Related papers: The continuous dependence for the Hal-MHD equation…
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…
This paper investigates the non-resistive compressible magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. We establish the global existence and stability of classical solutions for initial data sufficiently close to a constant…
The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations. Recently it has become common to study…
We consider the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] where $n\in \mathbb N$, $0<s<\min \{ n,\; 1+n/2\} $, $0<b<\min \{…
Hall magnetohydrodynamics (MHD) properties near a two-dimensional (2D) X-type magnetic neutral line in the steady state are considered via heuristic and rigorous developments. Upon considering the steady-state as the asymptotic limit of the…
We establish a complete Widder Theory for the fractional fast diffusion equation. Our work focuses on nonnegative solutions satisfying a certain integral size condition at infinity. We prove that these solutions possess a Radon measure as…
This article represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and although the equation has formal similarities with the…
The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example,…
We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: $D_t^\alpha u(t, x)=\textit{B}u+u\cdot W^H$, where $D_t^\alpha$ is the fractional…
We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.
This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at subboundary. Uniqueness result is obtained by using the…
We consider the homogenization for time-fractional diffusion equations in a periodic structure and we derive the homogenized time-fractional diffusion equation. Then we discuss the determination of the constant diffusion coefficient by…
Whether or not the solution to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system starting from smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on…
We study the numerical evaluation of the integral fractional Laplacian and its application in solving fractional diffusion equations. We derive a pseudo-spectral formula for the integral fractional Laplacian operator based on fractional…
In this work we study arbitrary-order hybrid discretizations of Friedrichs systems. Friedrichs systems provide a framework that goes beyond the standard classification of partial differential equations into hyperbolic or elliptic, and are…
In this paper, we give a rigorous justification of the deviation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
Fundamental rules and definitions of Fractional Differintegrals are outlined. Factorizing 1-D and 2-D Helmholtz equations four fractional eigenfunctions are determined. The functions exhibit incident and reflected plane waves as well as…
Turbulence in the magnetized plasma is well understood to be the consequence of wave interactions. When the Hall effect is added to the minimum magnetohydrodynamics (MHD), the MHD waves become dispersive and different nonlinear interactions…
We investigate a role of the Hall-effect in the current sheet evolution and onset of the secondary tearing (plasmoid) instability in the framework of the incompressible resistive Hall-magnetohydrodynamics (MHD). The model under…