A note on the fractional Schrodinger differential equations

Abstract

Purpose - The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrodinger differential equation: {idu/dt + Au + integral(t)(0)alpha(s)D(s)(1/2)u(s)ds = f(t), 0 < t < 1, u(0) = 0 in a Hilbert space H with a self-adjoint positive definite operator A under the condition vertical bar alpha(s)vertical bar < M-1/s(1/2) and the first order of accuracy difference scheme for the approximate solution of this initial value problem. Design/methodology/approach - The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem. Findings - The paper finds the stability for the fractional Schrodinger differential equation and the first order of accuracy difference scheme of that equation by applications to one-dimensional fractional Schrodinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrodinger differential equation with Dirichlet conditions. Originality/value - The paper is a significant work on stability of fractional Schrodinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrodinger differential equations.