Mathematics and Computers in Simulation最新文献

The non-fragile control issue of discrete-time conic-type nonlinear Markov jump systems under deception attacks has been investigated using an event-triggered method. The nonlinear terms satisfy the conic-type nonlinear constraint condition that lies in a known hypersphere with an uncertain center is employed. The deception attack may obstruct normal communication in an effort to obtain confidential information. In addition, a non-fragile event-triggered controller is suggested to further conserve communication resources. As a stochastic process, a deception attack is manageable by the established controller. Also, by choosing an appropriate Lyapunov-Krasovskii functional, a set of necessary conditions is found in terms of linear matrix inequalities (LMIs) that guarantee mean square stability of the discrete-time conic-type nonlinear Markov jump system in the presence of deception attacks. Finally, the proposed non-fragile event-triggered control techniques is validated with a DC-DC motor application system and another numerical example.

Semilinear hyperbolic stochastic partial differential equations (SPDEs) find widespread applications in the natural and engineering sciences. However, the traditional Gaussian setting may prove too restrictive, as phenomena in mathematical finance, porous media, and pollution models often exhibit noise of a different nature. To capture temporal discontinuities and accommodate heavy-tailed distributions, Hilbert space-valued Lévy processes or Lévy fields are employed as driving noise terms. The numerical discretization of such SPDEs presents several challenges. The low regularity of the solution in space and time leads to slow convergence rates and instability in space/time discretization schemes. Furthermore, the Lévy process can take values in an infinite-dimensional Hilbert space, necessitating projections onto finite-dimensional subspaces at each discrete time point. Additionally, unbiased sampling from the resulting Lévy field may not be feasible. In this study, we introduce a novel fully discrete approximation scheme that tackles these difficulties. Our main contribution is a discontinuous Galerkin scheme for spatial approximation, derived naturally from the weak formulation of the SPDE. We establish optimal convergence properties for this approach and combine it with a suitable time stepping scheme to prevent numerical oscillations. Furthermore, we approximate the driving noise process using truncated Karhunen-Loève expansions. This approximation yields a sum of scaled and uncorrelated one-dimensional Lévy processes, which can be simulated with controlled bias using Fourier inversion techniques.

This research focuses on devising a new fast difference scheme to simulate the Caputo tempered fractional derivative (TFD). We introduce a fast tempered ${}^{\lambda }F\pounds 2-1\sigma$ difference method featuring second-order precision for a tempered time fractional Burgers equation (TFBE) with tempered parameter $\lambda$ and fractional derivative of order $\alpha$ ($0<\alpha <1$). The model emerges in characterizing the propagation of waves in porous material with the power law kernel and exponential attenuation. To circumvent iteratively resolving the discretized algebraic system, we introduce a linearized difference operator for approximating the nonlinear terms appearing in the model. The second-order fast tempered scheme relies on the sum of exponents (SOE) technique. The method’s convergence and stability are analyzed theoretically, establishing unconditional stability and maintaining the accuracy of order $O({\tau }^2+h^2+ϵ)$, where $\tau$ denotes the temporal step size, $ϵ$ is the tolerance error and $h$ is the spatial step size. Moreover, a novel compact finite difference (CFD) scheme of high order is developed for tempered TFBE. We investigate the stability and convergence of this fourth-order compact scheme utilizing the energy method. Numerical simulations indicate convergence to $O({\tau }^2+h^4+ϵ)$ under robust regularity assumptions. Our computational results align with theoretical analysis, demonstrating good accuracy while reducing computational complexity and storage needs compared to the standard tempered ${}^{\lambda }\pounds 2-1\sigma$ scheme, with significant reduction in CPU time. Numerical outcomes showcase the competitive performance of the fast tempered ${}^{\lambda }F\pounds 2-1\sigma$ scheme relative to the standard ${}^{\lambda }\pounds 2-1\sigma$.

This paper investigates a robust anti-disturbance interval type-2 (IT2) fuzzy control for interconnected nonlinear partial differential equation (PDE) systems subject to parameter uncertainties by the conjunct observer. First, an IT2 fuzzy model is adopted to remodel the target system. Second, a state observer with mismatched premise variables is constructed to solve the problem that the original system and the observer do not share a uniform premise variable. Moreover, a disturbance observer is designed to estimate the unknown external disturbances, which can be modeled by exogenous PDE systems. Then, utilizing the conjunct observation information, an anti-disturbance IT2 fuzzy control strategy is proposed to attenuate the effect of disturbances on the system performance while ensuring that the closed-loop system is stable. Finally, simulation results verify the effectiveness of the proposed method.

This paper introduces an innovative model for infectious diseases in predator–prey populations. We not only prove the existence of global non-negative solutions but also establish essential criteria for the system’s decline and sustainability. Furthermore, we demonstrate the presence of a Borel invariant measure, adding a new dimension to our understanding of the system. To illustrate the practical implications of our findings, we present numerical results. With our model’s comprehensive approach, we aim to provide valuable insights into the dynamics of infectious diseases and their impact on predator–prey populations.

This paper proposes finite-difference schemes based on triangular stencils to approximate partial derivatives using bivariate Lagrange polynomials. We use first-order partial derivative approximations on triangles to introduce a novel hexagonal scheme for the second-order partial derivative on any rotated parallelogram grid. Numerical analysis of the local truncation errors shows that first-order partial derivative approximations depend strongly on the triangle vertices getting at least a first-order method. On the other hand, we prove that the proposed hexagonal scheme is always second-order accurate. Simulations performed at different triangular configurations reveal that numerical errors agree with our theoretical results. Results demonstrate that the proposed method is second-order accurate for the Poisson and Helmholtz equation. Furthermore, this paper shows that the hexagonal scheme with equilateral triangles results in a fourth-order accurate method to the Laplace equation. Finally, we study two-dimensional elliptic differential equations on different triangular grids and domains.

In the article is determined the exact order of limiting error of inaccurate information in the problem of recovery functions from Sobolev classes according to the information received from all possible linear functionals. The speed of recovery is the same as for accurate information, although this property is lost when we multiply the limiting error for the any increasing sequence. As a consequence of this result, in the context of the Computational (numerical) diameter, it is shown that Lagrange spline interpolation is the most effective among all possible computing methods, according to the information by value at points. Computational experiments confirm this conclusion.

The memristive chaotic systems have attracted much attention and have been thoroughly discussed. The analysis shows that the fractional-order system is closer to the real system, and the time-delay chaotic systems are of an infinite-dimension with high randomness and unpredictability. Here we consider these three concepts (time delay, fractional order and memristive) to propose a 1D time delay fractional order memristive chaotic system. The analysis of the system is investigated by theoretical analyses and numerical simulations using the Kumar algorithm based on the Caputo definition for fractional order. The results indicate that the system parameter can significantly affect the dynamic behavior, which can be indicated by bifurcation diagrams, Lyapunov exponent diagrams, and phase portraits. A numerical method is adapted here to simulate the system, enabling short-memory implementation using a field-programmable gate array (FPGA). The experimental results were in good agreement with the numerical simulation results. Furthermore, an image encryption scheme based on multi-level diffusion and multi-round diffusion–confusion was developed, which involves diffusion and confusion operations. Security analysis shows the effectiveness of the proposed algorithm in terms of high security and excellent encryption performance.

This paper presents a new approach that uses the Reservoir Computing Algorithm to solve Fokker-Planck-Kolmogorov (FPK) equation excited by both Gaussian white noise and non-Gaussian noise. Unlike typical numerical methods, this methodology does not necessitate spatial reconstruction or numerical supplementation. The novelty of this paper lies in the modifications made to the conventional Reservoir Computing algorithm. We altered the approach for calculating values of the input weight matrix and incorporated autoregressive techniques in the reservoir layer. In addition, we applied data normalization to the training data before training the algorithm to avoid a zero solution. The efficacy of this approach was verified through multiple arithmetic examples, showcasing its practicality and efficiency in solving FPK equations. Moreover, the Reservoir Computing-FPK algorithm is capable of solving high-dimensional and fractional-order FPK equations with a smaller training set than earlier algorithms. Finally, we analyzed how values of the input weight matrix and regularization parameter affected the performance of the algorithm. The findings suggest that the careful selection of hyperparameters can greatly improve the performance of the Reservoir Computing algorithm.

We propose a novel method of fundamental solutions (MFS) formulation for solving boundary value problems (BVPs) governed by the polyharmonic equation ${\Delta }^{N}u=0,N\in N\setminus \left\{1\right\}$, in $\Omega \subset R^d,d=2,3$. The solution is approximated by a linear combination of the fundamental solution of the operator ${\Delta }^N$ and its first $N-1$ derivatives along the outward normal vector to the MFS pseudo-boundary. The optimal position of the pseudo-boundary on which the source points are placed is found using the effective condition number technique. Moreover, the proposed technique, when applied to polyharmonic BVPs in radially symmetric domains, lends itself to the application of matrix decomposition algorithms. The effectiveness of the method is demonstrated on several numerical examples.