Nonlinear Gross-Pitaevskii-type models are frequently seen in the fields of Bose-Einstein condensation and quantum mechanics. We derive error estimates for the Lie-Trotter operator splitting spectral method for semiclassical sub-diffusive Gross-Pitaevskii equation in the unbounded domain or with the periodic boundary condition. After establishing a priori estimates for the analytic solution in fractional Sobolev space, the local error estimates for the Lie-Trotter splitting operator method are derived. The related estimates for the Lie commutator of nonlocal linear operator and nonlinear operator play key roles in deriving the local error estimates. We then obtain the global error bounds for the fully discrete scheme based on the space approximation with mapped Chebyshev spectral-Galerkin methods in the case of the unbounded domain and with Fourier spectral methods in the case of the periodic boundary condition. Especially, their convergence orders with respect to the small (scaled) Planck constant ε are obtained for the first time under the framework of Wentzel-Kramers-Brillouin analysis. Numerical experiments verify and complement our theoretical results.

In this paper, we propose and develop a novel hue-preserving color image restoration model. In the proposed model, saturation-value total variation is used as the regularization term. In order to reduce hue loss during regularization, the huepreserving technique is incorporated into the proposed energy functional. We then rewrite the proposed model equivalently into a three-block separable convex version and apply an alternating direction method of multipliers based algorithm to solve the proposed equivalent model numerically. We also give the convergence analysis of the proposed algorithm. Numerical examples are presented to demonstrate that the performance of the proposed restoration model is better than that of other testing methods in terms of visual quality and some criteria such as peak signal-to-noise ratio, structure similarity, and spatial-CIELAB color error.

Biodiesel, a sustainable and renewable energy source, is a promising alternative to fossil fuels. The transesterification process of biodiesel production effectively captures memory effects in reaction kinetics. In this study, we developed two fractional order models of the chemical catalytic transesterification reaction to explore the memory effects of the reaction kinetics utilizing two different non-singular kernel methods: Caputo-Fabrizio and Atangana-Baleanu in the Caputo sense. We compared the results with experimental data of biodiesel production and demonstrated the existence and uniqueness of the solution for the fractional system. A sensitivity analysis is performed using the Latin hypercube sampling method to evaluate the impact of various parameters on biodiesel production, followed by the computation of partial rank correlation coefficients based on Pearson’s correlation coefficient. We exhibit the dynamic behavior of all reactants corresponding to these fractional models with the variation of fractional order and the memory rate parameter. Additionally, we display the memory effect through the surface plots for biodiesel production by varying fractional order, molar ratio, and ultrasound frequency. Our numerical comparison with experimental data identifies the fractional-order value for the best fit of biodiesel production and can be increased by applying optimal control on ultrasound frequency.