Mathematical Methods in the Applied Sciences最新文献

In this paper, we consider the existence and multiplicity of normalized solutions for critical Choquard equations involving logarithmic nonlinearity in the Heisenberg group. Under suitable assumptions, combined with the truncation technique, the concentration-compactness principle, and the genus theory, we obtain the existence and multiplicity of the normalized solutions in the $��{�L�}^{�p�}�$-subcritical case. As far as we know, the result of the paper is completely new in the Euclidean case.

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi-scale four-dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three-dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two-dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three-dimensional system through investigating the dynamics on the two-dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non-monotonic fronts with oscillatory tails. In the second case, the normal to the one-dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three-dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three-dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

Optimal use of available resources is one of the major issues in many situations. Apart from this, age is also an important factor in modeling social interaction in a society. In this article, we consider an age-dependent social interaction model and study the stability and optimal control. Law enforcement is an important factor in controlling crime, and the deployment of police to enforce it is important. Now cost is a major issue, so optimal deployment is very important to study. Stability results are derived in terms of threshold parameters. Basic reproduction number is calculated for stability analysis. Using the adjoint system, the form of the law enforcement factor is obtained in terms of state space variables. We see that the cost functional increases with the increase in the population density of criminal population. Numerical results are also added to visually illustrate our theoretical results.

This research delves into a comprehensive investigation of a class of $��\psi �$-Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter $��\xi �$ in appropriate fractional $��\psi �$-Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional $��\psi �$-Hilfer generalized capillary phenomena.

The aim of this paper is to discover analytically the interactional responses of populations in a dynamic region where the reaction–diffusion process with forcing effects takes place through traveling wave solutions. An expansion method is considered here to properly capture the responses for the first time. In order to profoundly analyze the physical and mathematical discussions, some illustrative behavioral results are exhibited for various values of physical parameters. Especially for the different values of diffusion coefficients in the model under consideration, their effects on the behavior of the solitary wave are discussed and observationally supported by considering various illustrations. It is also seen that the solutions representing the diffusion seen to be in the form of the behavior of hexagonal Turing patterns in different time periods. The application of this study in mathematical biology is to analyze the relationship between the population density of certain species in any local region and the specific population density with invasion characteristics. In addition, the formation of the extinction vortex of the invading population, depending on the characteristics of the solutions presented, is also descriptively discussed.

This paper is concerned with the study of a Dirichlet boundary value problem for a system of two coupled anisotropic Darcy–Forchheimer–Brinkman equations on a bounded Lipschitz domain in $��{�ℝ�}^{�n�}�(�n�=�\mathrm{2,3}�)�$. Using variational methods and fixed point techniques, we obtain a well-posedness result in $��{�L�}^{�2�}�$-based Sobolev spaces for sufficiently small data. As an application, we investigate numerically the lid-driven flow problem in a square cavity saturated with a bidisperse porous medium and analyze the effect of various physical parameters on the fluid motion.

This study addressed the oscillation problems of half-linear differential equations with periodic damping. The solution space of any linear equation is homogeneous and additive. Generally, by contrast, the solution space of half-linear differential equations is homogeneous but not additive. Numerous oscillation and nonoscillation theorems have been devised for half-linear differential equations featuring periodic functions as coefficients. However, in certain cases, such as applying Mathieu-type differential equations to control engineering, which is a typical example of the Hill equation, some oscillation theorems cannot be applied. In this study, we established oscillation and nonoscillation theorems for half-linear Hill-type differential equations with periodic damping. To prove the results, we used the Riccati technique and the composite function method, which focuses on the composite function of the indefinite integral of the coefficients of the target equation and an appropriate multivalued continuously differentiable function. Furthermore, we discuss the special case of the oscillation constant of a damped half-linear Mathieu equation.

Mathematics, mathematical modeling of real systems, and mathematical and computer methodologies aimed at the qualitative and quantitative study of real physical systems interact in a nontrivial way. This work aims to examine a new class of inverse problems for a fractional partial differential equation with order fractional $��0�<�\rho �\le �1�$, which leads to the spectral problem involving Hermite's differential equation. We introduce proven theorems on the existence and uniqueness of solutions to the current problem. We obtain solutions in the form of series expansion using the Hermite orthogonal basis. Finally, we discuss the convergence analysis of the obtained solutions.

In this paper, we consider the 2D hyperbolic Prandtl equations on the half plane. Firstly, we obtain the global existence of solutions by using the classical energy methods in an analytic framework. Then, we prove the uniqueness of solutions. Besides, we also obtain a time exponential decay in analytic regularity norm of the solutions for any time $��t�\ge �0�$.

This paper strives to investigate the time fractional system that characterizes the ion sound wave influenced by the ponderomotive force induced by a high-frequency field, as well as the Langmuir wave in plasma. Initially, based on the qualitative theory for planar integrable systems, four-phase portraits are found in the $��(�u�,�y�)�$ phase plane under certain conditions on the physical parameters. These conditions are used to prove analytically the existence of solitary, kink (anti-kink), periodic, super-periodic, and unbounded wave solutions. The correspondence between the energy levels, phase orbits, and consequently the type of the solution is announced. We derived the bounded wave solutions associated with the phase orbits, which are shown to be consistent with the qualitative analysis of the types of solutions. Moreover, we studied the consistency between the obtained solutions by investigating the degeneracy of the solutions through the transmission between the phase orbits, or equivalently, through the dependence on the initial conditions. With the presence of perturbed periodic terms, the quasi-periodic behavior and chaotic patterns are investigated.