Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two-dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem.

The low Mach number limit of the nonisentropic compressible Hookean elastodynamic equations is rigorously proved with respect to well-prepared initial data. We introduce certain suitable seminorms to obtain the uniform estimate of solutions, for which the critical point is to cancel the higher order derivate terms caused by the coupling of velocity and deformation gradient.

Abstract. In this manuscript we consider the internal control problem for the fifth order KdV type equation, commonly called the Kawahara equation, on unbounded domains. Precisely, under certain hypotheses over the initial and boundary data, we are able to prove that there exists an internal control input such that solutions of the Kawahara equation satisfies an integral overdetermination condition. This condition is satisfied when the domain of the Kawahara equation is posed in the real line, left half-line and right half-line. Moreover, we are also able to prove that there exists a minimal time in which the integral overdetermination condition is satisfied. Finally, we show a type of exact controllability associated with the “mass” of the Kawahara equation posed in the half-line.

The dynamic transitions of the Brusselator model has been recently analyzed in Y. Choi et’al (2021) and T. Ma, S. Wang (2011). Our aim in this paper is to address the relation between the pattern formation and dynamic transition results left open in those papers. We consider the problem in the setting of a 2D rectangular box where an instability of the homogeneous steady state occurs due to the perturbations in the direction of several modes becoming critical simultaneously. Our main results are two folds: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states and (2) the relation between the dynamic transitions and the pattern formations. We observe that the Brusselator model exhibits different transition types and patterns depending on the nonlinear interactions of the pattern of the critical modes.

In this paper, we investigate a well-balanced and positive-preserving non-staggered central scheme, which has second-order accuracy on both time and spatial scales, for open channel flows with variable channel width and non-flat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross-sectional wet area is positive by introducing a draining time-step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady-state solutions and the moving steady-state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations in the moving steady-state solution and avoid generating spurious oscillations. It is also capable of showing that the scheme is positive-preserving and robust in solving the dam-break problem.

In this manuscript,we present a parametric family of derivative-free 3-steps iterative methods with a weight function for solving nonlinear equations. We study various ways of introducing memory to this parametric family in order to increase the order of convergence without new functional evaluations. We also performed numerical experiments to compare the iterative methods fromdifferent points of view.

This paper is concerned with a class of singular prey-taxis models in a smooth bounded domain under homogeneous Neumann boundary conditions. The main challenge of analysis is the possible singularity as the prey density vanishes. Employing the technique of a priori assumption, the comparison principle of differential equations and semigroup estimates, we show that the singularity can be precluded if the intrinsic growth rate of prey is suitably large and hence obtain the existence of global classical bounded solutions. Moreover, the global stability of co-existence and prey-only steady states with convergence rates is established by the method of Lyapunov functionals.

With the help of the Meir-Keeler contraction method and the Mann-type method, two adaptive inertial iterative schemes are introduced for finding solutions of the split variational inclusion problem in Hilbert spaces. The strong convergence of the suggested algorithms are guaranteed by a new stepsize criterion that does not require calculation of the bounded linear operator norm. Some numerical experiments and applications in signal recovery problems are given to demonstrate the efficiency of the proposed algorithms.

In this paper, we propose an inertial algorithm for solving split equality of monotone inclusion and $f$-fixed point of Bregman relatively $f$-nonexpansive mapping problems in reflexive real Banach spaces. Using the Bregman distance function, we prove a strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. As an application, we provide several applications of our method. Furthermore, we give a numerical example to demonstrate the behavior of the convergence of the algorithm.

This paper studies the non-autonomous non-Newtonian micropolar fluids in two-dimensional bounded domains. We first establish that the generated continuous process of the solutions operator possesses a pullback attractor. Then we verify the existence of statistical solutions by constructing the invariant Borel probability measures. Further, we prove that the statistical solutions possess the degenerated regularity of Lusin's type provided that the Grasshof number associated to the external forces is small enough.

In this paper, we study the stabilization of a coupled wave system formed by one localized non-regular fractional viscoelastic damping of Kelvin-Voigt type and localized non-smooth coefficients. Our main aim is to prove that the C0-semigroup associated with this model is strong stability and decays polynomially at a rate of t−1. By introducing a new system to deal with fractional Kelvin-Voigt damping, we obtain a new equivalent augmented system, so as to show the well-posedness of the system based on Lumer-Phillips theorem. We achieve the strong stability for the C0-semigroup associated with this new model by using a general criteria of Arendt-Batty, and then turn out a polynomial energy decay rate of order t−1 with the help of a frequency domain approach.

{In the present paper, a $\mu$-delayed Mittag-Leffler type function is introduced as a fundamental function. By means of $\mu$-delayed Mittag-Leffler type function, an exact analytical solution formula to non-homogeneous linear delayed Langevin equations involving two distinct $\mu$-Caputo type fractional derivatives of general orders is given. Also, a global solution of nonlinear version of delayed Langevin equations is inferred from the findings on hand and is verified with the aid of the functional(substitutional) operator. In terms of exponential function, we estimate $\mu$-delayed Mittag-Leffler type function. Existence uniqueness of solutions to nonlinear delayed Langevin fractional differential equations are obtained with regard to the weighted norm defined in accordance with exponential function. The notion of stability analysis in the sense of solutions to the described Langevin equations is discussed on the grounds of the fixed point approach. Numerical and simulated examples are shared to exemplify the theoretical findings. This paper provides novel results.

A Riemann-Liouville fractional Robin boundary-value problem is proposed to describe the fast heat transfer law both within isotropic materials and through the boundary of the materials in high temperature environment. The variational formulation of the fractional model is given, and further the energy estimation of the weak solution is deduced. The uniqueness theorem of weak solution is proved. A valid finite difference scheme is developed for the fractional model and numerical experiment is implemented. Numerical results indicate that the fractional model is applicable to discover the thermal superdiffusion in the thermal protective clothing(TPC) system and numerical algorithms are effective to improve the intelligence of TPC design.

In this paper, we discuss the Turing-Hopf bifurcation of a diffusive Holling-Tanner model with nonlocal effect and digestion time delay. The stability, Turing bifurcation, Hopf bifurcation and Turing-Hopf bifurcation are first researched. Then we derive the algorithm for calculating the normal form of Turing-Hopf bifurcation of a diffusive Holling-Tanner model with nonlocal effect and digestion time delay. At last, we carry out some numerical simulations to verify our theoretical analysis results. The stable positive constant steady state and the stable spatially inhomogeneous periodic solutions are found. Furthermore, the evolution process from unstable spatially inhomogeneous steady states to stable positive constant steady state, the evolution process from unstable spatially inhomogeneous steady states to stable spatially inhomogeneous periodic solutions, the evolution process from one unstable spatially inhomogeneous periodic solution to another stable spatially inhomogeneous periodic solution and the evolution process from unstable spatially inhomogeneous periodic solution to stable positive constant steady state are also found.

Let $H_c$ be a $(2n)\times(2n)$ symmetric tridiagonal matrix with diagonal elements $c \in \mathbb{R}$ and off-diagonal elements one, and $S$ be a $(2n)\times(2n)$ diagonal matrix with the first $n$ diagonal elements being plus ones and the last $n$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil $\mathcal{A}_c=H_c-\lambda S$ as $2n$ approaches to infinity. It was conjectured by DL that for any $n \in \mathbb{N}$ the non-real eigenvalues $\lambda$ of $\mathcal{A}_c$ satisfy both $|\lambda + c|<2$ and $|\lambda - c|<2$. The conjecture has been verified numerically for a wide range of $n$ and $c$, but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non-real eigenvalues of $\mathcal{A}_c$. We provide a proof of the conjecture for $n \leq 3$, and also in the case where $|\lambda + c|=|\lambda - c|$. In addition, numerics indicate that some phenomena may occur for more general linear pencils.

In this paper, we introduce a new traveling wave with a buffer zone, which is approached by an asymmetric credit migration model with fixed migration boundary. The asymptotic behavior of the solution of the model is discussed. By constructing two sets of sub and super solutions sequences, it is proved that the solution of the credit rating migration model approaches the new traveling wave with buffer zone as time goes to infinite in a direction. Additionally, some numerical results are presented.

The inappropriate use of insecticides may lead to disastrous pest outbreaks. Aiming at avoiding the outbreak of pests by optimizing the control strategies, we propose a novel mathematical model base on the idea of pulse in this study, namely a discrete-time model of pest-natural enemies, which considers the implementation of spraying insecticides within the time interval between two generations of pests. We first investigates the existence and stability of fixed points, and then using the central manifold theory, we proved that the system can exist period-doubling bifurcation. The main theoretical results are verified by numerical simulations. The experiments show that if the insecticidal time is within a certain range, the lower insecticidal rate stimulates the growth of pests and natural enemies, while the larger insecticidal rate can inhibit the growth of pests and natural enemies. In addition, the effects of pest growth rate, timing of pesticide spraying, and pesticide intensity on the system are comprehensively discussed. The main research provides good reference significance to choose an appropriate time to prevent the pest outbreaks.

Consider the nonautonomous semilinear evolution equation of type: $(\star) \; u’(t)=A(t)u(t)+f(t,u(t)), \; t \in \mathbb{R},$ where $ A(t), \ t\in \mathbb{R} $ is a family of closed linear operators on a Banach space $X$, the nonlinear term $f$, acting on some real interpolation spaces, is assumed to be almost periodic only in a weak sense (i.e. in Stepanov sense) with respect to $t$ and Lipschitzian in bounded sets with respect to the second variable. We prove the existence and uniqueness of positive almost periodic solutions in the strong sense (Bohr sense) for equation $ (\star) $ using the exponential dichotomy approach. Then, we establish a new composition result of Stepanov almost periodic functions by assuming only the continuity of $f$ in the second variable. Moreover, we provide an application to a nonautonomous system of reaction–diffusion equations describing a Lotka–Volterra predator–prey model with diffusion and time–dependent parameters in a generalized almost periodic environment.