Two competent novel techniques based on two-dimensional wavelets for nonlinear variable-order Riesz space-fractional Schrödinger equations

In this paper, two efficient semi-discrete techniques based on the two-dimensional shifted Legendre and Boubaker wavelets are proposed to devise the approximate solutions for nonlinear variable-order Riesz space fractional Schrödinger equations. Firstly, in order to implement these proposed techniques, new operational integration matrices for variable-order fractional derivatives were derived using wavelet functions vector for two considered cases. The main advantage behind the proposed approach is that the problems under consideration are transformed into the system of algebraic equations. Then, these systems of algebraic equations can be solved simply to obtain the approximate solutions for two considered cases. In addition, in order to determine the convergence analysis and error estimate of the proposed numerical techniques, some useful theorems are also analyzed rigorously. To illustrate the applicability, accuracy and efficiency of the proposed semi-discrete techniques, some concrete examples are solved using the suggested wavelets methods. The achieved numerical results reveal that the proposed methods based on the two-dimensional shifted Legendre and Boubaker wavelets very easy to implement, efficient and accurate.