An efficient computational scheme for solving coupled time-fractional Schrödinger equation via cubic B-spline functions.

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions.

Numerical investigation of the fractional diffusion wave equation with exponential kernel via cubic B-Spline approach.

Splines are piecewise polynomials that are as smooth as they can be without forming a single polynomial. They are linked at specific points known as knots. Splines are useful for a variety of problems in numerical analysis and applied mathematics because they are simple to store and manipulate on a computer. These include, for example, numerical quadrature, function approximation, data fitting, etc. In this study, cubic B-spline (CBS) functions are used to numerically solve the time fractional diffusion wave equation (TFDWE) with Caputo-Fabrizio derivative. To discretize the spatial and temporal derivatives, CBS with θ-weighted scheme and the finite difference approach are utilized, respectively. Convergence analysis and stability of the presented method are analyzed. Some examples are used to validate the suggested scheme, and they show that it is feasible and fairly accurate.