The proposed fuzzy composition-based filtering method aims to remove a presence of fractal Brownian noise in the MR brain images. The fractional Brownian motion (FBM) noise is a continuous time Gaussian processed noise and its very difficult to identify the positions and range of noise density level, due to a smoothed noise. The projected fuzzy scheme encloses an equivalent fuzzy interference scheme, a fuzzy average procedure and a fuzzy composition procedure. The noise subtraction scheme has been confirmed to be the finest while the depiction is tainted by means of fractional Brownian motion. With an average o/p Peak Signal to Noise Ratio (PSNR) of 37.22 and an average noisy image PSNR of 20.28, the average PSNR rate has improved by 16.94. In addition, the average mean square error (MSE) rate has decreased from 609.48 to 12.33 percent. An experimental result confirms that the fuzzy filtering achieves an outstanding eminence of reinstated images in terms of PSNR and MSE without the assistance of noiseless depiction.

By combining geometric singular perturbation theory (GSPT) with qualitative method, this paper analyzes the phenomenon of successive canard explosions in a singularly perturbed Spruce-Budworm model with Holling-II functional response. We select suitable parameters such that the critical curve is $S$-shaped, and the full model only admits a unique equilibrium. Then, under the variation of the breaking parameter, it is found that a canard explosion followed by an inverse canard explosion successively occurs in this model. That is, a relaxation oscillation arises via the first canard explosion, which persists for a large interval of parameter until it vanishes via the so-called inverse canard explosion. All these theoretical predictions are verified by numerical simulations.

Based on the existence of pullback attractors for the non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping defined in the entire space $\mathbb{R}^n$ by Xiaobin Yao in [15], in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem.

In this study, the numerical solutions of the Fornberg-Whitham (FW) equation modeling the qualitative behavior of wave refraction and the modified Fornberg-Whitham (mFW) equation describing the solitary wave and peakon waves with a discontinuous first derivative at the peak have been obtained. To obtain numerical results, the collocation finite element method has been combined with quintic B-spline bases. Although there are solutions to these equations by semi-analytical and analytical methods in the literature, there are very few studies using numerical methods. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. We have considered four test problems with nonhomogeneous boundary conditions that have analytical solutions to show the performance of the method. The numerical results of the two problems are compared with some studies in the literature. Additionally, peakon wave solutions and some new numerical results of the mFW equation, which are not available in the literature, are given in the last two problems. No comparison has been made since there are no numerical results in the literature for the last two problems. The error norms $L_2$ and $L_∞$ are calculated to demonstrate the presented numerical scheme’s accuracy and efficiency. The advantage of the scheme is that it produces accurate and reliable solutions even for modest values of space and time step lengths, rather than small values that cause excessive data storage in the computation process. In general, large step lengths in the space and time directions result in smaller matrices. This means less storage on the computer and results in faster outcomes. In addition, the present method gives more accurate results than some methods given in the literature.

The study of fractional Langevin equation has obtained abundant results in recent years. However, there are few studies on resonant fractional Langevin equation. In this paper, we investigate boundary value problems for fractional Langevin equation at resonance. By virtue of Banach contraction mapping principle and Leray-Schauder fixed point theorem, we obtain the uniqueness and existence of solutions. In addition, we get different stability results, including Ulam-Hyres stability and generalized Ulam-Hyres stability. Finally, give relevant examples to demonstrate the main results.

For the Cauchy problem of the 3D compressible MHD equations with Coulomb force, the large time behavior of this model is further investigated in this article. Compared to the previous related works in Tan-Tong-Wang [J. Math. Anal. Appl. 427 (2015) 600–617], the main novelty of this paper is that we prove the optimal decay rates for the highest-order spatial derivatives of the solutions to the compressible MHD equations with Coulomb force, which are the same as those of the heat equation.

This paper is devoted to the study of the 3D incompressible magnetohydrodynamic system. We prove the local in time well-posedness for any large initial data in $\dot{H}^1_{a,1}(\mathbb{R}^3)$ or $H^1_{a,1}(\mathbb{R}^3).$ Furthermore, the global well-posedness of a strong solution in $\tilde{L}^∞(0, T; H^1_{ a,1}(\mathbb{R}^3)) ∩ L^2 (0, T; \dot{H}^1_{a,1}(\mathbb{R}^3) ∩ \dot{H}^2_{a,1}(\mathbb{R}^3))$ with initial data satisfying a smallness condition is established.

Recovering system model from noisy data is a key challenge in the analysis of dynamical systems. Based on a data-driven identification approach, we develop a model selection algorithm called Entropy Regression Bayesian Information Criterion (ER-BIC). First, the entropy regression identification algorithm (ER) is used to obtain candidate models that are close to the Pareto optimum and combine as a library of candidate models. Second, BIC score in the candidate models library is calculated using the Bayesian information criterion (BIC) and ranked from smallest to largest. Third, the model with the smallest BIC score is selected as the one we need to optimize. Finally, the ER-BIC algorithm is applied to several classical dynamical systems, including one-dimensional polynomial and RC circuit systems, two-dimensional Duffing and classical ODE systems, three-dimensional Lorenz 63 and Lorenz 84 systems. The results show that the new algorithm accurately identifies the system model under noise and time variable $t,$ laying the foundation for nonlinear analysis.

This paper is concerned with the initial boundary problem for the three-dimensional density-dependent Boussinesq equations with vacuum. We obtain the existence of the global strong solution under the initial density in the norm $L^∞$ is small enough without any smallness condition of $u$ and $θ.$ Furthermore, the exponential decay rates of the solution and their derivatives in some norm was established. In addition, we show that the solution and their derivatives are monotonically decreasing with respect to time $t$ on $[0, T].$

This paper investigates the limit cycle bifurcation problem of the pendulum equation on the cylinder of the form $\dot{x} = y, \dot{y} = − {\rm sin} x$ under perturbations of polynomials of ${\rm sin} x,$ ${\rm cos} x$ and $y$ of degree $n$ with a switching line $y = 0.$ We first prove that the corresponding first order Melnikov functions can be expressed as combinations of anti-trigonometric functions and the complete elliptic functions of first and second kind with polynomial coefficients in both the oscillatory and rotary regions for arbitrary $n.$ We also verify the independence of coefficients of these polynomials. Then, the lower bounds for the number of limit cycles are obtained using their asymptotic expansions with $n = 1, 2, 3.$

This paper deals with dynamics of a predator-prey model with Allee effect and herd behavior. We first study the stability of non-negative constant solutions for such system. We also establish the existence of Hopf bifurcation solutions for such predator-prey model. The stability and bifurcation direction of Hopf bifurcation solution in the case of spatial homogeneity are further discussed. At the same time, several examples are given by MATLAB. Finally, the numerical simulations of the system are carried out through MATLAB, which intuitively verifies and supplements the theoretical analysis results.

In this paper, we apply $\overline{\partial}$-steepest descent method to analyze the long-time asymptotics of complex mKdV equation with the initial value belonging to weighted Sobolev spaces. Firstly, the Cauchy problem of the complex mKdV equation is transformed into the corresponding Riemann-Hilbert problem on the basis of the Lax pair and the scattering data. Then the long-time asymptotics of complex mKdV equation is obtained by studying the solution of the Riemann-Hilbert problem.

The effective degree SIR model is a precise model for the SIR disease dynamics on a network. The original ODE model is only applicable for a network with finite degree distributions. The new generating function approach rewrites with model as a PDE and allows infinite degree distributions. In this paper, we first prove the existence of a global solution. Then we analyze the linear and nonlinear stability of the disease-free steady state of the PDE effective degree model, and show that the basic reproduction number still determines both the linear and the nonlinear stability. Our method also provides a new tool to study the effective degree SIS model, whose basic reproduction number has been elusive so far.

In this paper, we consider a class of nonlinear differential equations with delays described by conformable fractional derivative. This type of differential equations can be used to describe dynamics of various practical models including biological and artificial neural networks with heterogeneous time-varying delays. By novel comparison techniques via fractional differential and integral inequalities, we prove under assumptions involving the order-preserving property of nonlinear vector fields that, with nonnegative initial states and inputs, the system state trajectories are always nonnegative for all time. This feature, called positivity, induces a special character, namely the monotonicity of the system. We then derive tractable conditions in terms of linear programming and prove, by utilizing the Brouwer’s fixed point theorem and comparisons induced by the monotonicity, that the system possesses a unique positive equilibrium point which attracts exponentially all state trajectories. An application to the exponential stability of fractional linear time-delay systems is also discussed. Numerical examples with simulations are given to illustrate the theoretical results.

In [6] , the authors studied an SIR epidemic model with free boundary. They first proved the global existence and uniqueness of solution, and then gave the criteria for spreading and vanishing. They also obtained the longtime behavior for the case of vanishing. However, the longtime behavior when spreading happens remains open. In this short paper, we aim to solve this open problem.

There are several main challenges in solving nonlinear differential equations with artificial neural networks (ANNs), such as a nonlinear system’s sensitivity to its initial values, discretization, and strategies for incorporating physics-based information into ANNs. As for the first issue, this paper addresses the initial value problems of nonlinear dynamical systems (a Duffing oscillator and a Burger’s equation), which cause large global truncation errors in sub-domains with a significant reduction in the influence of initial constraints, using meta-learning-based physics-informed neural networks (MPINNs). The MPINNs with dual learners outperform physics-informed neural networks with a single learner (no fine reinitialization capability). As a result, the former approach improves solution convergence by 98.83% in the sub-time domain (III) of a Duffing oscillator, and by 85.89% at $t = 45$ in a Burger’s equation problem, compared to the latter one. Model accuracy is highly dependent on the adaptability of the initial parameters in the first hidden layers of the meta-models. From correlation analyses, it is obvious that the parameters become less (the Duffing oscillator) or more (the Burger’s equation) correlated during fine reinitialization, as the update manner differs or is similar to the one used in pre-initialization. In the first example, the MPINN achieves both the mitigation of model sensitivity to its output and the improvement of model accuracy. Conversely, the second example shows that the proposed approach is not enough to solve both issues simultaneously, as increased model sensitivity to its output leads to higher model accuracy. The application of transfer learning reduces the number of iterative pre-meta-trainings.