In this paper, a deep dynamical analysis is made using tools from multidimensional real discrete dynamics of some derivative-free iterative methods with memory. They all have good qualitative properties, but one of them (due to Traub) shows the same behavior as Newton’s method on quadratic polynomials. Then, the same techniques are employed to analyze the performance of several multipoint schemes with memory, whose first step is Traub’s method, but their construction was made using different procedures. Therefore, their stability is analyzed, showing which is the best in terms of the wideness of basins of convergence or the existence of free critical points that would yield convergence towards different elements from the desired zeros of the nonlinear function. Therefore, the best stability properties are linked with the best estimations made in the iterative expressions rather than their simplicity. These results have been checked with a numerical and graphical comparison with many other known methods with and without memory, with different orders of convergence, with excellent performance.

In this paper, we consider the low Mach number limit problem for the compressible viscous micropolar fluid model based on the concept of dissipative measure-valued solutions. We prove that the dissipative measure-valued solutions of the compressible micropolar fluid model converge to the smooth solution of the incompressible micropolar system in the case of well-prepared initial data when the Mach number tends to zero.

This paper investigates the long time well-posedness of 2-D MHD boundary layer equation without resistivity. It is proved that if the initial data satisfies \begin{align*} \|(u_0,h_0-1)\|_{H_{\mu}^{3,0}}+\|(u_0,h_0-1)\|_{H_{\mu}^{1,2}}+\|(u_0,h_0-1)\|_{H_{\mu}^{2,1}}\le \varepsilon, \end{align*} then the life span of the solution is at least of order $\varepsilon^{-\f43}$.

The article is dedicated towards the study of fractional-order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions in Banach spaces. First, mild solution of the system is constructed and subsequently, its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using the concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out corrections in examples presented in previous articles.

In this paper, we are concerned with the non-relativistic limit of a class of computable approximation models for radiation hydrodynamics. The models consist of the compressible Euler equations coupled with moment closure approximations to the radiative transfer equation. They are first-order partial differential equations with source terms. As hyperbolic relaxation systems, they are showed to satisfy the structural stability condition proposed by W.-A. Yong (1999). Base on this, we verify the non-relativistic limit by combining an energy method with a formal asymptotic analysis.

In this paper we study the existence of a Nash-type equlibrium for a non-potential nonlinear system by combining variational methods with the monotonicity approach. The advance over existing research is that we can consider systems of Dirichlet problems in which the operator is not necessarily linear.

Abstract: In this paper, the nonlinear Schrödinger-type equation −(∇ + iA) ^2 u + u + λ[I_α*(K|u|^2)]Ku=af(|u|)u/|u| in R ^3 is considered in the presence of magnetic field, where A ∈ C ^1 (R ^3 ,R^ 3 ), α ∈ (0,3), I_α denotes the Riesz potential, K ∈ L^ p (R ^3 ,(0,∞)) for some p ∈ (6/(1 + α),∞], a ∈ L^ q (R 3 ,[0,∞)) \ {0} for some q ∈ (3/2,∞], and f ∈ C(R,[0,∞)) is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding A, K, a, and f, variational methods are used to establish the existence of ground-state solutions of the above equation for sufficiently small values of the parameter λ.

In this paper, we derive an Euler-Maruyama (EM) method for a class of multi-term fractional stochastic nonlinear differential equations, and prove its strong convergence. The strong convergence order of this EM method is $\min\{\alpha_{m}-0.5,~\alpha_{m}-\alpha_{m-1}\}$, where $\{\alpha_{i}\}_{i=1}^{m}$ is the order of Caputo fractional derivative satisfying that $1>\alpha_{m}>\alpha_{m-1}>\cdots>\alpha_{2}>\alpha_{1}>0$, $\alpha_{m}>0.5$, and $\alpha_{m}+\alpha_{m-1}>1$. Then, a fast implementation of this proposed EM method is also presented based on the sum-of-exponentials approximation technique. Finally, some numerical experiments are given to verify the theoretical results and computational efficiency of our EM method.

We are happy to announce our work on MMAS special issue “Integro-differential models of natural and anthropogenic processes and phenomena” is successfully completed, so please enjoy original and high-quality contributions related to the mathematical theory of such processes and phenomena including the models, computational algorithms and analysis and mathematical methods regarded as new and promising for the understanding the problems arise both in natural and anthropogenic conditions.

In studying the stability of B\’{e}nard problem we usually have to solve a variational problem to determine the critical Rayleigh number for linear or nonlinear stability. To solve the variational problem one usually transform it to an eigenvalue problem which is called Euler-Lagrange equations. An operator related to the Euler-Lagrange equations is usually referred to as Euler-Lagrange operator whose spectrum is investigated in this paper. We have shown that the operator possesses only the point spectrum consisting of real number, which forms a countable set. Moreover, it is found that the spectrum of the Euler-Lagrange operator depends on the thickness of the fluid layer.

We investigate Schwarz lemma in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. The bicomplex is a generalization of complex which has closed relation with Frac- tal geometry, Minkowski Space-Time, Maxwell’s equations, Schrödinger equation and Gaussian pulse wave. In this paper, we first construct a type of bicomplex Möbius transformation and obtain some results : the mapping properties on bicom- plex spheres and bicomplex ball, preserving the inverse points with respect to the bicomplex sphere (0,1). Then we obtain the Poisson integral formula in bicom- plex setting, and by using the Poisson integral formula, we give the Schwarz lemma for bicomplex holomorphic functions in bicomplex setting. Finally, we shall give the Schwarz lemma and the Schwarz-Pick type lemma for holomorphic functions in bicomplex analysis. These results may give new energy for the development of quan- tum mechanics.

In this research article, we construct a q-analogue of the operators defined by Betus and Usta (Numer. Methods Partial Differential Eq. 1-12, (2020)) and study approximation properties in a polynomial weighted space. Further, we modify these operators to study the approximation properties of differentiable functions in the same space and show that the mofidied operators give a better rate of convergence.

In this paper, we study the existence of global $L^2$-constrained minimizers related to the following Kirchhoff type equation: $$ -\left(a+b\ds\int_{\R^N}|\nabla u|^2\right)\Delta u-f(u)=\lambda u,~~~x\in \R^N,~\lambda\in\R,$$ where $N\leq3$, $a,$ $b>0$ are constants, $f(u)$ is a general $L^2$-subcritical nonlinearity. By using the concentration compactness principle, we prove the sharp existence and nonexistence of global $L^2$-constraint minimizers.

In this article, the generalized (3+1)-dimensional Korteweg-de Vries-Zakharov-Kuznetsov equation is investigated, which describes the influence of magnetic fields on weak ion-acoustic waves in plasma made up of cool and hot electrons and may be regarded as a nonlinear complicated physical model. To find out some new traveling waves solutions and other exact solutions, the improved $F$-expansion approach and the $\exp(-\phi (\zeta))$-expansion approach is applied to above mentioned nonlinear higher dimensional model. Several solutions have been found, including dark soliton, periodic type solitons, bell shaped solitons, single bell shaped solitons. We also show a graphical representation of a number of exact solutions to the equation, together with a description of their behaviour. The proposed techniques can also be used to solve a range of nonlinear evolution problems in mathematical physics and plasma physics.

This work brings a one-dimensional logistic harvesting model with Allee effect to the time-varying framework. This new framework is more sober than the autonomous version of the system because it; the framework, permits all environment-dependent coefficients to depend on time. Based on these coefficients, we derive sets of conditions that drive population to “mathematical” extinction. More precisely, we investigate various local and global stability notions including uniform stability, attractivity, asymptotic stability and the (uniform) exponential stability.

We consider the following X-ray free electron lasers Schr\”{o}dinger equation \begin{equation*} (i\nabla-A)^2u+V(x)u-\frac{\mu}{|x|} u=\left(\frac{1}{|x|}*|u|^2\right) u-K(x)|u|^{q-2} u, \,\, x\in \mathbb{R}^3, \end{equation*} where $A\in L_{loc}^2(\mathbb{R}^3,\mathbb{R}^3)$ denotes the magnetic potential such that the magnetic field $B=\text{curl} \, A$ is $\mathbb{Z}^{3}$-periodic, $\mu\in \mathbb{R}$, $K \in L^{\infty}\left(\mathbb{R}^3\right)$ is $\mathbb{Z}^{3}$ -periodic and non-negative, $q\in(2,4)$. Using the variational method, based on a profile decomposition of the Cerami sequence in $H^1_A\left(\mathbb{R}^3\right)$, we obtain the existence of the ground state solution for suitable $\mu\geq0$. When $\mu<0$ is small, we also obtain the non-existence. Furthermore, we give a description for the asymptotic behaviour of the ground states as $\mu \to 0^+$.

In this paper, a backward problem for a nonlinear time-fractional diffusion equation in an axis-symmetric cylinder has been considered. Under some assumptions, we prove the existence and uniqueness of the solution to the nonlinear problem. The ill-posedness of the backward problem is established and we obtain the error estimates by a generalized quasi-boundary value regularization method.

We present Wong-type oscillation criteria for nonlinear impulsive differential equations having discontinuous solutions and involving both negative and positive coefficients. We use a technique that involves the use of a nonprincipal solution of the associated linear homogeneous equation. The existence of principal and nonpricipal solutions was recently obtained by the present authors in [J. Math. Anal. Appl. 503 (2021) 125311]. As special cases, we have superlinear and sublinear Emden-Fowler equations under impulse effects. It is shown that the oscillation behavior changes due to impulses, in particular impulses acting on the solution itself, not on its derivative. An example is also given to illustrate the importance of the results.