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

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.


In this paper, an efficient numerical method is used to provide the approximate solution 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 approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions 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 numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, efficiency, and applicability of the proposed method.