Convergence analysis of an <i>L</i>1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations
Mahmoud A. Zaky, Ahmed S. Hendy
Abstract
This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrödinger equations with Riesz space- and Caputo time-fractional derivatives. The L1 finite difference approximation is used for the discretization of the Caputo fractional derivative and the Legendre-Galerkin spectral method is used for the spatial approximation. Additionally, by using a proper form of discrete Grönwall inequality, the scheme is proved to be unconditionally stable and convergent with 2−β accuracy in time and spectral accuracy in space in case of smooth solutions. Finally, some numerical tests are preformed to distinguish the validity of our theoretical results.
Topics & Concepts
MathematicsDiscretizationGalerkin methodFractional calculusLegendre polynomialsMathematical analysisNonlinear systemSpectral methodConvergence (economics)Applied mathematicsSpace (punctuation)Finite differenceFinite difference methodPhilosophyEconomicsEconomic growthLinguisticsPhysicsQuantum mechanicsFractional Differential Equations SolutionsDifferential Equations and Numerical MethodsNumerical methods in engineering