This paper is devoted to the study of Caputo-type fractional FitzHugh-Nagumo equations driven by fractional Brownian motion (fBm). We establish the existence and uniqueness of mild solution under some conditions on the coefficients. The exponential synchronization and finite-time synchronization for the stochastic FitzHugh-Nagumo equations are provided. The analysis of synchronization phenomenon for time-fractional FitzHugh-Nagumo equations perturbed by fBm are provided.

We consider the system of linear equations arisen from the finite element discretization of the distributed optimal control problem with time-periodic parabolic equations. A block alternating splitting iteration (BASI) method is presented for solving the obtained system. We prove that the BASI method is unconditionally convergent. We derive the BASI preconditioner and present an estimation formula for the parameter of the BASI preconditioner. Numerical results are presented to verify the efficiency of both the BASI method and the BASI preconditioner. Comparison with some existing methods are also given.

The stability analysis of dynamic continuous structural system (DCSS) has often been investigated by discretizing it into several low-dimensional elements. The integrated results of all elements are employed to describe the whole dynamic behavior of DCSS. In this paper, DCSS is regarded as the complex dynamic network with the discretized elements as the dynamic nodes and the time-varying stiffness as the dynamic link relations between them, by which the DCSS can be regarded to be the large-scale system composed of the node subsystem (NS) and link subsystem (LS). Therefore, the dynamic model of DCSS is proposed as the combination of dynamic equations of NS and LS, in which their state variables are coupled mutually. By using the model, this paper investigates the stability of DCSS. The research results show that the state variables of NS and LS are uniformly ultimately bounded (UUB) associated with the synthesized coupling terms in LS. Finally, the simulation example is utilized to demonstrate the validity of method in this paper.

In this paper, we investigate the global well-posedness of the system of incompressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no nonlinear compatibility conditions on the initial data.

This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation with a small parameter. The existence of a boundary layer as the parameter goes to zero is obtained together with the estimation on the thickness of the boundary layer. The main result extends an earlier work of Frid and Shelukhin (1999).

In this work, we extend the direct method in [26] to identify the marine sources and scatterers simultaneously from the far-field pattern in a stratified ocean waveguide. The proposed approach is essentially direct and does not involve any optimizations, solution procedures or matrix inversions, thus computationally rather efficient and simple. Some numerical simulations are carried out to exhibit the robustness and effectiveness of the proposed method in the reconstruction procedure. The extended direct method can not only identify the sources in different locations but also can reconstruct the scatterers in different shapes and positions, therefore it can be considered as an efficient numerical approach for providing reliable estimates of the marine inhomogeneities in the marine acoustics.

In this paper, we take a fresh look at the determination of the transition matrix for a homogeneous, biisotropic particle employing the Null-field approach. The condition for passivity of the material is discussed. In particular, we focus on some previously unproved properties of the Beltrami spherical vector waves, such as completeness, orthogonality, and linear independence, which are all instrumental in the analysis of the scattering problem with the Null-field approach. Some numerical examples illustrate the approach.

We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the $L^2$-norm of the solution. In Appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring-profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min-max problem. In the case of the constrained minimization problem solutions are shown to be positive but the wave propagation constants undetermined, but in the min-max approach the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi-trivial solutions, meaning, that optical solitons cannot be observed when, say, one of the beams are off. Additionally, the second harmonic always remains positive.

In this work, we will design unexpected configurations for the optical soliton propagation in lossy fiber system in presence the dispersion term solitons via two distinct and impressive techniques. The first one is the (G’/G)-expansion method, while the second is solitary wave ansatze method. The two methods are implemented in same vein and parallel. The obtained perceptions are new and weren’t achieved before. The comparison between our achieved visions and that achieved by other authors who used different schemas has been documented.

The well-posedness and regularity properties of diffusion-aggregation equations, emerging from interacting particle systems, are established on the whole space for bounded interaction force kernels by utilizing a compactness convergence argument to treat the non-linearity as well as a Moser iteration. Moreover, we prove a quantitative estimate in probability with arbitrary algebraic rate between the approximative interacting particle systems and the approximative McKean--Vlasov SDEs, which implies propagation of chaos for the interacting particle systems.

We study the temporal equation of radiating stars by using three powerful methods for the analysis of nonlinear differential equations. Specifically, we investigate the global dynamics for the given master ordinary differential equation to understand the evolution of solutions for various initial conditions as also to investigate the existence of asymptotic solutions. Moreover, with the application of Lie’s theory, we can reduce the order of the master differential equation, while an exact similarity solution is determined. Finally, the master equation possesses the Painlev\’{e} property, which means that the analytic solution can be expressed in terms of a Laurent expansion.

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol-Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler’s identity, we give relations among theis characteristic function, the Apostol-Bernoulli polynomials and numbers, and also trigonometric functions including sin w and cos w . A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol-Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas.

The classical Stefan problem, concerning mere heat-transfer during solid-liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba-Jaumann corotational time derivatives, linearized by exploiting the additive Green-Naghdi’s decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase-field model, exploiting a concept of slightly (so-called “semi”) compressible materials. The $L^1$-theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid-liquid phase transition. A rigorous proof of existence of week solutions is provided for an incomplete melting, exploiting a time-discretisation approximation.

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem \[ \frac{d}{dx}(1+\delta y+\gamma y^{2})^{n}\frac{dy}{dx}]+2x\frac{dy}{dx}=0,\,\,\,x>0,\,\,y(0)=0,\,\,\,y(\infty)=1, \] which was proposed in 1974 by [1] to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0;1). The modified error function of two parameters $\varphi_{\delta,\gamma}$ is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in [3, 4].

A method for practical realization of the inverse scattering transform method for the Korteweg-de Vries equation is proposed. It is based on analytical representations for Jost solutions and for integral kernels of transformation operators obtained recently by the authors. The representations have the form of functional series in which the first coefficient plays a crucial role both in solving the direct scattering and the inverse scattering problems. The direct scattering problem reduces to computation of a number of the coefficients following a simple recurrent integration procedure with a posterior calculation of scattering data by well known formulas. The inverse scattering problem reduces to a system of linear algebraic equations from which the first component of the solution vector leads to the recovery of the potential. We prove the applicability of the finite section method to the system of linear algebraic equations and discuss numerical aspects of the proposed method. Numerical examples are given, which reveal the accuracy and speed of the method.

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

The main aim of this paper is to study hyponormal and dissipative correct restrictions and extensions as well as their applications to differential operators.