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This study discusses a class of linear systems of fractional differential equations with non-constant coefficients, with a particular focus on problems exhibiting highly oscillatory and non-smooth behavior. We first establish the regularity…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
This paper is concerned with the study of Green's functions for one dimensional diffusions with constant diffusion coefficient and linear time inhomogeneous drift. It is well know that the whole line Green's function is given by a Gaussian.…
We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize…
In this paper, a method to solve functionally commutative time- dependent linear homogeneous differential equation is discussed. We apply this technique to solve some dynamical quantum problems.
We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is…
We propose an iterative finite element method for solving non-linear hydromagnetic and steady Euler's equations. Some three-dimensional computational tests are given to confirm the convergence and the high efficiency of the method.
An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the…
Given a spatially dependent mass we obtain the two-point Green's function for exactly solvable nonrelativistic problems. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrodinger…
We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
A new method for the stabilization of the sign problem in the Green Function Monte Carlo technique is proposed. The method is devised for real lattice Hamiltonians and is based on an iterative ''stochastic reconfiguration'' scheme which…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
We analyze solvability of a special form of distributed order fractional differential equations within the space of tempered distributions supported by the positive half-line.
This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward…
A new scheme has been proposed to solve the B.E. condenstates in terms of Green's function approach. It has been shown that the radial wave function of two interacting atoms, moving in a common harmonic oscillator potential modified by an…