The linearized Vlasov equation for a plasma system in a uniform magnetic field and the corresponding linear Vlasov operator are studied. The spectrum and the corresponding eigenfunctions of the Vlasov operator are found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and complex. Interestingly, the real eigenvalues are infinitely degenerat...

Relativistic shocks provide an efficient method for high-energy particle acceleration in many astrophysical sources. Multiple shock systems are even more effective and of importance, for example, in the internal shock model of gamma-ray bursts. We investigate the reacceleration of pre-existing energetic particles at such relativistic internal shocks by the first order Fermi process of pitch ang...

We study the initial–boundary value problem of the viscous diffusion equations which include GBBM equations, Sobolov–Galpern equations and some standard nonlinear diffusion equations as special cases. By using the integral estimate method and the eigenfunction method we prove that when the nonlinear terms satisfy some conditions the solutions of the problem decay to zero according to the expone...

Abstract We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. first prove existence eigenfunctions. then set up a framework for to discretize equation defining eigenfunctions, obtain numerical approximations eigenvalue eigenfunction using finite element method. Finally, we show that conve...

The boundary-value problem for the linear horizontally-forced sloshing problem can be solved using two different classes of eigenfunction expansions. The first will be referred to as the ”cosine” expansion since the organizing centre is a cosine series in the x−direction (the horizontal direction), and the second is called the “vertical eigenfunction expansion” since the organizing centre is a ...

Abstract Inspired by the squared eigenfunction symmetry constraint, we introduce a new τk-flow by “extending” a specific tn-flow of discrete KP hierarchy (DKPH). We construct extended discrete KPH (exDKPH), which consists of tnflow, τk-flow and tn evolution of eigenfunction and adjoint eigenfunctions, and its Lax representation. The exDKPH contains two types of discrete KP equation with self-co...

These notes describe the multiconfigurational self-consistent-field (MCSCF) method, which is a general approach for describing chemical systems in which a single electron configuration is no longer an adequate description of the electronic structure. This commonly happens in reactions that break or form chemical bonds, diradicals, and metals of the first transition row. A single-determinant, re...