Nonlinear Diffusion Term Research Articles

A multiscale view of nonlinear diffusion in biology: From cells to tissues

This paper presents a review on the mathematical tools for the derivation of macroscopic models in biology from the underlying description at the scale of cells as it is delivered by a kinetic theory model. The survey is followed by an overview of research perspectives. The derivation is inspired by the Hilbert’s method, known in classic kinetic theory, which is here applied to a broad class of kinetic equations modeling multicellular dynamics. The main difference between this class of equations with respect to the classical kinetic theory consists in the modeling of cell interactions which is developed by theoretical tools of stochastic game theory rather than by laws of classical mechanics. The survey is focused on the study of nonlinear diffusion and source terms.

Nonlinear edge-preserving diffusion with adaptive source for document images binarization

This paper proposes a nonlinear edge-preserving diffusion equation with an adaptive source term for binarization of degraded document images. The role of nonlinear diffusion term is to smooth images with preservation of text edges and corners, while the source term is responsible for the desired binarization. Unlike other binarization techniques (such as clustering-based and threshold-based), the idea behind the proposed method is that a sequence of gradually binarized images is obtained by solving the evolution equation starting with the image to be binarized, and tends to the slightly smoothed version of the desired binary image at infinity. A semi-implicit parallel splitting-up method is developed for solving the proposed model effectively. The proposed model with algorithm is tested on the DIBCO (Document Image Binarization Competitions) series datasets. The results show that it has generally the best performance, compared to four PDE (partial differential equation)-based binarization models, and six recent and benchmark binarization algorithms (non-PDE based).

Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process.

This paper is devoted to the justification of the macroscopic, mean-field nutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva et al. (Phys A 392:5644-5662, 2013) to model the complex spatio-temporal dynamics exhibited by the bacterium Bacillus subtilis during experiments run in vitro. This justification is based on a microscopic description of the movement of individual cells whose changes in velocity (in both speed and orientation) obey a velocity jump process governed by a transport equation of Boltzmann type. For that purpose, the asymptotic method introduced by Hillen and Othmer (SIAM J Appl Math 61:751-775, 2000; SIAM J Appl Math 62:1222-1250, 2002) is applied, which consists of the computation of the leading order term in a regular Hilbert expansion for the solution to the transport equation, under an appropriate parabolic scaling and a first order perturbation of the turning rate of Schnitzer type (Schnitzer in Phys Rev E 48:2553-2568, 1993). The resulting parabolic limit equation at leading order for the bacterial cell density recovers the degenerate nonlinear cross diffusion term and the associated chemotactic drift appearing in the original system of equations. Although the bacterium B. subtilis is used as a prototype, the method and results apply in more generality.

Convergence and center manifolds for differential equations driven by colored noise

In this paper, we study the convergence and pathwise dynamics of random differential equations driven by colored noise. We first show that the solutions of the random differential equations driven by colored noise with a nonlinear diffusion term uniformly converge in mean square to the solutions of the corresponding Stratonovich stochastic differential equation as the correlation time of colored noise approaches zero. Then, we construct random center manifolds for such random differential equations and prove that these manifolds converge to the random center manifolds of the corresponding Stratonovich equation when the noise is linear and multiplicative as the correlation time approaches zero.

An efficient semi‐implicit temporal scheme for boundary‐layer vertical diffusion

Time integration of the boundary‐layer vertical diffusion equation has been investigated. The nonlinearity associated with the diffusion coefficient makes the implicit approach impractical, while the use of an explicit scheme limits the stable time‐step sizes and consequently would be inefficient. By using a diagonally implicit Runge–Kutta scheme, a new approach has been proposed in which the diffusion coefficients at each internal stage are calculated by a weight‐averaged combination of solutions. Using the weight coefficient α offers more robust calculations due to involving implicit solutions and, as shown, it could improve the accuracy due to more engaging the explicit solutions. It has been found that the proposed semi‐implicit method is more accurate and computationally less expensive than the implicit scheme. Moreover, in terms of stability and accuracy improvement, the advantage of the proposed DIRK scheme, compared to the scheme proposed by Diamantakis et al. (), has been revealed, particularly for a highly nonlinear diffusion term.

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

The random dynamics in \begin{document}$ H^s(\mathbb{R}^n) $\end{document} with \begin{document}$ s\in (0,1) $\end{document} is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diffusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain.

A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems

A discontinuous Galerkin approximation for the space variables with the backward Euler time discretisation for a kind of parabolic problems with the nonlinear diffusion, convection and source terms is investigated. To solve the strongly nonlinear algebra system, a two-grid method is proposed. With this algorithm, solving a nonlinear system on the fine discontinuous finite element space is reduced into solving a nonlinear problem on a coarse gird of size and solving a linear problem on a fine grid of size. Convergence estimates in H1-norm are obtained. The numerical experiments are provided to confirm our theoretical analysis.

On Use of Expanding Parameters and Auxiliary Term in Homotopy Perturbation Method for Boussinesq Equation with Tidal Condition

This paper uses the homotopy perturbation method for the analytical solution of groundwater table fluctuations, in response to the tidal boundary condition, for a coastal unconfined aquifer with sloping beach face. The Boussinesq equation for sloping beach contains two non-linear terms. The governing equation is reconstructed in homotopic form with two virtual perturbation parameters and an auxiliary term. The secular terms generated from the non-linear diffusion term and the slope term are eliminated by using parameter expansions based on two virtual parameters. Two non-dimensional parameters emerge from the solution in the process of eliminating secular terms: (i) parameter equivalent to amplitude parameter and (ii) parameter representing beach slope. The second-order (starting from zeroth-order) solution is presented. The higher-order solution efficiently captures the non-linearity of the problem.

Weak Pullback Attractors for Mean Random Dynamical Systems in Bochner Spaces

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.

Approximation-based decentralized output-feedback control for uncertain stochastic interconnected nonlinear time-delay systems with input delay and asymmetric input saturation

This paper proposes an adaptive observer-based neural controller for a class of uncertain large-scale stochastic nonlinear systems with actuator delay and time-delay nonlinear interactions, where drift and diffusion terms contain all state variables of their own subsystem. First, a state observer is established for estimating the unmeasured states, and a predictor-like term is utilized to transform the input delayed system into the delay-free system. Second, novel appropriate Lyapunov–Krasovskii functionals are used to compensate the time-delay terms, and neural networks are employed to approximate unknown nonlinear functions. At last, an output-feedback adaptive neural control scheme is constructed by using Lyapunov stability theory and backstepping technique. It is shown that the designed neural controller can ensure that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) and the tracking error is driven to a small neighborhood of the origin. The simulation results are presented to further show the effectiveness of the proposed approach.

Reactions, diffusion, and volume exclusion in a conserved system of interacting particles.

Complex biological and physical transport processes are often described through systems of interacting particles. The effect of excluded volume on these transport processes has been well studied; however, the interplay between volume exclusion and reactions between heterogenous particles is less well studied. In this paper we develop a framework for modeling reaction-diffusion processes which directly incorporates volume exclusion. We consider simple reactions (unimolecular and bimolecular) that conserve the total number of particles. From an off-lattice microscopic individual-based model we use the Fokker-Planck equation and the method of matched asymptotic expansions to derive a low-dimensional macroscopic system of nonlinear partial differential equations describing the evolution of the particles. A biologically motivated, hybrid model of chemotaxis with volume exclusion is explored, where reactions occur at rates dependent upon the chemotactic environment. Further, we show that for reactions that require particle contact the appropriate reaction term in the macroscopic model is of lower order in the asymptotic expansion than the nonlinear diffusion term. However, we find that the next reaction term in the expansion is needed to ensure good agreement with simulations of the microscopic model. Our macroscopic model allows for more direct parametrization to experimental data than existing models.

Stability estimates for systems with small cross-diffusion

We discuss the analysis and stability of a family of cross-diffusion boundary value problems with nonlinear diffusion and drift terms. We assume that these systems are close, in a suitable sense, to a set of decoupled and linear problems. We focus on stability estimates, that is, continuous dependence of solutions with respect to the nonlinearities in the diffusion and in the drift terms. We establish well-posedness and stability estimates in an appropriate Banach space. Under additional assumptions we show that these estimates are time independent. These results apply to several problems from mathematical biology; they allow comparisons between the solutions of different models a priori. For specific cell motility models from the literature, we illustrate the limit of the stability estimates we have derived numerically, and we document the behaviour of the solutions for extremal values of the parameters.

New criteria for stochastic suppression and stabilization of hybrid functional differential systems

SummaryThis paper establishes new criteria for stochastic suppression and stabilization of hybrid functional differential systems with general 1‐sided polynomial growth condition. For an unstable nonlinear hybrid functional differential system with general 1‐sided polynomial growth condition, 2 independent Brownian noise processes are used to perturb the system into the stochastic hybrid differential system. Theoretical analysis shows that one of the nonlinear diffusion terms may suppress the explosive solution of deterministic system, and the other one can make the perturbed hybrid system almost surely stable with general decay rate.

Stochastic suppression and stabilisation of non‐linear hybrid delay systems with general one‐sided polynomial growth condition and decay rate

This study investigates the stochastic suppression and stabilisation of non-linear hybrid delay systems with general one-sided polynomial growth condition and decay rate. Given an unstable non-linear hybrid delay system x ˙ ( t ) = f ( x ( t ) , x ( t − τ ( t ) ) , r ( t ) , t ) with general one-sided polynomial growth condition, the authors introduce two independent Brownian noise and perturb this system into stochastic hybrid delay system d x ( t ) = f ( x ( t ) , x ( t − τ ( t ) ) , r ( t ) , t ) d t + h ( x ( t ) , r ( t ) , t ) d B 1 ( t ) + g ( x ( t ) , r ( t ) , t ) d B 2 ( t ) . It is shown that the non-linear diffusion term g ( x ( t ) , r ( t ) , t ) may suppress the potential explosion of hybrid delay system. Under a stronger condition, another linear diffusion term h ( x ( t ) , r ( t ) , t ) will make the perturbed stochastic hybrid delay system is almost surely stable with general decay rate.

Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.

Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise

In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.

Uncoupled PID Control of Multi-Agent Nonlinear Uncertain Stochastic Systems

This paper proposes a PID (proportional-integral-derivative) control design method to solve the regulation problem for a class of coupled multi-agent nonlinear uncertain stochastic systems, where each agent only has access to its own regulation error without communicating with others. A three-dimensional manifold will be constructed based on the information about the Lipschitz constants of both the unknown nonlinear drift and diffusion terms, such that the three parameters of each agent’s PID controller can be chosen arbitrarily from this common manifold. It will be shown that such an uncoupled PID design method can globally stabilize the whole nonlinear uncertain stochastic multi-agent system with the regulation error of each agent approaching to zero asymptotically.

Probabilistic Representation for Solutions to Nonlinear Fokker--Planck Equations

One obtains a probabilistic representation for the entropic generalized solutions to a nonlinear Fokker--Planck equation in $\mathbb{R}^d$ with a multivalued nonlinear diffusion term as density probabilities of solutions to a nonlinear stochastic differential equation. The case of a nonlinear Fokker--Planck equation with linear space dependent drift is also studied.

Financial market dynamics: superdiffusive or not?

The behavior of stock market returns over a period of 1–60 d has been investigated for S&P 500 and Nasdaq within the framework of nonextensive Tsallis statistics. Even for such long terms, the distributions of the returns are non-Gaussian. They have fat tails indicating that the stock returns do not follow a random walk model. In this work, a good fit to a Tsallis q-Gaussian distribution is obtained for the distributions of all the returns using the method of Maximum Likelihood Estimate. For all the regions of data considered, the values of the scaling parameter q, estimated from 1 d returns, lie in the range 1.4–1.65. The estimated inverse mean square deviations (beta) show a power law behavior in time with exponent values between −0.91 and −1.1 indicating normal to mildly subdiffusive behavior. Quite often, the dynamics of market return distributions is modelled by a Fokker–Plank (FP) equation either with a linear drift and a nonlinear diffusion term or with just a nonlinear diffusion term. Both of these cases support a q-Gaussian distribution as a solution. The distributions obtained from current estimated parameters are compared with the solutions of the FP equations. For negligible drift term, the inverse mean square deviations (betaFP) from the FP model follow a power law with exponent values between −1.25 and −1.48 indicating superdiffusion. When the drift term is non-negligible, the corresponding betaFP do not follow a power law and become stationary after certain characteristic times that depend on the values of the drift parameter and q. Neither of these behaviors is supported by the results of the empirical fit.

Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux

In this paper, we investigate the 3D Keller–Segel–Stokes (K–S–S) system with nonlinear diffusion term \(\Delta n^{m}\) (\(m>0\)) and rotational flux posed in a bounded domain \(\Omega \) with smooth boundary. Under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity S(x, n, c) satisfies \(|S(x,n,c)|\le C_{S}(1+n)^{-\alpha }\), by seeking some new functionals and using the bootstrap arguments on the regularized system, we establish the existence and boundedness of global weak solutions to K–S–S system for arbitrarily large initial data under the assumption \(m+2\alpha >2\) and \(m>\frac{3}{4}\), which includes both the degenerate \((m>1)\) and the singular \((m<1)\) case.