Magnetic Force-Free Theory: Nonlinear Case

In this paper, a theory of force-free magnetic field useful for explaining the formation of convex closed sets, bounded by a magnetic separatrix in the plasma, is developed. This question is not new and has been addressed by many authors. Force-free magnetic fields appear in many laboratory and astrophysical plasmas. These fields are defined by the solution of the problem ∇×B=ΛB with some field conditions B∂Ω on the boundary ∂Ω of the plasma region. In many physical situations, it has been noticed that Λ is not constant but may vary in the domain Ω giving rise to many different interesting physical situations. We set Λ=Λ(ψ) with ψ being the poloidal magnetic flux function. Then, an analytic method, based on a first-order expansion of ψ with respect to a small parameter α, is developed. The Grad–Shafranov equation for ψ is solved by expanding the solution in the eigenfunctions of the zero-order operator. An analytic expression for the solution is obtained deriving results on the transition through resonances, the amplification with respect to the gun inflow. Thus, the formation of Spheromaks or Protosphera structure of the plasma is determined in the case of nonconstant Λ.

Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

Transition to Fully or Partially Arrested State in Coupled Logistic Maps on a Ladder

Layered structures are an object of interest for theoretical and experimental reasons. In this work, we study coupled map lattice on a ladder. The ladder consists of two one-dimensional chains coupled at every point. We study linearly and nonlinearly coupled logistic maps in this system and study transition to nonzero persistence, in particular. We coarse-grain the variable value by assigning spin [Formula: see text] ([Formula: see text]) to sites that have value greater (less) than the fixed point and compute the number of sites that have not changed their spin values at all even times till the given time [Formula: see text]. The fraction of such sites at a given time [Formula: see text] is known as persistence. In our system, we observe a power-law of persistence at the critical value of coupling. This transition is also accompanied by long-range antiferromagnetic ordering for nonlinear coupling and long-range ferromagnetic ordering for linear coupling. The number of domain walls decay as [Formula: see text] at the critical point in both cases. The persistence exponent is 0.375 for a nonlinear case with two layers which is an exponent for the voter model on the ladder as well as for the Ising model at zero temperature or voter model in 1D. For linear coupling, we obtain a smaller persistence exponent.

Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinear case up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this framework. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework.

Highly Accurate Experimental Heave Decay Tests with a Floating Sphere: A Public Benchmark Dataset for Model Validation of Fluid–Structure Interaction

Highly accurate and precise heave decay tests on a sphere with a diameter of 300 mm were completed in a meticulously designed test setup in the wave basin in the Ocean and Coastal Engineering Laboratory at Aalborg University, Denmark. The tests were dedicated to providing a rigorous benchmark dataset for numerical model validation. The sphere was ballasted to half submergence, thereby floating with the waterline at the equator when at rest in calm water. Heave decay tests were conducted, wherein the sphere was held stationary and dropped from three drop heights: a small drop height, which can be considered a linear case, a moderately nonlinear case, and a highly nonlinear case with a drop height from a position where the whole sphere was initially above the water. The precision of the heave decay time series was calculated from random and systematic standard uncertainties. At a 95% confidence level, uncertainties were found to be very low—on average only about 0.3% of the respective drop heights. Physical parameters of the test setup and associated uncertainties were quantified. A test case was formulated that closely represents the physical tests, enabling the reader to do his/her own numerical tests. The paper includes a comparison of the physical test results to the results from several independent numerical models based on linear potential flow, fully nonlinear potential flow, and the Reynolds-averaged Navier–Stokes (RANS) equations. A high correlation between physical and numerical test results is shown. The physical test results are very suitable for numerical model validation and are public as a benchmark dataset.

A Regularization Method for Nonlinear Integro-differential Equations with a Zero Operator of the Differential Part and with Several Rapidly Varying Kernels

A nonlinear integro-differential equation with a zero operator of the differential part and several rapidly changing kernels is considered. The work is a continuation of the research conducted earlier for a single rapidly changing core. The main ideas of this generalization and the subtleties that arise in the development of the corresponding algorithm for the regularization method are fully visible in the case of two rapidly changing kernels, so for the sake of reducing the calculations, this particular case is taken. A similar problem with a single spectral value of the kernel of an integral operator is analyzed in one of the authors ' papers. In this case, the singularities in the solution of the problem are described only by the spectral value of the kernel. However, the influence of the zero operator of the differential part affects the fact that in the first approximation, the asymptotics of the solution of the problem under consideration will not contain the functions of the boundary layer, and the limit operator itself will become degenerate (but not zero). The conditions for the solvability of the corresponding iterative problems, as in the linear case, will not be in the form of differential equations (as was the case in problems with a non-zero operator of the differential part), but integro-differential equations, and the formation of these equations is significantly influenced by nonlinearity. Note that, in contrast to the linear case, there is no inhomogeneity of the corresponding linear problem in the right part of the problem under study. As it was shown earlier, its presence in the problem would lead to the appearance in the asymptotic solution of terms with negative powers of a small parameter, and in the nonlinear case there would be innumerable such powers, and the corresponding formal asymptotic solution would have the form of a Laurent series. This would make the creation of an algorithm for asymptotic solutions problematic, so in this paper, wanting to remain within the framework of asymptotic solutions of the type of Taylor series, inhomogeneity is excluded. In addition, in the nonlinear case, so-called resonances may occur, which significantly complicate the development of the corresponding algorithm for the regularization method. This publication deals with the non-resonant case. It is assumed that the study of an alternative variant (a more complex resonant problem) will be carried out in the future.

A Covariance Feedback Approach to Covariance Control of Nonlinear Stochastic Systems

In this paper, the covariance control algorithm for nonlinear stochastic systems using covariance feedback is studied. Covariance control of nonlinear systems scenario involves the theory of covariance control based on the idea of the covariance feedback. Therefore, the proposed covariance control algorithm is derived for our case, firstly by applying the covariance control method and linear approximation of nonlinear systems, and then it is achieved by adopting this method for a class of nonlinear stochastic systems by using feedback linearization idea and a covariance feedback controller. The effectiveness of the proposed covariance feedback algorithm is studied using numerous simulations concerning different nonlinear case studies.

Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop (1955, “The Treatment of Damping Forces in Vibration Theory,” Aeronaut. J., 59(539), pp. 738–742). We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as t→∞. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid (1956, “Free Vibration and Hysteretic Damping,” Aeronaut. J., 60(544), pp. 283–283), which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.

About Two-dimensional Mathematical Model of Contaminant Migration in Unsaturated Catalytic Porous Media with Traps

This paper proposes an approach for the computer simulation of complex physical problem of contaminant migration through unsaturated catalytic porous media to the filter-trap. The corresponding mathematical model in the two-dimensional nonlinear case is presented. The model includes detailed considerations of the moisture transfer of saline solutions under the generalized Darcy’s and Cluta’s laws in different subregions of soil. The numerical solution of the boundary value problem was found by the finite difference method and proposed the algorithm for computer implementation. The proposed algorithm may be used for creating software with effective risk assessment strategies and predicting the cleaning and further useful use of the soil massifs.