This paper deals with the approximation of discrete probability measure-valued data by a new subdivision scheme. Its construction relies on a coupling between linear subdivision and optimal transport. A mathematical analysis is performed to study its convergence. Two test cases are finally described to emphasize its capability: the first one is related to point cloud interpolation while the second one is a first attempt in the framework of image approximation.
In this paper, the indirect signal production system with nonlinear transmission is considered in a bounded smooth domain () associated with homogeneous Neumann boundary conditions, where satisfies with . It is known from [1] that the system possesses a global bounded solution if when . In the case and if we consider superlinear transmission, no regularity of or can be derived directly. In this work, we show that if , the solution is global and bounded via an approach based on the maximal Sobolev regularity.
The resonant Schrödinger equation of the third order is studied. The Painlevé test for nonlinear partial differential equations is used to determine integrability of equation. It is shown that the necessary condition for integrability of partial differential equations by the inverse scattering transform is fulfilled at certain parameter restriction. Analytical solutions in the form of periodic and solitary wave are presented.
This paper deals with a fully parabolic chemotaxis system in a bounded domain with smooth boundary, where Under the condition it is shown that the corresponding system possesses a globally bounded classical solution, which extends the previous boundedness result from to <
The weighted essentially non-oscillatory (WENO) interpolation-based schemes (the alternative WENO (AWENO) scheme and the explicit weighted compact nonlinear (WCNS-E) scheme) are limited to the fifth-order despite the sixth-order Taylor expansion of the numerical flux used. This order reduction is due to the fifth-order accuracy inherent in the WENO interpolation. We investigate the perturbed WENO interpolation with a free parameter and affine-invariant WENO weights to recover sixth-order accuracy in smooth regions. A cutoff function of the free parameter with a threshold, determined by approximate dispersion relation analysis, is applied to enhance the ENO property. The proposed schemes perform better in accuracy, dissipation, resolution, shock-capturing, and efficiency in 1D and 2D benchmark problems.
This paper addresses the stability of a coupled system that consists of a conservative Rayleigh beam and a damped elastic string, the latter being equipped with frictional damping. By estimating the growth of the resolvent along the imaginary axis, we employ the frequency domain method to demonstrate that the coupled Rayleigh beam–string system exhibits polynomial stability, characterized by a decay rate . Furthermore, we establish the optimality of the decay rate, , through a rigorous spectral analysis. Finally, a numerical example is present to validate the theoretical findings.
We consider Biot’s equations of poroelasticity where the development of viscous boundary layers in the pores is allowed for by using a dynamic permeability convolution operator in the time domain. This system with memory effects is also referred to as the dynamic Biot–Allard model. We use a series representation of the dynamic permeability in the frequency domain to rewrite the equations in the time domain in a coupled system without convolution integrals, which is also suitable for designing efficient numerical approximation schemes. The main result here is the well-posedness of the system, rewritten in evolutionary form, which is proved by an abstract theory for evolutionary problems.
A single species spatial population model with nonlocal memory diffusion and distributed delay is formulated. Taking the memory delay and distributed delay as bifurcation parameters, we study the stability of the positive homogeneous steady state and bifurcation behaviors. The results show that the joint effect of distributed delay and memory delay induces the occurrence of multiple stability switches, which is impossible for the system without the distributed delay.
The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and -norm and obtain uniform convergence of the CDG method for the first time.
In this paper, a semi-discrete matrix coupled dispersionless system is presented. A Lax pair is proposed, and the Darboux transformation is employed to construct exact solutions to the semi-discrete matrix coupled dispersionless system. These solutions numerically exhibit a variety of exact phenomena, including periodic patterns, breathers, rogue waves, and bright and dark solitons.
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