A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation

Masoumeh Hosseininia, ,Faculty of Mathematics, Yazd University, Yazd, 89195-741, Iran, Mohammad Hossein Heydari, Carlo Cattani, ,Department of Mathematics, Shiraz University of Technology, Shiraz, 71555-313, Iran, ,Engineering School (DEIM), University of Tuscia, Viterbo, 01100, Italy
2018 Discrete and Continuous Dynamical Systems. Series S  
In this study, an efficient semi-discrete method based on the twodimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions for nonlinear variable-order time fractional two-dimensional (2D) Schrödinger equation. First, the variable-order time fractional derivative involved in the considered problem is approximated via the finite difference technique. Then, by help of the finite difference scheme and the theta-weighted method, a recursive algorithm is derived for the
more » ... blem under examination. After that, the real functions available in the real and imaginary parts of the unknown solution of the problem are expanded via the 2D LWs. Finally, by applying the operational matrices of derivative, the solution of the problem is transformed to the solution of a linear system of algebraic equations in each time step which can simply be solved. In the proposed method, acceptable approximate solutions are achieved by employing only a small number of the basis functions. To illustrate the applicability, validity and accuracy of the wavelet method, some numerical test examples are solved using the suggested method. The achieved numerical results reveal that the method established based on the 2D LWs is very easy to implement, appropriate and accurate in solving the proposed model.
doi:10.3934/dcdss.2020295 fatcat:k5prdiesqzaspjhr6hph6ecu3q