A fractional-order generalized Taylor wavelet method for nonlinear fractional delay and nonlinear fractional pantograph differential equations

We study the numerical solutions of nonlinear fractional delay differential equations (DEs) and nonlinear fractional pantograph DEs. We introduce a new class of functions called fractional-order generalized Taylor wavelets (FOGTW). We provide an exact formula for computing the Riemann-Liouville fractional integral operator for FOGTW by using the regularized beta functions. By applying the formula and collocation method, we reduce the given nonlinear fractional delay DEs and nonlinear fractional pantograph DEs to a system of algebraic equations. The FOGTW method together with the exact formula is very efficient for solving the nonlinear fractional delay DEs and nonlinear fractional pantograph DEs and give very accurate results. Several examples are given to demonstrate the effectiveness of the present method.