Nonlinear Diffusion Equation Research Articles

Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delay

ABSTRACT This paper deals with the existence of traveling waves for doubly nonlinear degenerate diffusion equations with Nicholson's blowflies-type source term and time delay. We prove the existence of monotone and non-monotone traveling waves by the monotone iteration and Schauder's fixed point theorem respectively. Furthermore, the non-critical traveling waves are proven to be globally -stable. The adopted approach for the proof is the weighted energy method combining the compactness analysis.

On the Omori Law in the Physics of Earthquakes

This paper proposes phenomenological equations that describe various aspects of aftershock evolution: elementary master equation, logistic equation, stochastic equation, and nonlinear diffusion equation. The elementary master equation is a first-order differential equation with a quadratic term. It is completely equivalent to Omori’s law. The equation allows us to introduce the idea of proper time of earthquake source “cooling down” after the main shock. Using the elementary master equation, one can pose and solve an inverse problem, the purpose of which is to measure the deactivation coefficient of an earthquake source. It has been found for the first time that the deactivation coefficient decreases with increasing magnitude of the main shock. The logistic equation is used to construct a phase portrait of a dynamical system simulating the evolution of aftershocks. The stochastic equation can be used to model fluctuation phenomena, and the nonlinear diffusion equation provides a framework for understanding the spatiotemporal distribution of aftershocks. Earthquake triads, which are a natural trinity of foreshocks, main shock, and aftershocks, are considered. Examples of the classical triad, the mirror triad, the symmetrical triad, as well as the Grande Terremoto Solitario, which can be considered as an anomalous symmetrical triad, are given. Prospects for further development of the phenomenology of earthquakes are outlined.

Fast meshfree methods for nonlinear radiation diffusion equation

Radiation diffusion is a phenomenon of interest in the field of astrophysics, inertial confinement fusion and so on. Since it is modeled by nonlinear equations that are usually solved in complex domains, it is difficult to solve by means of finite element method and finite difference method and so on. In the paper, we will provide a kind of new fast meshfree methods based on radial basis functions. At first, the part of diffusion terms for 1D and 2D radiation diffusion equations are linearized directly on time to form the new implicit schemes, and Kansa’s non-symmetric collocation method with the compactly supported radial basis function is used to solve the radiation diffusion problem. Second, the successive permutation iterative algorithms for full-implicit discretization on time are constructed furtherly, which are more efficient than the former algorithm. In the end, the accuracy and efficiency of the presented algorithms are verified by 1D and 2D numerical experiments. The new meshfree methods not only avoid the complexity of mesh generation, but also solve the radiation diffusion problem with high accuracy.

Time-dependent condensation of bosonic potassium

We calculate the time-dependent formation of Bose–Einstein condensates (BECs) in potassium vapours based on a previously derived exactly solvable nonlinear boson diffusion equation (NBDE). Thermalization following a sudden energy quench from an initial temperature T_mathrm {i} to a final temperature T_mathrm {f} below the critical value and BEC formation are accounted for using closed-form analytical solutions of the NBDE. The time-dependent condensate fraction is compared with available ^{39}K data for various scattering lengths.Graphic

Counter-current imbibition and non-linear diffusion in fractured porous media: Analysis of early- and late-time regimes and application to inter-porosity flux

The displacement of non-aqueous phase liquid (NAPL) in permeable porous rock by water saturating the surrounding fractures is studied. In many situations of practical interest, capillarity is the dominant driving force. Using the global pressure concept, it is shown that the water saturation is driven by a generic non-linear diffusion equation governing the matrix block counter-current spontaneous imbibition. In addition, in most cases, the saturation-dependent diffusion coefficient vanishes at the saturation end points, that renders the driving equation highly singular.In this paper, two exact asymptotic solutions valid for short and long times are presented, under the assumption that the conductivity vanishes as a power-law of both phase saturations at the extreme values of the fluid saturation.Focusing on the late-time domain, the asymptotic solution is derived using an Ansatz that is written under the form of a power-law time decay of the NAPL saturation. In the spatial domain, this solution is an eigenvector of the non-linear diffusion operator driving the saturation, for a problem with Dirichlet boundary conditions. If the diffusion coefficient varies as a power law of the NAPL saturation, the spatial variations of the solution are given analytically for a one-dimensional porous medium corresponding to parallel fracture planes. The analytical solution is in very good agreement with results of numerical simulations involving various realistic sets of input transport parameters.Generalization to the case of two- or three-dimensional matrix blocks of arbitrary shape is proposed using a similar Ansatz, solution of a non-linear eigenvalue problem. A fast converging algorithm based on a fixed-point sequence starting from a suitable first guess was developed. Comparisons with full-time simulations for several typical block geometries show an excellent agreement.These results permit to set-up an analytical formulation generalizing linear single-phase representation of matrix-to-fracture exchange term. It accounts for the non-linearity of the local flow equations using the power-law dependence of the conductivity for low NAPL saturation. The corresponding exponent can be predicted from the input conductivity parameters. Similar findings are also presented and validated numerically for two- or three-dimensional matrix blocks. That original approach paves the way to research leading to a more faithful description of matrix-to-fracture exchanges when considering a realistic fractured medium composed of a population of matrix blocks of various size and shapes.

Solutions of a Nonlinear Diffusion Equation with a Regularized Hyper-Bessel Operator

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces.

Convergence rate for the incompressible limit of nonlinear diffusion–advection equations

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cellpopulation models to free boundary problems of Hele–Shaw type. Although a vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows

We demonstrate that the reformulation of renormalization group (RG) flow equations as nonlinear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional ${\mathbb{Z}}_{2}$-symmetric model that the dissipative character of generic nonlinear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semigroup property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dissipative character of RG flows, its irreversibility and the entropy production during the RG flow may be linked to the existence of a so-called $\mathcal{C}\text{\ensuremath{-}}/\mathcal{A}$-function. In total, this introduces an asymmetry in the so-called RG time---in complete analogy to the thermodynamic arrow of time---and allows for an interpretation of infrared actions as equilibrium solutions of dissipative RG flows equations. The impossibility of resolving microphysics from macrophysics is evident in this framework. Furthermore, we directly link the irreversibility and the entropy production in RG flows to an explicit numerical entropy production, which is manifest in diffusive and non-linear partial differential equations (PDEs) and a standard mathematical tool for the analysis of PDEs. Using exactly solvable zero-dimensional ${\mathbb{Z}}_{2}$-symmetric models, we explicitly compute the (numerical) entropy production related to the total variation nonincreasing property of the PDE during RG flows toward the infrared limit. Finally, we discuss generalizations of our findings and relations to the $\mathcal{C}\text{\ensuremath{-}}/\mathcal{A}$-theorem as well as how our work may help to construct truncations of RG flow equations in the future, including numerically stable schemes for solving the corresponding PDEs.

Classification of blow-up and global existence of solutions to an initial Neumann problem

The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous Neumann problem of a nonlinear diffusion equation driven by the p(x)-Laplace operator. Complete classification of global existence and blow-up in finite time of solutions is given when the initial data satisfies different conditions. We first obtain a threshold result for the solution to exist globally or to blow up in finite time when the initial energy is subcritical or critical. We also establish a decay rate of the L2 norm for global solutions. Furthermore, we provide some sufficient conditions for the existence of global and blow-up solutions for supercritical initial energy. We finally give two-sided estimates of asymptotic behavior when the diffusion term dominates the source. This is a continuation of our previous work [17].

A time-splitting local meshfree approach for time-fractional anisotropic diffusion equation: application in image denoising

Image denoising approaches based on partial differential modeling have attracted a lot of attention in image processing due to their high performance. The nonlinear anisotropic diffusion equations, specially Perona–Malik model, are powerful tools that improve the quality of the image by removing noise while preserving details and edges. In this paper, we propose a powerful and accurate local meshless algorithm to solve the time-fractional Perona–Malik model which has an adjustable fractional derivative making the control of the diffusion process more convenient than the classical one. In order to overcome the complexities of the problem, a suitable combination of the compactly supported radial basis function method and operator splitting technique is proposed to convert a complex time-fractional partial differential equation into sparse linear algebraic systems that standard solvers can solve. The numerical results of classical and fractional models are explored in different metrics to demonstrate the proposed scheme’s effectiveness. The numerical experiments confirm that the method is suitable to denoise digital images and show that the fractional derivative increases the model’s ability to remove noise in images.

Modelling of Irreversible Homogeneous Reaction on Finite Diffusion Layers

The mathematical model proposed by Chapman and Antano (Electrochimica Acta, 56 (2010), 128–132) for the catalytic electrochemical–chemical (EC’) processes in an irreversible second-order homogeneous reaction in a microelectrode is discussed. The mass-transfer boundary layer neighbouring an electrode can contribute to the electrode’s measured AC impedance. This model can be used to analyse membrane-transport studies and other instances of ionic transport in semiconductors and other materials. Two efficient and easily accessible analytical techniques, AGM and DTM, were used to solve the steady-state non-linear diffusion equation’s infinite layers. Herein, we present the generalized approximate analytical solution for the solute, product, and reactant concentrations and current for the small experimental values of kinetic and diffusion parameters. Using the Matlab/Scilab program, we also derive the numerical solution to this problem. The comparison of the analytical and numerical/computational results reveals a satisfactory level of agreement.

An Efficient Local Meshfree Method for Signal Smoothing by a Time–Fractional Nonlinear Diffusion Equation

An Efficient Local Meshfree Method for Signal Smoothing by a Time–Fractional Nonlinear Diffusion Equation

A new and efficient approach for solving linear and nonlinear time-fractional diffusion equations of distributed order

A new and efficient approach for solving linear and nonlinear time-fractional diffusion equations of distributed order

A meshless method to solve the variable-order fractional diffusion problems with fourth-order derivative term

Using the meshless collocation method, a scheme for solving nonlinear variable-order fractional diffusion equations with a fourth-order derivative term is presented. Here the fractional derivative term is approximated by weighted and shifted Grünwald difference (WSGD) approximation formula. The difficulty caused by the nonlinear terms is carefully handled by quasilinearization technique. Using the radial basis functions, the solution of the problem is written in terms of the primary approximation, and the related correcting functions at each time step. Then the approximation is substituted back to the governing equations where the unknown parameters can be determined. Finally, the method is supported by several numerical experiments on irregular domains.

Lie Symmetries of the Nonlinear Fokker-Planck Equation Based on Weighted Kaniadakis Entropy

The paper studies the Lie symmetries of the nonlinear Fokker-Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted Kaniadakis entropy, are found. The MaxEnt problem associated to the weighted Kaniadakis entropy is given a complete solution, together with the thermodynamic relations which extend the known ones from the non-weighted case. Several different, but related, arguments point out a subtle dichotomous behavior of the Kaniadakis constant k, distinguishing between the cases k∈(−1,1) and k=±1. By comparison, the Lie symmetries of the NFPEs based on Tsallis q-entropies point out six “exceptional” cases, for: q=12, q=32, q=43, q=73, q=2 and q=3.

Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations

In this article, an efficient two-grid finite element method is proposed for solving the nonlinear time fractional variable coefficient diffusion equations. This algorithm firstly solves a nonlinear system to get the numerical solution uHn on the coarse grid with size H, then based on the initial iterative solution uHn on the coarse grid, the linearized finite element system is solved on the fine grid with size h to get the numerical solution Uhn, in which the temporal direction is approximated by the L2−1σ scheme. Besides, the stability and priori error estimates of standard finite element method and two-grid method are given. Finally, the validity and efficiency of the two-grid algorithm are verified by two numerical experiments.

Large deviations for stochastic equations in Hilbert spaces with non-Lipschitz drift

We prove a Freidlin–Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space–time white noise.

Stability Results for Backward Nonlinear Diffusion Equations with Temporal Coupling Operator of Local and Nonlocal Type

Stability Results for Backward Nonlinear Diffusion Equations with Temporal Coupling Operator of Local and Nonlocal Type

H-theorems for systems of coupled nonlinear Fokker-Planck equations

Nonlinear diffusion and Fokker-Planck equations constitute valuable tools in the study of diverse phenomena in complex systems. Processes described by these equations are closely related to thermostatistical formalisms based on generalized entropic functionals. Inspired by these relations, we explore the behavior of systems of coupled, nonlinear Fokker-Planck equations. In particular, we establish an H-theorem for a wide family of this type of systems. This H-theorem is formulated in terms of an appropriate free-energy–like functional. The nonlinear evolution equations discussed here include, as particular instances, those governing the dynamics of interacting multi-species, many-body systems in the overdamped-motion regime.

A comparison between nonlinear and constant thermal properties approaches to estimate the temperature in LASER welding simulation

The nonlinear thermophysical properties significantly affect the temperature field and the appearance of the weld bead in the LASER Beam Welding (LBW). Then, it is vital to have a well-defined numerical model for analyzing the thermal behavior of the welded material. Nonetheless, many papers still address the welding simulation using constant thermal properties. In this way, this paper proposes a three-dimensional thermal analysis of an unsteady LBW aiming to compare the difference between the constant and nonlinear thermophysical properties approaches. It applied the Finite Volume Method (FVM) to solve the nonlinear three-dimensional heat diffusion equation with an enthalpy function to model the phase change using a fully implicit scheme. In traditional models, these considerations promote a significant increase in computational time for the convergence of the method. Thus, CUDA-C in-house parallel routines were implemented and executed in a Graphics Processing Unit (GPU) to solve this problem. Lab-controlled experiments validated the proposed methodology. The results highlighted the importance of using the nonlinear approach. Furthermore, a detailed study demonstrated the difficulty of knowing precisely the placement of thermocouples, given the high-temperature gradient in the welding processes. The proposed methodology demonstrated to be a faster, cheaper, and efficient way to simulate the LBW.