Mathematical Methods in the Applied Sciences最新文献

The main attention of this paper focuses on studying the dynamical behaviors, cubic–quartic optical solitons, chaotic pattern, and phase portraits for the nonlinear coupled Kaup–Newell equation in birefringent fibers. First of all, by means of traveling wave transforms and homogeneous balance principle, the coupled Kaup–Newell equation in birefringent fibers is simplified into an ordinary differential equation. Secondly, the dynamical properties of two-dimensional system and the corresponding perturbed system have been studied. Finally, with the assistance of the complete discriminant system method, the optical soliton solutions of the coupled Kaup–Newell equation in birefringent fibers can be derived, which include solitary wave solutions, rational function solutions, Jacobian elliptic function solutions, and hyperbolic function solutions. In addition, two-dimensional portraits, three-dimensional portraits, contour plots, density plots, and two-dimensional gradplots of the obtained solutions are also given by explaining the propagation of optical solitons for the coupled Kaup–Newell equation in birefringent fibers.

This paper considers the stability of the 3D incompressible MHD-Boussinesq system with mixed dissipation in $��{�ℝ�}^{�3�}�$. The main purpose of this paper is to prove the global stability of perturbations near the hydrostatic equilibrium for the 3D incompressible MHD-Boussinesq system with mixed dissipation and damping. By using the energy methods, we obtain that this system possesses a global solution for initial data in $��{�H�}^{�3�}�$.

This paper investigates an initial boundary value problem for the relaxed one-dimensional compressible fluid equations, where the Newton's law of viscosity and Fourier's law of heat conduction are replaced by Maxwell's law and Cattaneo's law, respectively. By transforming the system into Lagrangian coordinates, the resulting formulation exhibits a uniform characteristic boundary structure. We first construct an approximate system with noncharacteristic boundaries and establish its local well-posedness by verifying the maximal nonnegative boundary conditions. Subsequently, through the construction of a suitable weighted energy functional and careful treatment of boundary terms, we derive uniform a priori estimates, thereby proving the global well-posedness of smooth solutions for the approximate system. Utilizing these uniform estimates and standard compactness arguments, we further obtain the existence and uniqueness of global solutions for the original system. In addition, the global relaxation limit is established. The analysis is fundamentally based on energy estimates.

In this study, the complex spatiotemporal dynamical behaviors of a diffusive toxic plankton system, subject to time delay and prey-taxis with a Ricker-type sensitivity function, are systematically investigated. Firstly, an analysis of the existence of Codimension-1 Turing bifurcation, Hopf bifurcation, and Codimension-2 Turing–Hopf bifurcation and double Hopf bifurcation are conducted. In particular, we investigate that the addition of chemotaxis term enhances the spatial heterogeneity of the system, thereby inducing Turing instability. Then, a key contribution of this paper lies in departing from the traditional center manifold approach and employing the multiple-timescale method to derive the amplitude equations near the nonzero-mode double Hopf bifurcation point. Subsequently, based on the derived normal form, we analyze the topological structure of orbital distributions near the double Hopf bifurcation point and identify the corresponding spatiotemporal patterns in the original system. The results show that a high chemotactic sensitivity can lead to spatial heterogeneity in the system. The coupling between chemotaxis and toxin delay can induce spatial complexity, such as stability switching, spatially inhomogeneous periodic oscillations, and spatially inhomogeneous aperiodic oscillations.

In this study, we construct the generalized Lyapunov- and Hartman-type inequalities for both linear and nonlinear generalized Caputo fractional differential equations under Dirichlet-type boundary conditions. By leveraging these inequalities, we can establish important results related to disconjugacy for the fractional differential equations. The inequalities we propose not only generalize but also enrich existing research in the literature, effectively addressing exceptional cases of fractional differential equations.

In this paper, we study some families of right modules of quaternionic slice regular functions induced by a generalized fractal–fractional derivative with respect to a truncated quaternionic exponential function on slices. Important Banach spaces of slice regular functions, namely, the Bergman and the Dirichlet modules, are two important elements of one of our families.

Initiation of convection in the horizontal layer is studied in the Casson nanofluids. The current investigation examines the onset of convective instability in a Casson nanofluid subjected to a helical force, employing both linear and weakly nonlinear analyses. Two distinct methodologies are employed to investigate the linear and weakly nonlinear behaviors. The linear theory is investigated using the Galerkin approach, while the weakly nonlinear theory is explored through multiple scale analysis. In this, the Prandtl and Lewis numbers exhibit no discernible impact on stationary convection. But in the Casson parameter, the nanoparticle Rayleigh number, adjusted diffusivity proportion (modified diffusivity ratio), and helical force parameter destabilize the flow. While in oscillatory convection, it is found that the nanoparticle Rayleigh number, the Lewis number, and helical force parameter stabilize the flow, but the Prandtl number, adjusted diffusivity proportion (modified diffusivity ratio), and the Casson parameter are the destabilizing effects on the flow. Amplitude equation is derived in weakly nonlinear analysis.

This paper presents an efficient computational framework for solving convection-diffusion obstacle problems, designed for convection-dominated regimes while ensuring local and global mass conservation. The method relies on an operator-splitting strategy that decouples the problem into convection and diffusion sub-problems, treated, respectively, in Lagrangian and Eulerian settings. The convective transport is handled by a particle-in-cell method, while the diffusion, formulated as a parabolic variational inequality, is discretized using mixed finite elements. This leads to symmetric saddle-point systems with complementarity conditions, solved efficiently via a primal-dual active set algorithm. To ensure conservative coupling between particles and mesh, a PDE-constrained $��{�\ell �}_{�2�}�$ projection is employed. The effectiveness and performance of the overall approach have been established by rigorous benchmarks with analytical solutions from the literature, covering both structured and unstructured meshes.

In this paper, we investigate the combined nonequilibrium diffusion and low Mach number limits of the compressible Navier–Stokes–Fourier–P1 (NSF–P1) model with general initial data, which arises in the radiation hydrodynamics. Compared to the classical compressible Navier–Stokes–Fourier system, the NSF–P1 model has an asymmetric singular structure caused by the radiation field. To handle these singular terms, we introduce an equivalent pressure and an equivalent velocity to balance the order of singularity and establish the uniform estimates of solutions by designating appropriate weighted norms as well as carrying out delicate energy analysis. We conclude that, for partial general initial data and the strong scattering effect, the NSF–P1 model converges to the system of low Mach number heat-conducting viscous flows coupled with a diffusion equation. We also discuss the variations of the limit equations as the scattering intensity changes. Furthermore, when the scattering effect is sufficiently weak, we can obtain the singular limits of the NSF–P1 model with full general initial data.