In this paper, the existence of solutions of fractional differential equations with integral boundary conditions is investigated. The upper and lower solutions combined with monotone iterative technique is applied. Problems of existence and unique solutions are discussed.

Abstract. Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggeredgrid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in * Corresponding author.

No abstract

An electron thermal transport theory based on the super-Gaussian electron distribution function f0∝e−vm is investigated for magnetized laser plasmas in order to obtain accurate transport coefficients used in the radiation hydrodynamic codes. It is found that the super-Gaussian distribution suppresses the diffusive heat flow and the Righi-Leduc heat flow. The diffusive heat flow and Righi-Leduc heat flow can be suppressed by as much as 50% and 75% under the typical hohlraum plasma condition, respectively. The super-Gaussian distribution introduces isothermal heat flows associated with the gradients of electron density and the super-Gaussian exponential factor m. And the isothermal heat flows induce the anomalous Nernst effects. Moreover, the self-generated magnetic field in laser plasmas can be generated not only by the thermalelectric effect but also by the nonparallel gradients of electron temperature and the super-Gaussian exponential factor m, the nonparallel gradients of electron density, and the super-Gaussian exponential factor m.

No abstract