An efficient algorithm for solving the variable-order time-fractional generalized Burgers' equation

A numerical scheme based on the Haar wavelets coupled with the nonstandard finite difference scheme is presented to solve the variable-order time-fractional generalized Burgers' equation (VO-TFGBE). In the proposed technique, firstly, we approximate the time-fractional derivative by the nonstandard finite difference (NSFD) scheme and convert the VO-TFGBE into the nonlinear ordinary differential equation at each time level, and then we apply the Haar wavelet series approximation for the space derivatives. The proposed technique requires only one dimensional Haar wavelet approximation with a significantly smaller number of Haar coefficients to solve time-dependent partial differential equations. The presence of the NSFD scheme provides flexibility to choose different denominator functions and also provides high accuracy for large temporal step sizes. The convergence and stability of the proposed technique are discussed. Some test examples are solved to demonstrate the effectiveness of the technique and validate the theoretical results.