Solving of partial differential equations with distributed order in time using fractional-order Bernoulli-Legendre functions

In this paper, an effcient numerical method is used to provide the approximate so- lution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by ap- proximating functions that appear in the DFPDEs by fractional-order Bernoulli func- tions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the nu- merical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, effciency, and applicability of the proposed method.