Mathematical Methods in the Applied Sciences最新文献

In this article, we introduce a new nonlocal operator $��{�H�}^{�\alpha �}�$ defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator $��{�R�}^{�\alpha �}�$ is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the $��Z�$-transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag-Leffler sequences.

A filtration system comprising a Biot poroelastic solid coupled to an incompressible Stokes free-flow is considered in 3D. Across the flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. In the inertial, linear, and non-degenerate case, the hyperbolic-parabolic coupled problem is posed through a dynamics operator on a chosen energy space, adapted from Stokes-Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. The generation of a strongly continuous semigroup for the dynamics operator is obtained via a non-standard maximality argument. The latter employs a mixed-variational formulation in order to invoke the Babuška-Brezzi theorem. The Lumer-Philips theorem then yields semigroup generation, and thereby, strong and generalized solutions are obtained. For the linear dynamics, density obtains the existence of weak solutions; we extend to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial linear Biot-Stokes filtration system, we provide a clear elucidation of strong solutions and a construction of weak solutions, as well as their regularity through associated estimates.

We study the inverse problem for a differential equation of $��2�b�$-order with the Caputo fractional derivative over time and Schwartz-type distribution in its right-hand side. The generalized solution of the Cauchy problem for such an equation, space-dependent part of a source, and a time-dependent reaction coefficient in the equation are unknown. We find sufficient conditions for unique local in time solvability of the inverse problem under time- and space-integral overdetermination conditions.

In this paper, we apply the second-order averaging theory for obtaining an explicit expression of the small amplitude periodic solution that bifurcates from a zero-Hopf equilibrium point of a tritrophic food chain model. This model considers logistic growth rate for the lowest trophic level, Holling functional responses type III for the middle and for the highest level. We first prove that this model has a zero-Hopf equilibrium point, after we show that from this equilibrium bifurcates a small limit cycle, and finally, we provide the explicit expression of the first two terms in the power series of this limit cycle. These differential systems for which the equilibrium point is non hyperbolic are not easy to study, in particular, if the equilibrium is zero-Hopf. As far as we know, this is the first time that the averaging theory has been used to exhibit the first and second terms of the power series expansion of the analytical expression of a limit cycle that bifurcates from a zero-Hopf equilibrium in the food chain models.

We study the positive solutions for a class of Kirchhoff-type problems involving the nonlinearity $��\lambda �f�\left(�x�\right)�{�u�}^{�p�-�1�}�+�g�\left(�x�\right)�{�u�}^{�5�}�(�2�<�p�<�4�)�$ on a bounded domain. The major difficulty of such problems is the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition; especially, we cannot use Pohozaev's identity directly since our domain is bounded and the weight potentials are not $��{�C�}^{�1�}�$-smoothness. Another difficulty is caused by the absence of compactness as the appearance of the critical Sobolev growth. In this work, we shall combine the Nehari manifold and some novel analytical skills to overcome the above difficulties and then obtain some existence results. Furthermore, we show some asymptotic behaviors of the solutions.

Chemical reactions are embedded in spatiotemporal fluctuations instead of a constant environment. Here, we aimed to assess reaction–diffusion (RD) with dichotomous noise-controlling system parameters in the Brusselator and examine the effect of these fluctuations on the dynamic behavior of chemical reactions. By performing a multiscale perturbation analysis, we demonstrated that the correlated noise can broaden the Turing region even if molecular memory (autocorrelation time) exists. However, for small noise, short-term memory promotes Turing instability. The instability of the Brusselator is determined by the noise strength, which belongs to the optimal region if the diffusion coefficient is fixed. Turing pattern selection and stability are also governed by the dynamic character of the amplitude equation, and the entire Turing instability region shifts to the right in the phase space with noise perturbation. Finally, numerical simulations validate the theoretical derivation that correlated noise can amplify Turing pattern formation to maintain distinct patterns.

This research investigates 2D benchmark flow around a circular cylinder, utilizing the incompressible Navier–Stokes equations alongside the continuity and energy equations. The numerical solution is achieved through finite element discretization for space variable, combined with a second-order Crank–Nicolson scheme for time integration. The computational results are derived using the FEATFLOW finite element based library package. Our study focuses on the dimensionless form of the flow equations and examines three key dimensionless parameters: drag, lift, and pressure drop. Upon applying finite element method (FEM) discretization, the system is converted into a set of linear or nonlinear ordinary differential equations or algebraic equations for steady-state scenarios. We then apply the Newton–Raphson method as the outer nonlinear solver, and the multigrid method for efficiently resolving the linear subproblems. To ensure numerical accuracy, we evaluated the $��{�L�}_{�2�}�$ and $��{�H�}^{�1�}�$ errors, confirming that the experimental order of convergence matches the theoretical predictions. Flow profiles were both graphically represented and tabulated, offering a detailed understanding of the simulation results.