Related papers: A differentiation formula, with application to the…
In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by…
Discretizations of differential equations are often studied through their modified equation. This is a differential equation, usually obtained as a power series, with solutions that exactly interpolate the discretization. By comparing the…
We consider the nonlinear Schr\"odinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha,2}$ for some $\alpha\in (0,1)$. Due to the loss of…
A discrete version of the two-dimensional inverse scattering problem is considered. On this basis, algebraic transformations for the two-dimensional finite-difference Schredinger equation are elaborated.
We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the $d$-dimensional fractional equation is reformulated as a local partial differential equation in…
In this paper it is shown how to solve numerically eigenvalue problems associated to second order linear ordinary differential equations, containing also terms which depend on the variable. A didactic presentation of the Numerov Method is…
We analyze the Schr\"odingerization method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schr\"odingerization technique, introduced in [31], transforms any linear ordinary and partial…
In this note we solve theoretically the Schrodingers differential equation using results based on our previous work which concern semigroup operators. Our method does not use eigenvectors or eigenvalues and the solution depends only from…
An exact discretization method is being developed for solving linear systems of ordinary fractional-derivative differential equations with constant matrix coefficients (LSOFDDECMC). It is shown that the obtained linear discrete system in…
In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and…
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…
We consider a nonlinear parabolic equation of fractional order in space and propose its numerical discretization. The fractional derivative is defined through a functional analytic setting, rather than the traditional definition of…
We apply general difference calculus in order to obtain solutions to the functional equations of the second order. We show that factorization method can be successfully applied to the functional case. This method is equivariant under the…
In the context of the analysis of measured data, one is often faced with the task to differentiate data numerically. Typically, this occurs when measured data are concerned or data are evaluated numerically during the evolution of partial…
In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schr\"odinger equation. The technique is based on the use of first order operators intertwining two difference…
A general formula is presented for any order derivative of Chebyshev polynomials instead of the existing recursive relationship. Hence, the Chebyshev finite difference method is made applicable not only to second order problems but also to…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…
Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple…
We use a discrete multiscale analysis to study the asymptotic integrability of differential-difference equations. In particular, we show that multiscale perturbation techniques provide an analytic tool to derive necessary integrability…