Mathematical Methods in the Applied Sciences最新文献

We show a quantitative analysis of the perturbed electric field of a 3D anisotropic conductive rod geometry embedded in homogeneous background, which is an extension of the work in 2D case. The solution to the perturbed problem is presented by layer potential techniques, and dedicated asymptotic analysis is employed for characterization of the density function. The asymptotic result shows that near the high curvature boundary of the rod, the electric field is much strong, compared with other parts adjacent to the rod.

In this paper, we propose a new numerical method to simulate acoustic scattering problems in two-dimensional periodic structures with non-periodic incident fields. Applying the Floquet-Bloch transform to the scattering problem yields a family of quasi-periodic boundary value problems dependent on the Floquet-Bloch parameter. Consequently, the solution of the original scattering problem is written as the inverse Floquet-Bloch transform of the solutions to these boundary value problems. The key step in our method is the numerical approximation of this integral transform by a quadrature rule with a nonuniform choice of quadrature points adapted to the regularity of the family of quasi-periodic solutions. This achieved by a graded subdivision of the full interval for the Floquet-Bloch parameter and applying a Gauss-Legrendre quadrature rule on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.

This paper investigates a stochastic two predators–one prey system with ratio-dependent functional response under regime switching. The stochastic extinction of species and the existence of ergodic stationary distribution for the system are established, and the transition probability of the solution converging to the stationary distribution also is obtained. To illustrate our theoretical results, the numerical examples and simulations are presented. Our findings also demonstrate that the stationary distribution and extinction of species for the stochastic two predators–one prey system are affected by random perturbations, leading to an imbalance in ecology.

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as temporal networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff-type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first-order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self-adjoint boundary value problem on the tree for some second-order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree.

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely, the probabilistic degenerate Whitney numbers of the second kind associated with $��Y�$ and the probabilistic degenerate Dowling polynomials associated with $��Y�$. We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely, the probabilistic degenerate $��r�$-Whitney numbers of the second kind associated with $��Y�$ and the probabilistic degenerate $��r�$-Dowling polynomials associated with $��Y�$, and get similar results to the aforementioned numbers and polynomials.

In this paper, we consider the following fourth-order equation of Kirchhoff type

The main aim of this article is to focus on the fundamental properties of the gyrator transform on the Schwartz spaces. A more generalization Hörmander symbol class is introduced and studied the boundedness properties of pseudo-differential operators associated with the gyrator transform. Integral representation and kernel of the pseudo-differential operators are derived. It is shown that pseudo-differential operators satisfies certain norm inequalities on Sobolev space. As an application, we have successfully applied the gyrator transform to solve heat equation and Fredholm integral equation.

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic-power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell-shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

In this paper, we introduce an innovative hybrid discrete-continuum model that provides a comprehensive framework for understanding the intricate interactions between tumor cells, immune cells, and the effects of immunotherapy. This study distinguishes itself by addressing the limitations of traditional diffusion models, which often fail to capture the irregular and nonlocal movements of cells within the complex tumor microenvironment. To overcome these challenges, we employ fractional time derivatives and the fractional Laplacian operator, offering a more accurate representation of anomalous diffusion processes that are critical in cancer dynamics. Our research begins with a rigorous mathematical analysis, where we establish the global existence of a unique mild solution, laying a solid theoretical foundation for the model. A key innovation of our study is the introduction of the “invasion threshold,” a critical parameter inspired by the next-generation operator used in epidemiological modeling. This threshold provides a powerful tool for determining the existence and stability of equilibrium points, offering deep insights into the conditions that either promote or inhibit tumor growth under immunotherapeutic interventions. By integrating a hybrid discrete-continuum approach, we capture the individual behavior of each cell, allowing for a more detailed exploration of the dynamic interplay within the tumor microenvironment. This model not only advances our understanding of tumor-immune interactions but also holds potential for informing more effective therapeutic strategies, making a significant contribution to the field of mathematical oncology.