Nonlinear Diffusion System Research Articles

On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions

We study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays a significant part in analyzing the qualitative characteristics of nonlinear models of reaction-diffusion, heat transfer, filtration, and other physical, chemical, and biological processes. In the global solvability case, the key components of the asymptotic solutions are obtained. Iterative methods, which quickly converge to the exact solution while maintaining the qualitative characteristics of the nonlinear processes under study, are known to require the presence of an appropriate initial approximation. This presents a significant challenge for the numerical solution of nonlinear problems. A successful selection of initial approximations allows for the resolution of this challenge, which depends on the value of the numerical parameters of the equation, which are primarily in the computations recommended using an asymptotic formula. Using the asymptotics of self-similar solutions as the initial approximation for the iterative process, numerical calculations and analysis of the results are carried out. The outcomes of numerical experiments demonstrate that the results are in excellent accord with the physics of the process under consideration of the nonlinear diffusion system.

A mode of convergence arising in diffusive relaxation

Abstract In this work, a mode of convergence for measurable functions is introduced. A related notion of Cauchy sequence is given, and it is proved that this notion of convergence is complete in the sense that Cauchy sequences converge. Moreover, the preservation of convergence under composition is investigated. The origin of this mode of convergence lies in the path of proving that the density of a Euler system converges almost everywhere (up to subsequences) towards the density of a non-linear diffusion system, as a consequence of the convergence in the relaxation limit.

Nonlinear diffusion system for simultaneous restoration and binarization of degraded document images

Existing diffusion models can only do tasks for either restoration or binarization of degraded document images; in this paper, we pay close attention to the problem of simultaneous restoration and binarization. We first introduce a model of image formation for degraded document images, which decomposes a degraded document image into the product of background and foreground components. Then a weakly coupled nonlinear diffusion system is developed for simultaneous restoration and binarization of degraded document images. In this diffusion system, the first equation is used to estimate background component and the second equation is responsible for the foreground component. The restored and binarized images are finally obtained by the estimated background and foreground components, respectively. Two coupled diffusion equations are solved alternately by explicit finite differencing. The proposed model is tested on some degraded document images that represent five types of typical degradations including bleed-through, complex background, page stains, stamps and smeared ink, and is compared with eight closely related models for degraded document binarization, qualitatively and quantitatively. Experimental results illustrate that the proposed model has shown promising results in terms of restoration and binarization of degrade document images.

Image Enhancement Model Based on Fractional Time-Delay and Diffusion Tensor

Image enhancement is one of the bases of image processing technology, which can enhance useful features and suppress useless information of images according to the specified task. In order to ensure coherent enhancement for images with oriented flow-like structures, we propose a nonlinear diffusion system model based on time-fractional delay. By combining the nonlinear isotropic diffusion equation with fractional time-delay regularization, we construct a structure tensor. Meanwhile, the introduction of source terms enhances the contrast of the image, making it effective for denoising images with high-level noise. Based on compactness principles, the existence of weak solutions for the model is proved by using the Galerkin method. In addition, various experimental results verify the enhancement ability of the proposed model.

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

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.

A Legendre wavelet collocation method for 1D and 2D coupled time-fractional nonlinear diffusion system

A Legendre wavelet collocation method is proposed for solving a nonlinear coupled time fractional diffusion system. We have formulated a Riemann-Liouville fractional integral operator for Legendre wavelet (RLFIO-L) adopting the definition of Riemann-Liouville fractional integral operator combined with the Laplace transformation. Both the time and space variables are discretized in terms of the Legendre wavelet and RLFIO-L. The nonlinear coupled diffusion system is quasi-linearized by making use of the Newton's method. For theoretical concerns, the upper bound of error norm of the proposed method is estimated. Some numerical experiments are presented to authenticate the computational efficacy of the method.

Global existence in a two-dimensional nonlinear diffusion model for urban crime propagation

In this paper, we will investigate the urban crime system with the nonlinear diffusion ∂tu=∇⋅um−1∇u−χ∇⋅uv∇v−uv+B1(x,t),x∈Ω,t>0,∂tv=Δv+uv−v+B2(x,t),x∈Ω,t>0under no-flux initial–boundary conditions in a bounded convex domain Ω⊂R2 with smooth boundary. Our first result asserts that when m>32, χ∈(0,∞) and when 1<m≤32, χ∈0,32, the nonlinear diffusion system possesses locally bounded solutions for arbitrary initial data and sufficiently regular source terms B1 and B2. Our second result further reveals that the solutions can be demonstrated to be globally bounded if the source terms satisfy mild additional conditions. In particular, this improves the related result for m>32 (Rodríguez and Winkler, 2020) to the case of arbitrary m>1 under the additional assumption on χ.

Spatio-temporal dynamic solitary wave solutions and diffusion effects to the nonlinear diffusive predator-prey system and the diffusion-reaction equations

The coupled parabolic nonlinear evolution equations that describe the temporal-spatio dynamics of predator-prey systems and the nonlinear diffusion-reaction equations with diffusion effects have broad applications in biology, population dynamics, ecology, and many other fields where nonlinearity and diffusivity are key features of wave evolution. As a result, exact traveling wave solutions of such models are highly effective in numerical and analytical theories. In this article, we find the soliton solutions of the above stated biological models by contriving the enhanced modified simple equation method which is related to the exponential, hyperbolic, and trigonometric functions by balancing the highest order nonlinearity and diffusivity associated with biological groups. Graphical representations of some of the obtained solutions in three- and two-dimensional layouts are provided to estimate their behavior. The results revealed that the stated method is categorical, robust, and efficient in finding exact solutions to a variety of nonlinear evolution equations.

An anisotropic diffusion system with nonlinear time-delay structure tensor for image enhancement and segmentation

Anisotropic diffusion model has been established as one of the most significant image processing techniques, because it not only allows for space-variant filtering, but also realizes real direction-specific anisotropic smoothing of image structures. In this paper, an anisotropic nonlinear diffusion system is proposed for image enhancement and segmentation by taking into account the structure tensor constructed from the time-delay regularization and a nonlinear isotropic diffusion equation. Meanwhile, the diffusion system makes full use of the source term which increases the contrast and makes the image be less susceptible to noise, even in high level noise images. The source term also causes the restored image to tend to the binary image, which has proven to be an effective strategy for image segmentation. We analyze the proposed model in the view of the existence and uniqueness of the weak solution by using Galerkin's method. Some other theoretical analysis such as the maximum property is also discussed in the paper. With the appropriate and adaptive parameters, the anisotropic diffusion system achieves a better trade-off between computation time and image processing task. Furthermore, various image experiments illustrate that the proposed algorithm achieves much better efficiency and universality.

Globally unsolvability of integro-differential diffusion equation and system with exponential nonlinearities

In this paper, the Cauchy problem for an integro-differential diffusion equation and a system with nonlocal nonlinear sources are considered. The results on the existence of local integral solutions and the nonexistence of global weak solutions to the nonlinear integro-differential diffusion equation and system are presented.

Tamed-Euler method for nonlinear switching diffusion systems with locally Hölder diffusion coefficients

It is widely known that stochastic differential equations with Markovian switching, involving terms without Lipschitz continuity like |u|1/2+α for α∈[0,1/2), are of great practical value in many fields such as finance and biology. In this paper, we develop the tamed Euler-Maruyama schemes for switching diffusion systems modulated by a Markov chain, under the circumstances that drift coefficient satisfies the locally Lipschitz condition and diffusion coefficient satisfies the locally Hölder continuous condition. Moreover, we obtain the rate of convergence of the numerical algorithm not only at time T but also over the time interval [0,T]. Finally we give the numerical experiments to illustrate the theoretical results.

Opinion formation systems via deterministic particles approximation

We propose an ODE-based derivation for a generalized class of opinion formation models either for single and multiple species (followers, leaders, trolls). The approach is purely deterministic and the evolution of the single opinion is determined by the competition between two mechanisms: the opinion diffusion and the compromise process. Such deterministic approach allows to recover in the limit an aggregation/(nonlinear)diffusion system of PDEs for the macroscopic opinion densities. Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

Hydrodynamic limit for a chain with thermal and mechanical boundary forces

We prove the hydrodynamic limit for an harmonic chain with a random exchange of momentum that conserves the kinetic energy but not the momentum. The system is open and subject to two thermostats at the boundaries and to external tension. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a non-linear diffusive systems of conservative partial differential equations.

Noncooperative and Cooperative Strategy Designs for Nonlinear Stochastic Jump Diffusion Systems With External Disturbance: T–S Fuzzy Approach

In this article, we consider the multiplayer $H_{\infty }$ noncooperative and cooperative game strategy designs for a class of stochastic jump diffusion systems with external disturbance. To attenuate the effect from the competitive strategies of other players and unpredictable external disturbance on the desired target tracking performance of each player, a multiplayer $H_{\infty }$ noncooperative game strategy design problem is proposed and formulated as an equivalent multiobjective optimization problem (MOP) with a Nash equilibrium solution. Also, the multiplayer $H_{\infty }$ cooperative game strategy design problem is discussed and formulated as an equivalent single-objective optimization problem. To overcome the difficulties in solving multiple Hamilton–Jacobi–Issacs inequalities (HJIIs) for noncooperative and cooperative game strategy designs, the Takagi–Sugeno fuzzy model is introduced to approximate the nonlinear stochastic system and HJII could be transformed into a set of linear matrix inequalities (LMIs). Besides, an LMI-constrained multiobjective evolution algorithm is developed to efficiently solve the MOP of a noncooperative multiplayer $H_{\infty }$ stochastic game strategy design problem. A financial market with multiple investors is provided as a simulation example to demonstrate the effectiveness of the proposed noncooperative and cooperative game investment strategies.

Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems

We consider a Cauchy semilinear problem for a time-fractional diffusion system∂αu∂tα+Au=F(u,v),∂αv∂tα+Bv=G(u,v),which involves symmetric uniformly elliptic operators A,B on a bounded domain Ω in Rd with sufficiently smooth boundary. The problem is equipped with final value conditions (FVCs), i.e., (u;v)|t=T are given. We derive a spectral representation of solutions with FVCs where the solution operators are not bounded on L2(Ω). Our work focuses on establishing existence and uniqueness of a solution in a suitable space.

Galerkin finite element schemes with fractional Crank–Nicolson method for the coupled time-fractional nonlinear diffusion system

This paper deals with two fractional Crank–Nicolson–Galerkin finite element schemes for coupled time-fractional nonlinear diffusion system. The first scheme is iterative and is based on Newton’s method, while the other one is a linearized scheme. Existence-uniqueness results of the fully discrete solution for both schemes are discussed. In addition, a priori bounds and a priori error estimates are derived for proposed schemes using a new discrete fractional Gronwall-type inequality. Both the schemes yield $$O({\varDelta } t^2)$$ accuracy in time and hence, superior to $$O({\varDelta } t^{2-\alpha })$$ accurate L1 scheme existing in the literature. Moreover, three different numerical examples are provided to illustrate the theoretical estimates .

Finite element analysis for coupled time-fractional nonlinear diffusion system

This paper is concerned with the L1-Galerkin finite element analysis of the coupled time-fractional nonlinear diffusion system. Existence and uniqueness results are discussed at continuous as well as discrete levels. Two different methods for linearizing the nonlinear fully discrete problems are used. We provide a priori bounds and convergence estimates in L2(Ω) norm for fully-discrete problems. Numerical results based on the presented finite element methods are provided to validate the theoretical estimates.

Information Diffusion Nonlinear Dynamics Modeling and Evolution Analysis in Online Social Network Based on Emergency Events

A nonlinear dynamic emergency public event information diffusion system and mathematical model for public events are proposed based on the propagation dynamics. First, public emergency information communication is analyzed and designed, the opinion evolution is divided into and the viewpoint of value, influence, interest, conformity, intimacy five factors of information diffusion for public emergencies; second, the dynamic diffusion network is designed; information propagation mathematical public emergencies the final model is constructed; the proposed model and social reality and the true statistics of empirical experiments are simulated and analyzed. The experimental results show that the model and the real emergency public event propagation process are consistent, the development trend can predict public events, and the proposed mathematical model is reasonable and effective.