Application of fractional-order Bernoulli functions for solving fractional Riccati differential equation

Ordokhani, Yadollah; Rahimkhani, Parisa; Babolian, Esmail

doi:10.22075/ijnaa.2017.1476.1379

Application of fractional-order Bernoulli functions for solving fractional Riccati differential equation

Document Type : Research Paper

Authors

Yadollah Ordokhani ¹

Parisa Rahimkhani ²

Esmail Babolian ³

¹ Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

² Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran National Elites Foundation, Tehran, Iran

³ Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

10.22075/ijnaa.2017.1476.1379

Abstract

In this paper, a new numerical method for solving the fractional Riccati differential equation is presented. The fractional derivatives are described in the Caputo sense. The method is based upon fractional-order Bernoulli functions approximations. First, the fractional-order Bernoulli functions and their properties are presented. Then, an operational matrix of fractional order integration is derived and is utilized to reduce the under study problem to a system of algebraic equations. Error analysis included the residual error estimation and the upper bound of the absolute errors are introduced for this method. The technique and the error analysis are applied to some problems to demonstrate the validity and applicability of our method.

Keywords

Fractional Riccati differential equation

Fractional-order Bernoulli functions

Caputo derivative

Operational matrix

Collocation method