Stochastic model for ultraslow diffusion

Ultraslow diffusion is a physical model in which a plume of diffusing particles spreads at a logarithmic rate. Governing partial differential equations for ultraslow diffusion involve fractional time derivatives whose order is distributed over the interval from zero to one. This paper develops the stochastic foundations for ultraslow diffusion based on random walks with a random waiting time between jumps whose probability tail falls off at a logarithmic rate. Scaling limits of these random walks are subordinated random processes whose density functions solve the ultraslow diffusion equation. Along the way, we also show that the density function of any stable subordinator solves an integral equation (5.15) that can be used to efficiently compute this function.     

A population model with nonlinear diffusion

A model is presented for a single species population moving in a limited one-dimensional environment. The birth-death process is specialized by assuming a constant death modulus and a birth modulus which is an exponential in the age. The diffusion mechanism is nonlinear and results in a problem for the space population density which has a degenerate parabolic form and is similarly to the porous media equation. It is shown that the effect of the nonlinearity in the diffusion is to produce an approach to steady state even when the process is birth dominant. The interaction of the birth-death and diffusion processes is studied and is shown to yield a modified birth-death mechanism which is both time and space dependent.     

Coexistence solutions of a periodic competition model with nonlinear diffusion

This paper is concerned with a spatially heterogeneous Lotka–Volterra competition model with nonlinear diffusion and nonlocal terms, under the Dirichlet boundary condition. Based on the theory of Leray–Schauder’s degree, we give sufficient conditions to assure the existence of coexistence periodic solutions, which extends some results of G. Fragnelli et al.     

Finite element approximations of a nonlinear diffusion model with memory

The convergence of a finite element scheme approximating a nonlinear system of integro-differential equations is proven. This system arises in mathematical modeling of the process of a magnetic field penetrating into a substance. Properties of existence, uniqueness and asymptotic behavior of the solutions are briefly described. The decay of the numerical solution is compared with both the theoretical and finite difference results.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

Optimal investment and premium control in a nonlinear diffusion model

This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases: (i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.     

A nonlinear diffusion model for image restoration

In this paper, we study the ill posed Perona-Malik equation of image processing^[14] and the regularized P-M model i.e. C-model proposed by Catte et al.^[4]. The authors present the convex compound of these two models in the form of the system of partial differential equations. The weak solution for the equations is proved in detail. The additive operator splitting (AOS) algorithm for the proposed model is also given. Finally, we show some numeric experimental results on images.     

Galerkin methods for a model of population dynamics with nonlinear diffusion

A numerical method is proposed to approximate the solution of a nonlinear and nonlocal system of integro-differential equations describing age-dependent population dynamics with spatial diffusion. We use a finite difference method along the characteristic age-time direction combined with finite elements in the spatial variable. Optimal order error estimates are derived for this approximation. © 1996 John Wiley & Sons, Inc.     

A nonlinear age-dependent model with spatial diffusion

The main goal of this paper is the study of the existence and uniqueness of a positive solution for a nonlinear age-dependent equation with spatial diffusion. For that, we mainly use the properties of an eigenvalue problem related to the equation and the sub-supersolution method. We justify that this method works for this equation, in which there is a potential blow-up and a nonlocal initial condition.     

Stability analysis of a stochastic logistic model with nonlinear diffusion term

This paper presents an asymptotic analysis of a stochastic logistic population model with nonlinear diffusion term. The classical probability method is applied to obtain the criteria of asymptotic behavior for the considered model. The numerical simulations validate the efficiency of the theory analysis.     

A model for diffusion of mulucomponent information

The epidemic model of diffusion of news (or disease) is generalized to describe the diffusion of a multi‐component information. The multivaluedness of information in our model arises due to the large number (k) of constituent components or items of the information in question. When the different components of information are assumed to bear no hierarchy, the master equation of the model contains an intractably large number of variables (2 ^k ). The dynamics of the model, however, displays some simplifying features, one of which is the conservation of homogeneity of distribution of population over the different information vectors (in the sense defined in the text). The homogenized version of the model is found to be numerically tractable. The growth curves for large k continue to display sigmoid shapes, but with large ‘saturation times’. The dependence of ‘saturation time’ (i.e. the time required for spread of all the information in almost the entire population) on various parameters of the model, for uniform initial distributions, is numerically investigated. The ‘saturation time’ is found to vary inversely with the intensity of interaction (ß) and the size of population (N), as expected. An important numerical feature that emerges is that the ‘saturation time’ seems to be in linear proportion to the number of information items (k).     

STABILITY OF INNOVATION DIFFUSION MODEL WITH NONLINEAR ACCEPTANCE

In this article,an innovation diffusion model with the nonlinear acceptance is proposed to describe the dynamics of three competing products in a market.It is proved that the model admits a unique positive equilibrium,which is globally stable by excluding the existence of periodic solutions and by using the theory of three dimensional competition systems.     

Finite element analysis for a nonlinear diffusion model in image processing

The optimal error estimate O(h^k+1) for a popular nonlinear diffusion model widely used in image processing is proved for the standard k^th-order (k ≥ 1) conforming tensor-product finite elements in the L²-norm. The optimal L²-estimate is obtained by the integral identity technique [1–3] without using the classic Nitsche duality argument [4].     

Positive steady states for a prey-predator model with some nonlinear diffusion terms

This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof.     

An electrochemistry model with nonlinear diffusion: steady-state solutions

We formulate and study the steady-state solutions for an electrochemistrymodel with nonlinear diffusion. We establish the existenceand uniqueness of solutions, prove the global convergence ofa successive iteration scheme, and use examples to illustratethe formation of vacuum regions, which is not possible in alinear diffusion model.     

Noisy image segmentation based on nonlinear diffusion equation model

This paper addresses the segmentation problem in noisy image based on nonlinear diffusion equation model and proposes a new adaptive segmentation model based on gray-level image segmentation model. This model also can be extended to the vector value image segmentation. By virtue of the prior information of regions and boundary of image, a framework is established to construct different segmentation models using different probability density functions. A segmentation model exploiting Gauss probability density function is given in this paper. An efficient and unconditional stable algorithm based on locally one-dimensional (LOD) scheme is developed and it is used to segment the gray image and the vector values image. Comparing with existing classical models, the proposed approach gives the best performance.     

Stochastic evolution of innovation diffusion in heterogeneous groups: study of life cycle patterns

A theoretical framework has been proposed to study patternsof innovation diffusion in a heterogeneous population, withapplicability to a number of problem areas including marketing.The heterogeneity in the population is captured through randomlyvarying parameters, which have been modelled in terms of two-pointdistributions. The effect of heterogeneity leads to the generationof bi-modal life cycle patterns besides the conventional uni-modalpattern resulting from S-shaped curve. The stochastic evolutionof the mean and variance of the number of adopters is foundto depict a high level of relative fluctuation around the pointof inflexion. As a result of randomness in parameters, the resultingdifferential equation for the evolution of the mean of the adoptionprocess is characterized by a non-autonomous system having parameterswhich are no longer constant but become time dependent. Fordemonstrating the effectiveness of the proposed framework, areal data set which depicts a bi-modal life cycle curve is investigated.The fit is found to be extremely good while capturing appropriateproduct life cycle curve.