationsare a modification of the dissipative Navier-Stokes (NS) equations, first introduced by Gallavotti [Phys. Lett. A 223, 91 (1996)0375-960110.1016/S0375-9601(96)00729-3], in which the energy or the enstrophy is kept constant by adjusting the viscosity over time. Spectral direct numerical simulations of this model were performed by Shukla etal. [Phys. Rev. E 100, 043104 (2019)2470-004510.1103/PhysRevE.100.043104] and Margazoglou etal. [Phys. Rev. E 105, 065110 (2022)10.1103/PhysRevE.105.065110]. Here we consider a linear, forced reversible system obtained by projecting RNS equationson a log lattice rather than on a linearly spaced grid in Fourier space, as is done in regular spectral numerical simulations. We perform numerical simulations of the system at extremely large resolutions, allowing us to explore regimes of parameters that were out of reach of the direct numerical simulations of Shukla etal. Using the nondimensionalized forcing as a control parameter, and the square root of enstrophy as the order parameter, we confirm the existence of a second-order phase transition well described by a mean-field Landau theory. The log-lattice projection allows us to probe the impact of the resolution, highlighting an imperfect transition at small resolutions with exponents differing from the mean-field predictions. Our findings are in qualitative agreement with predictions of a 1D nonlinear diffusive model, the reversible Leith model of turbulence. We then compare the statistics of the solutions of RNS and NS, in order to shed light on an adaptation of the Gallavotti conjecture, in which there is equivalence of statistics between the reversible and irreversible models, to the case where our reversible model conserves either the enstrophy or the energy. We deduce the conditions in which the two are equivalent. Our results support the validity of the conjecture and represent an instance of nonequilibrium system where ensemble equivalence holds for mean quantities.

Unconditional superconvergence analysis of an energy-stable C–N fully discrete scheme for the nonlinear magnetic diffusion model with memory

The unconditional superconvergence behavior of an energy-stable Crank–Nicolson (C–N) fully discrete scheme with nonconforming EQ1rot finite element method (FEM) is investigated for the nonlinear integro-differential system associated with the diffusion of the magnetic field. The stability of the numerical solution in the broken H1-norm is proved by the Gagliardo–Nirenberg (G–N) inequality, and the unique solvability is demonstrated by the Brouwer fixed point theorem. In which the boundness of the L∞-norm and the inverse inequality are eliminated. The combination of the Green formula and two special properties of EQ1rot element leads to the unconditional superconvergence estimate without any restriction between the time step and space partition size. Finally, the validity of the theoretical findings is supported by some numerical results.

Solvability of nonlinear diffusion models with flux at the boundary driven by nonmonotone local and memory interactions

We investigate local and global solvability for nonlinear diffusion models with boundary flux driven by competing local and memory interactions. New local existence and comparison theories are developed for such models which are not amenable to a-priori arguments for nonnegative solutions, as well as not permitting standard monotonicity techniques of analysis. A transformation used to address nonnegativity subsequently results in classical solutions for strictly positive initial states. An alternating construction introduced to obtain solutions via monotone limits from above and below is somewhat reminiscent of the method of upper and lower solutions. Subsolution/supersolution comparison results that follow are applied to analyze boundedness, positivity, and blow up in finite time of solutions for power law cases of localized absorption and memory driven dissipation at the boundary. This connects the present work with our previous analyses of models having memory driven absorption at the boundary, shown to exactly parallel known results in cases of localized absorption, and begins to reveal comparative strengths of each.

Numerical solution for a nonlinear diffusion model with source terms

Abstract The nonlinear diffusion system with source terms is studied. This model is derived via Maxwell equations describing electromagnetic field propagation in media. The uniqueness of the solution and its behavior as t → ∞ t\to\infty are proved. We also prove that the finite difference approximation converges in space and time. Numerical implementation with various experiments for different values of the input parameters is performed to validate the theoretical conclusions.