A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations

A new fractional-order wavelet basis as generalization of the classical Legendre wavelet is defined. The operational matrices both for derivative and fractional derivative in the sense of Caputo for this fractional-order wavelet are derived. Then, a numerical scheme based on these operational matrices and the typical Tau method is proposed for solving some nonlinear fractional differential equations. Illustrative examples show that the present wavelet Tau method is numerically effective and convenient for solving fractional differential equations. Moreover, the obtained results confirm that, in comparison with the classical Legendre wavelet method, the fractional-order wavelet basis is more efficient and accurate for solving fractional differential equations. Error analysis and convergence of the proposed wavelet method are also provided.