We study the Green’s function for the linearized Boltzmann equation. For the short-time period, the Green’s function is dominated by the particle-like waves; and for large-time, by the fluid-like waves exhibiting the weak Huygens principle. The fluidlike waves are constructed by the spectral analysis and complex analytic techniques, making uses of the rotational symmetry of the equation in the ...

Since the advent of Taylor Series, polynomial methods have been used to solve di↵erential equations and learn the properties of di↵erential equations. There are many polynomial methods for solving di↵erential equations and understanding dynamical systems. For example, Taylor polynomials, Chebyshev Polynomials and Adomian Polynomials are used to generate approximate solutions. Automatic di↵erent...

in this paper, we implemented the modified iteration perturbation method (mipm) for approximating the periodic behavior of a tapered beam. this problem is formulated as a nonlinear ordinary differential equation with linear and nonlinear terms. the solution is quickly convergent and does not need to complicated calculations. comparing the results of the mipm with the exact solution shows that t...

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces b H s (R) defined by the norm ‖v0‖ b Hr s (R) := ‖〈ξ〉 b v0‖Lr′ ξ , 〈ξ〉 = (1 + ξ) 1 2 , 1 r + 1 r = 1. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ r > 1, s ≥ 2j−1 2r . The proof uses an appropriate variant of the Fourier restriction norm method. A coun...

Fractional-order differential equations are powerful tools for the representation of dynamic systems that exhibit long-term memory effects. The verified simulation such system models with help interval allows computation guaranteed enclosures domains reachable pseudo states over a certain finite time horizon. In previous work author, an iteration scheme—derived on basis Picard iteration—was pub...

In this paper, the optimal switching problem is proposed for one-dimensional reflected backward stochastic differential equations (BSDEs, for short) where the generators, the terminal values, and the barriers are all switched with positive costs. The value process is characterized by a system of multi-dimensional reflected BSDEs with oblique reflection, whose existence and uniqueness is by no m...

In this paper, we propose a modified prediction–correction method for structured monotone variational inequalities. This method is based on a reformulation of the normal prediction–correction method. Each iteration of the new method contains two predictions and a correction. At each iteration a new strategy is applied to obtain a proper step length. All the computing processes are easily implem...

Abstract. A simple numerical solution procedure – namely the method of lines combined with an off-the-shelf ordinary differential equation (ODE) solver was shown in previous work to provide efficient, mass-conservative solutions pressure-head form Richards' equation. We implement such a our model openRE. developed novel quantify boundary fluxes that reduce water balance errors without negative ...