Diffusion–substitution based gray image encryption scheme

In this paper, an encryption algorithm for gray images using a secret key of 128-bits size is proposed. Initially, visual quality of image is degraded by the mixing process. Resultant image is partitioned into key dependent dynamic blocks and, further, these blocks are passed through key dependent diffusion and substitution processes. Total sixteen rounds are used in the encryption algorithm. Proposed technique is simple to implement and has high encryption rate. Simulation experiment results have been given to validate the high security features and effectiveness of proposed system.     

Experimental Reaction–Diffusion Chemical Processors for Robot Path Planning

In this paper we discuss the experimental implementation of a chemical reaction–diffusion processor for robot motion planning in terms of finding the shortest collision-free path for a robot moving in an arena with obstacles. These reaction–diffusion chemical processors for robot navigation are not designed to compete with existing silicon-based controllers. These controllers are intended for the incorporation into future generations of soft-bodied robots built of electro- and chemo-active polymers. In this paper we consider the notion of processing as being implicit in the physical medium constituting the body of a soft robot. This work therefore represents some early steps in the employment of excitable media controllers. An image of the arena in which the robot is to navigate is mapped onto a thin-layer chemical medium using a method that allows obstacles to be represented as local changes in the reactant concentrations. Disturbances created by the objects generate diffusive and phase wave fronts. The spreading waves approximate to a repulsive field generated by the obstacles. This repulsive field is then inputted into a discrete model of an excitable reaction–diffusion medium, which computes a tree of shortest paths leading to a selected destination point. Two types of chemical processors are discussed: a disposable palladium processor, which executes arena mapping from a configuration of obstacles, given before an experiment and, a reusable Belousov–Zhabotinsky processor which allows for online path planning and adaptation for dynamically changing configurations of obstacles.     

Non-unified Compact Residual-Distribution Methods for Scalar Advection–Diffusion Problems

This paper solves the advection–diffusion equation by treating both advection and diffusion residuals in a separate (non-unified) manner. An alternative residual distribution (RD) method combined with the Galerkin method is proposed to solve the advection–diffusion problem. This Flux-Difference RD method maintains a compact-stencil and the whole process of solving advection–diffusion does not require additional equations to be solved. A general mathematical analysis reveals that the new RD method is linearity preserving on arbitrary grids for the steady-state advection–diffusion equation. The numerical results show that the flux difference RD method preserves second-order accuracy on various unstructured grids including highly randomized anisotropic grids on both the linear and nonlinear scalar advection–diffusion cases.     

Adaptive Stabilization of a Class of Reaction–Diffusion Systems

This paper is concerned with adaptive stabilization of a class of reaction–diffusion systems governed by a nonlinear partial differential equation of the first order in time but the fourth order in space. In the presence of bounded deterministic disturbances, the adaptive stabilizer is constructed by the concept of high-gain nonlinear output feedback and the estimation mechanism of the unknown parameters. In the control system the global asymptotic stability and the convergence of the system state to zero will be guaranteed.     

WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation

In this paper, a new numerical approximation is discussed for the two-dimensional distributed-order time fractional reaction–diffusion equation. Combining with the idea of weighted and shifted Grünwald difference (WSGD) approximation (Tian et al. in Math Comput 84:1703–1727, 2015; Wang and Vong in J Comput Phys 277:1–15, 2014) in time, we establish orthogonal spline collocation (OSC) method in space. A detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order \(\mathscr {O}(\tau ^2+\Delta \alpha ^2+h^{r+1})\), where \(\tau , \Delta \alpha , h\) and r are, respectively the time step size, step size in distributed-order variable, space step size, and polynomial degree of space. Interestingly, we prove that the proposed WSGD-OSC scheme converges with the second-order in time, where OSC schemes proposed previously (Fairweather et al. in J Sci Comput 65:1217–1239, 2015; Yang et al. in J Comput Phys 256:824–837, 2014) can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Some numerical results are also given to confirm our theoretical prediction.     

A Robust Residual-Type a Posteriori Error Estimator for Convection–Diffusion Equations

In this paper, a new robust residual type a posteriori error estimator is developed and analyzed for convection–diffusion equations. A novel dual norm is introduced, under which the error estimator is proved to be robust with respect to the singularly perturbed parameter \(\varepsilon \). Both theoretical and numerical results showed that the estimator performs better than the existing ones in literature.     

Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems

We give a survey on recent developments of stabilization methods based on local projection type. The considered class of problems covers scalar convection–diffusion equations, the Stokes problem and the linearized Navier–Stokes equations. A new link of local projection to the streamline diffusion method is shown. Numerical tests for different type of boundary layers arising in convection–diffusion problems illustrate the stabilizing properties of the method.     

$${\varvec{}}$$th Moment Exonential Stability of Hybrid Delayed Reaction–Diffusion Cohen–Grossberg Neural Networks

In this paper, we propose hybrid reaction–diffusion Cohen–Grossberg neural networks (RDCGNNs) with variable coefficients and mixed time delays. By using the Lyapunov–Krasovkii functional approach, stochastic analysis technique and Hardy inequality, some novel sufficient conditions are derived to ensure the pth moment exponential stability of hybrid RDCGNNs with mixed time delays. The obtained sufficient conditions are relevant to the diffusion terms. The results of this paper are novel and improve some of the previously known results. Finally, two numerical examples are provided to verify the usefulness of the obtained results.     

An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.     

Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction–diffusion systems (RDSs) on surfaces in \({\mathbb {R}}^3\) that evolve under a given velocity field. A fully-discrete method based on the implicit–explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction–diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM–IMEX Euler discretisation of a RDS on a logistically growing surface.     

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grünwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from \({\mathcal {O}}(N^2)\) to \({\mathcal {O}}(N)\) and the computational complexity from \({\mathcal {O}}(N^3)\) to \({\mathcal {O}}(N\log N)\) in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.     

Passivity and Pinning Passivity of Coupled Delayed Reaction–Diffusion Neural Networks with Dirichlet Boundary Conditions

This paper respectively considers passivity problem and pinning passivity problem for coupled delayed reaction–diffusion neural networks (CDRDNNs). By construction of appropriate Lyapunov functionals and utilization of inequality techniques, several passivity conditions are derived for the CDRDNNs. Moreover, the pinning control technique is developed to obtain some passivity criteria for CDRDNNs. Finally, two numerical examples are also provided to verify the correctness of the theoretical results.     

Existence,Uniqueness and Stability of Mild Solutions to a Stochastic Nonlocal Delayed Reaction–Diffusion Equation

Neural Processing Letters - The aim of this paper is to investigate the existence, uniqueness and stability of mild solutions to a stochastic delayed reaction–diffusion equation with spatial...     

Robust Exponential Synchronization for Stochastic Delayed Neural Networks with Reaction–Diffusion Terms and Markovian Jumping Parameters

This paper investigates robust exponential synchronization for stochastic delayed neural networks with reaction–diffusion terms and Markovian jumping parameters driven by infinite dimensional Wiener processes. The novelty of this paper lives in the use of a new Lyapunov–Krasovskii functional and Poincaré inequality to present some criteria for robust exponential synchronization in terms of linear matrix inequalities (LMIs) and matrix measure under Robin boundary conditions. Finally, two numerical examples are provided to illustrate the effectiveness of the easily verifiable synchronization LMIs in MATLAB toolbox.     

Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.