Bounded Solutions of Second Order of Accuracy Difference Schemes for Semilinear Fractional Schrödinger Equations

Abstract

AbstractThe present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation$$\begin{array}{} \displaystyle i\!\frac{du}{dt}+Au = \int\limits_{0}^{t}f\left( s,D_{s}^{\alpha }u(s)\right) ds,\, \, \, 0 \lt t \lt T,\, u\left( 0\right) = 0 \end{array} $$in a Hilbert space H with a self-adjoint positive definite (SAPD) operator A. Stable difference schemes (DSs) have significant interest in investigations of fractional partial differential equations. The main theorem concerns the existence and uniqueness of the uniformly bounded solutions (UBSs) with respect to step time of second order of accuracy DSs for this semilinear fractional Schrödinger differential problem. In practice, existence and uniqueness theorems for a UBS of the one-dimensional initial boundary value problem (BVP) with nonlocal condition and multi-dimensional problem with local condition on the boundary are proved. Numerical results and explanatory illustrations are presented to show the validation of the theoretical results.