Related papers: Solving the Dirac equation with nonlocal potential…
An approximate method is proposed to solve position dependent mass Schr\"odinger equation. The procedure suggested here leads to the solution of the PDM Schr\"odinger equation without transforming the potential function to the mass space or…
We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous…
A nonlocal quantum model is presented for calculating the atomic dielectric response to a strong laser electric field. By replacing the Coulomb potential with a nonlocal potential in the Schrodinger equation, a 3+1D calculation of the…
We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in one extra dimension. It is shown that this exactly solvable model can be obtained from a Schroedinger…
In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier…
Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schr\"odinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and…
We consider the three-dimensional Dirac equation in spherical coordinates with coupling to static electromagnetic potential. The space components of the potential have angular (non-central) dependence such that the Dirac equation is…
Using an explicit Euler substitution it was obtained a system of differential equations, which can be used to find the solution of time-dependent 1-dimentional Schr\H{o}dinger equation for a general form of the time-dependent potential.
We develop a quantum algorithm for solving high-dimensional time-fractional heat equations. By applying the dimension extension technique from [FKW23], the $d+1$-dimensional time-fractional equation is reformulated as a local partial…
In the incremental knapsack problem ($\IK$), we are given a knapsack whose capacity grows weakly as a function of time. There is a time horizon of $T$ periods and the capacity of the knapsack is $B_t$ in period $t$ for $t = 1, \ldots, T$.…
The asymptotic iteration method (AIM) is applied to obtain highly accurate eigenvalues of the radial Schroedinger equation with the singular potential V(r)=r^2+\lambda/r^\alpha (\alpha,\lambda> 0) in arbitrary dimensions. Certain…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…
We give a study of some molecular vibration potentials by solving the D-dimensional Schrodinger equation using the asymptotic iteration method (AIM). The eigenvalue values obtained by the AIM are found to agree with analytic solutions. The…
In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr\"odinger equation with the…
We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular,…
This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via…
In this article, we present a simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution and…
In this work, aiming to solve numerically the Schr\"odinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schr\"odinger equation with potentials of the form V(x) + deltap(x),…
We consider the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By…
The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant…