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A conservative scheme with optimal error estimates for a multidimensional space-fractional Gross–Pitaevskii equation

Hendy Ahmed S., Macías-Díaz Jorge E.

International Journal of Applied Mathematics and Computer Science

2019

Vol. 29, no. 4

713--723

The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system.

Fractional self adjoint operator Poincaré and Sobolev type inequalities

Anastassiou G. A.

Fasciculi Mathematici

2016

Nr 57

5--24

We present here many fractional self adjoint operator Poincaré and Sobolev type inequalities to various directions. Initially we give several fractional representation formulae in the self adjoint operator sense. Inequalities are based in the self adjoint operator order over a Hilbert space.

A logarithmic Sobolev inequality for one-dimensional multivalued stochastic differential equations

Djehiche B., Eddahbi M., Ouknine Y.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

13--18

We establish a logarithmic Sobolev inequality for a one-dimensional multivalued stochastic differential equation associated with the subdifferential of a convexlower semicontinuous function, using an explicit expression for the Malliavin derivative of the considered process. This result is given under some mild conditions on thecoefficients.