Related papers: Regularity for time fractional wave problems
We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction…
In this work, we propose and analyse a weak Galerkin method for the electrical impedance tomography based on a bounded variation regularization. We use the complete electrode model as the forward system that is approximated by a weak…
In this study, we consider a class of non-autonomous time-fractional partial advection-diffusion-reaction (TF-ADR) equations with Caputo type fractional derivative. To obtain the numerical solution of the model problem, we apply the…
We provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the…
These lecture notes introduce the Galerkin method to approximate solutions to partial differential and integral equations. We begin with some analysis background to introduce this method in a Hilbert Space setting, and subsequently…
Variable-order time-fractional wave equations provide a flexible model for wave phenomena with evolving memory effects and anomalous temporal dynamics. Their numerical approximation is challenging because the variable-order fractional…
This article presents a new primal-dual weak Galerkin finite element method for the div-curl system with tangential boundary conditions and low-regularity assumptions on the solution. The numerical scheme is based on a weak variational form…
In this paper, we propose a simple numerical algorithm based on the weak Galerkin (WG) finite element method for a class of fourth-order problems in fluorescence tomography (FT), eliminating the need for stabilizer terms required in…
We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear…
Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically…
An ill-posed Cauchy problem for the wave equation is considered: the solution is to be determined by the Cauchy data on some part of the time-space boundary. By means of Fourier method we obtain a regularization algorithm for this problem,…
We develop an approach for simulating acousto-elastic wave phenomena, including scattering from fluid-solid boundaries, where the solid is allowed to be anisotropic, with the Discontinuous Galerkin method. We use a coupled first-order…
This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k-1, and k(k>=1) for the velocity,…
This article introduces a weak Galerkin (WG) finite element method for linear elasticity interface problems on general polygonal/ployhedra partitions. The developed WG method has been proved to be stable and accurate with optimal order…
We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward…
The discontinuous Galerkin (DG) method is an established method for computing approximate solutions of partial differential equations in many applications. Unlike continuous finite elements, in DG methods, numerical fluxes are used to…
In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth…
We propose efficient and parallel algorithms for the implementation of the high-order continuous time Galerkin method for dissipative and wave propagation problems. By using Legendre polynomials as shape functions, we obtain a special…
In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value…
A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the…