Nonlinear Diffusion System Research Articles

Exact Solutions of Nonlinear Diffusive Predator-Prey System by New Extension of Tanh Method

Exact Solutions of Nonlinear Diffusive Predator-Prey System by New Extension of Tanh Method

A Nonlinear Coupled Diffusion System for Image Despeckling and Application to Ultrasound Images

Despite extensive availability of filters for denoising, speckle noise suppression remains a challenging task. Speckle noise also hinders tasks such as efficient extraction of features, recognition, analysis, detection of edges, etc. Therefore, in this paper, motivated by the impressive performance of time delay regularization in additive noise removal, we develop a class of nonlinear diffusion-based coupled partial differential equation models for multiplicative noise removal. This denoising framework considers separate partial differential equations to handle diffusion function as well as fidelity term. By using a maximum a posteriori regularization approach, we can derive an energy functional and study the associated evolution problem which corresponds to the denoised image we want to recover. We then evaluate the effectiveness of our model with several standard test images and real ultrasound images. Qualitative and quantitative studies confirm that the proposed model is robust in comparison with state-of-the-art approaches. The denoised images have appealing visual characteristics with different levels of noise and textures while preserving the important details of the original image.

Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons

Focusing on hybrid diffusion dynamics involving continuous dynamics as well as discrete events, this article investigates the explicit approximations for nonlinear switching diffusion systems modulated by a Markov chain. Different kinds of easily implementable explicit schemes have been proposed to approximate the dynamical behaviors of switching diffusion systems with locally Lipschitz continuous drift and diffusion coefficients in both finite and infinite intervals. Without additional restriction conditions except those which guarantee the exact solutions possess their dynamical properties, the numerical solutions converge strongly to the exact solutions in finite horizon, moreover, realize the approximation of long-time dynamical properties including the moment boundedness, stability and ergodicity. Some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach.

An Implicit Unified Gas Kinetic Scheme for Radiative Transfer with Equilibrium and Non-Equilibrium Diffusive Limits

AbstractThis paper is about the construction of a unified gas-kinetic scheme (UGKS) for a coupled system of radiative transport and material heat conduction with different diffusive limits. Different from the previous approach, instead of including absorption/emission only, the current method takes both scattering and absorption/emission mechanism into account in the radiative transport process. As a result, two asymptotic limiting solutions will appear in the diffusive regime. In the strong absorption/emission case, an equilibrium diffusion limit is obtained, where the system is mainly driven by a nonlinear diffusion equation for the equilibrium radiation and material temperature. However, in the strong scattering case, a non-equilibrium limit can be obtained, where coupled nonlinear diffusion system with different radiation and material temperature is obtained. In addition to including the scattering term in the transport equation, an implicit UGKS (IUGKS) will be developed in this paper as well. In the IUGKS, the numerical flux for the radiation intensity is constructed implicitly. Therefore, the conventional CFL constraint for the time step is released. With the use of a large time step for the radiative transport, it becomes possible to couple the IUGKS with the gas dynamic equations to develop an efficient numerical method for radiative hydrodynamics. The IUGKS is a valid method for all radiative transfer regimes. A few numerical examples will be presented to validate the current implicit method for both optical thin to optical thick cases.

Study on Bird Flu Infection Process within a Poultry Farm with Effects of Spatial Diffusion

We describe an ODE model and a PDE model. A nonlinear diffusion system of partial differential equations is analyzed for investigation of bird flu transmission process within a poultry farm. Effect of threshold number that gives measure of infectious intensity is investigated. It depends on three factors: removal of infected birds, vaccination, and capacity of the farm. Our analysis shows that the transmission of bird flu within a poultry farm is expressible in terms of traveling wave solutions. It also shows a spatial significance of diffusivity.

Observer design for a nonlinear diffusion system based on the Kirchhoff transformation

This paper deals with the state estimation for a nonlinear diffusion system. An observer that reconstructs the whole state, from the available measurements, is proposed based on an equivalent linear diffusion model obtained using the Kirchhoff tangent transformation. This bijective mapping allows to apply the available and powerful state estimation theory of linear distributed parameter systems and simplifies the observer design. Hence, an observer can be designed for the obtained equivalent linear diffusion system and by using the Kirchhoff transformation, the whole state of the original nonlinear diffusion system is recovered. The observability analysis of the nonlinear diffusion system and the convergence of the proposed observer are also investigated based on the equivalent linear diffusion system. The effectiveness of the proposed observer is shown, through numerical simulation runs, in the case of a heated steel rod by considering both an uniformly distributed and a punctual boundary sensing.

Existence and uniqueness for a class of nonlinear population diffusion system

ABSTRACTIn this paper, we introduce a class of nonlinear population diffusion system. Existence and uniqueness of strong solution are proved for stochastic age-dependent population diffusion system in Hilbert space by use of Gronwall's lemma and Burkholder–Davis–Gundy's inequality.

A tightly-coupled domain-decomposition approach for highly nonlinear stochastic multiphysics systems

Multiphysics simulations often involve nonlinear components that are driven by internally generated or externally imposed random fluctuations. When used with a domain-decomposition (DD) algorithm, such components have to be coupled in a way that both accurately propagates the noise between the subdomains and lends itself to a stable and cost-effective temporal integration. We develop a conservative DD approach in which tight coupling is obtained by using a Jacobian-free Newton–Krylov (JfNK) method with a generalized minimum residual iterative linear solver. This strategy is tested on a coupled nonlinear diffusion system forced by a truncated Gaussian noise at the boundary. Enforcement of path-wise continuity of the state variable and its flux, as opposed to continuity in the mean, at interfaces between subdomains enables the DD algorithm to correctly propagate boundary fluctuations throughout the computational domain. Reliance on a single Newton iteration (explicit coupling), rather than on the fully converged JfNK (implicit) coupling, may increase the solution error by an order of magnitude. Increase in communication frequency between the DD components reduces the explicit coupling's error, but makes it less efficient than the implicit coupling at comparable error levels for all noise strengths considered. Finally, the DD algorithm with the implicit JfNK coupling resolves temporally-correlated fluctuations of the boundary noise when the correlation time of the latter exceeds some multiple of an appropriately defined characteristic diffusion time.

Exact solutions and maximal dimension of invariant subspaces of time fractional coupled nonlinear partial differential equations

We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter–Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman–Broer–Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.

Global existence and nonlinear diffusion of classical solutions to non-isentropic Euler equations with damping in bounded domain

We consider classical solutions to the initial boundary value problem for the non-isentropic compressible Euler equations with damping in several space dimensions. We establish a global existence theory for classical solutions to this system when the initial data are sufficiently small. We prove that the pressure and the velocity converge exponentially toward some constants, while the entropy and the density in general do not approach constants. Finally, we prove that, as the time goes to infinity, the pressure and the velocity components of the non-isentropic Euler equations with damping converge exponentially toward those of an associated nonlinear diffusion system, and the global existence of classical solutions to these nonlinear diffusion equations is also established.

Blocked Particle Gibbs Schemes for High Dimensional Interacting Systems

This paper examines the use of blocking strategies for Particle Gibbs sampling schemes for high dimensional latent state space models with interacting components. Such strategies are particularly advantageous for high-dimensional systems because they allow multiple lower-dimensional blocks to be sampled, avoiding the of within the particle filter. This paper presents algorithms for blocked Particle Gibbs sampling, as well as examining several special cases. Analogies to the bootstrap particle filter are given, along with an optimal proposal in the case of Gaussian systems with linear Gaussian observations. The paper demonstrates blocking schemes on realistic example applications: tracking multiple interacting targets and data assimilation for a nonlinear diffusion system. The effect of block size and number of particles used on computational efficiency is also examined through experiments, which demonstrate that the optimal blocking strategy is both problem and algorithm dependent and results from a compromise between the improved efficiency of blocking in Gibbs schemes and the curse of dimensionality in the particle filter.

High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

The main purpose of the paper is to show how to use implicit---explicit Runge---Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a large class of problems. This approach gives a great flexibility, and allows, in many cases the construction of simple linearly implicit schemes without any Newton's iteration. This is obtained by identifying the (possibly linear) dependence on the unknown of the system which generates the stiffness. Only the stiff dependence is treated implicitly, then making the whole method much simpler than fully implicit ones. The resulting schemes are denoted as semi-implicit R---K. We adopt several semi-implicit R---K methods up to order three. We illustrate the effectiveness of the new approach with many applications to reaction---diffusion, convection diffusion and nonlinear diffusion system of equations.

Periodic solutions in reaction–diffusion equations with time delay

Spatial diffusion and time delay are two main factors in biological and chemical systems. However, the combined effects of them on diffusion systems are not well studied. As a result, we investigate a nonlinear diffusion system with delay and obtain the existence of the periodic solutions using coincidence degree theory. Moreover, two numerical examples confirm our theoretical results. The obtained results can also be applied in other related fields.

Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems

The equivalence relation between conditional Lie–Bäcklund symmetry, higher-order differential constraint and direct reduction is presented in the case of evolution systems. The second-order nonlinear conditional Lie–Bäcklund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional Lie–Bäcklund symmetry of evolution systems. Consequently, the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems.

Controllability of nonlinear diffusion system

AbstractIn this article, the semi‐group theory is exploited to study the controllability of a particular but important class of distributed parameter systems with a uniform distributed control variable. This class is described by a partial differential equation that characterizes the nonlinear diffusion phenomenon. The main idea consists of using the Cole–Hopf tangent transformation, under some assumptions, in order to derive an equivalent linear model of the nonlinear diffusion system. Then, the study of the controllability of the linear model, based on the semi‐group approach, allows us to conclude about the controllability of the nonlinear diffusion system. The study is illustrated by an application example.

Existence and uniqueness of an attractive nonlinear diffusion system

We establish the existence and uniqueness of an attractive fractional coupled system. Such a system has applications in biological populations of cells. We confirm that the fractional system under consideration admits a global solution in the Sobolev space. The solution is shown to be unique. The technique is founded on analytic method of the fixed point theory and the fractional differential operator is scrutinized from the view of the Riemann–Liouville differential operator. Finally, we illustrate some entropy fractional differential inequalities regarding the solution of the system.

A feedback control method for the stabilization of a nonlinear diffusion system on a graph

In this paper, we consider the internal stabilization problems of FitzHugh–Nagumo (FHN) systems on the locally finite connected weighted graphs, which describe the process of signal transmission across axons in neurobiology. We will establish the proper condition on the weighted Dirichlet–Laplace operator on a graph such that the nonlinear FHN system can be stabilized exponentially and globally only using internal actuation over a sub-domain with a linear feedback form.

Optimal high-order diagonally-implicit Runge–Kutta schemes for nonlinear diffusive systems on atmospheric boundary layer

Nonlinear diffusion equations are extensively applicable in diverse fields of science and engineering. Numerical stability is a common concern in this class of equations. In the present study, a three-stage third-order diagonally-implicit Runge–Kutta (DIRK) scheme is introduced by optimizing the error and linear stability analysis for a commonly used nonlinear diffusive system in atmospheric boundary layer. The proposed scheme is stable for a wide range of time steps and able to resolve different diffusive systems with diagnostic turbulence closures, or prognostic ones with a diagnostic length scale, with enhanced accuracy and stability compared to current schemes. It maintains A-stability, which makes it appropriate for the solution of stiff problems. The procedure implemented in this study is quite general and can be used in other diffusive systems as well.

Boundary Geometric Control of Nonlinear Diffusion Systems

The paper addresses the boundary control of a nonlinear diffusion system submitted to Neumann actuation. The control law is designed in the framework of geometric control theory using directly the nonlinear partial differential equation model without any previous reduction. First, an equivalent linear model using the Cole-Hopf transformation is obtained, then the manipulated variable is inserted in the state equation of this equivalent linear model by means of a Dirac delta function to make the boundary condition homogeneous. Based on the resulting final model, the control law is derived using the characteristic index notion and the closed loop stability is demonstrated using concepts from the powerful semigroup theory. The control law performance is evaluated through numerical simulation by considering a nonlinear heat conduction control problem.

The simultaneous and non-simultaneous blow-up criteria for a diffusion system

This paper investigates the finite time blow-up of nonnegative solutions for a nonlinear diffusion system with a more complicated source term, which is a product of localized source, local source, and weight function, and complemented by homogeneous Dirichlet boundary conditions. The criteria are proposed to identify simultaneous and non-simultaneous blow-up solutions. Moreover, the related classification for the four parameters in the model is optimal and complete. The results extend those in Zhang and Yang [12].