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This paper presents an efficient numerical method for solving the distributed fractional differential equations (FDEs). The suggested framework is based on a hybrid of block-pulse functions and Taylor polynomials. For the first time, the Riemann–Liouville fractional integral operator for the hybrid of block-pulse functions and Taylor polynomials has been derived directly and without any approximations. By taking into account the property of this operator, the problem under consideration is converted into a system of algebraic equations. The present method can be applied to both linear and nonlinear distributed FDEs. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed hybrid functions. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique, and a comparison is made with the existing results.

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.

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