Studies in Applied Mathematics最新文献

In the present work, we address some aspects related to the numerical solution of the Poisson equation within the framework of the Smoothed Particle Hydrodynamics (SPH). In this case, the Poisson equation is solved using nonlocal, kernel-based discretizations of the differential operators (i.e., gradient, divergence, and Laplacian operators) defined on scattered particle sets without topological connections. This paper clarifies two distinct sources of pressure artifacts that are often conflated in the SPH literature: (i) checkerboard-like spurious pressure modes caused by discrete operator incompatibility and pressure-velocity decoupling, and (ii) loss of accuracy due to particle disorder and kernel support truncation near boundaries. We propose diagnostics and procedures that separate these mechanisms and use representative benchmark tests to compare common SPH Poisson/Laplacian formulations in terms of robustness, convergence, and sensitivity to disorder and boundary deficiency. The resulting perspective reconciles conflicting claims and provides a theoretical guidance for practical applications.

We present a new class of models for assessing the cell dynamics characterizing muscular dystrophies. The proposed approach comprises a system of integro-differential equations for the statistical distributions, over a large patient cohort, of the densities of muscle fibers and immune cells implicated in muscle inflammation, degeneration, and regeneration, which underpin disease development. Considering an appropriately scaled version of this model, we formally derive, as the corresponding mean-field limit, a system of Fokker–Planck equations, from which we subsequently derive, as a macroscopic model counterpart, a system of differential equations for the mean densities of muscle and immune cells in the cohort of patients and the related variances. Then, we study long-time asymptotics for the mean-field model by determining the quasi-equilibrium cell distribution functions, which are in the form of probability density functions of inverse Gamma distributions, and proving the long-time convergence to such quasi-equilibrium distributions. The analytical results obtained are illustrated by means of a sample of results of numerical simulations. The modeling approach presented here has the potential to offer new insights into the balance between degeneration and regeneration mechanisms in the progression of muscular dystrophies, and provides a basis for future extensions, including the modeling of therapeutic interventions.

High-frequency seismic wave descriptions are essential for fundamental geophysical applications, including earthquake hypocenter location, magnitude estimation, and traveltime–amplitude tomography. These descriptions rely on two coupled PDEs: the eikonal equation (governing traveltimes) and the transport equation (governing amplitudes). Solving both equations together provides sharper subsurface images and more reliable magnitude/location estimates. Although the transport equation has been transformed into an ODE in the classical literature, this reduction relies crucially on the isotropy assumption (i.e., the wavefront normal is parallel to the ray direction). An explicit, characteristic-based ODE formulation analogous to the one used for the eikonal equation—and valid without imposing isotropy—has been lacking. This study addresses that limitation by deriving a unified first-order ODE framework, using the principles of contact geometry, that simultaneously transforms both the eikonal and transport equations into coupled ODE systems. This formulation allows ray paths, traveltimes, and amplitudes to be solved together and explicitly as functions of ray time. In addition, we introduce a general methodology for computing the traveltime Laplacian—a key quantity in the amplitude equation—without assuming isotropy. We first apply the chain rule to the ray-coordinate mappings to derive an exact analytical expression for the Laplacian. We then provide a geometric interpretation that links this analytical expression to the local structure of the ray tubes in both the physical and momentum spaces, offering a robust way to compute amplitude evolution in anisotropic media. To validate the framework, we solve the coupled ray-tracing and amplitude-tracing ODEs analytically for benchmark models. For a homogeneous medium, our method analytically derives the classical amplitude decay proportional to $��1�/�t�$ in three-dimensional space and $��1�/�\sqrt{�t�}�$ in two-dimensional space, explicitly linking our formulation to the physical principle of geometric spreading and energy conservation. For a linearly inhomogeneous and isotropic medium, we derive a new closed-form analytical solution for amplitude evolution along the ray. These analytical solutions demonstrate the accuracy and computational power of our unified framework.

A sharp upper bound is established for the amplitudes of finite-gap solutions of the focusing modified Korteweg–de Vries equation. The proof shows that the optimization can be carried out entirely within the finite-dimensional hierarchy, without explicit integration of the finite-gap solution. The maximal amplitude is expressed in terms of the roots of the invariant polynomial associated with the finite-gap solution. An analogous formula is established for a bounded class of solutions of the defocusing modified Korteweg–de Vries equation. Both upper bounds are sharp and are attained by explicit initial data.