Abstract In this work, we propose a numerical method to find an approximation of the variable-order integral of a given function using a generalization of the modified hat functions. First, an operational matrix of the basis functions corresponding to the variable-order integral operator is introduced. Then, using this matrix and an approximation of the given function, we find an approximation of the variable-order integral operator of the function. An error estimate is proved. Two test examples are included to show the efficiency and accuracy of our new technique. Finally, this new technique is used to solve the variable-order differential equations numerically and some illustrative problems are provided to validate the applicability and accuracy of this new scheme.
