Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations

Yanxin Wang,Li Zhu,Tianhe Yin

doi:10.1186/s13662-017-1270-7

Abstract

In this paper, the sine-cosine wavelet method is presented for solving Riccati differential equations. The sine-cosine wavelet operational matrix of fractional integration is derived and utilized to transform the equations to system of algebraic equations. Also, the error analysis of the sine-cosine wavelet bases is given. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. Some examples are included to demonstrate the validity and applicability of the technique.

Highlights

Fractional calculus is an extension of derivatives and integrals to non-integer orders and it has been widely used to model engineering and scientific problems
As Figure shows, our results are in well agreement with the exact solution, convergent regions of solution obtained by Adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), modified homotopy perturbation method (MHPM) and NHPM are small
This comparison indicates that the sine-cosine wavelet method is accurate and is able to solve this nonlinear Riccati differential equation in a very wider region

Summary

Introduction

Fractional calculus is an extension of derivatives and integrals to non-integer orders and it has been widely used to model engineering and scientific problems. Saha et al [ ] used the modified ADM method to solve the fractional Riccati differential equations, Odibat and Momani. [ ], Hosseinnia et al [ ] and Khan et al [ ] used the modified homotopy perturbation method (MHPM) to solve the fractional Riccati differential equations. Our main aim is to solve the Riccati differential equations by using sine-cosine wavelet. The sine-cosine wavelet has been used to solve the integro-differential equation [ ]. The sine-cosine wavelet operational matrix of fractional integration is derived firstly and used to solve the Riccati differential equations, the wavelet operational matrix method is computer oriented.

The Caputo definition of fractal derivative operator is given by

Conclusion