Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

S.C Shiralashetti,A.B Deshi

doi:10.18052/www.scipress.com/bmsa.17.46

Abstract

In this paper, numerical solutions of Riccati and fractional Riccati differential equations are obtained by the Haar wavelet collocation method. An operational matrix of integration based on the Haar wavelet is established, and the procedure for applying the matrix to solve these equations. The fundamental idea of Haar wavelet method is to convert the proposed differential equations into a group of non-linear algebraic equations. The accuracy of approximate solution can be further improved by increasing the level of resolution and an error analysis is computed. The examples are given to demonstrate the fast and flexibility of the method. The results obtained are in good agreement with the exact in comparison with existing ones and it is shown that the technique introduced here is robust, easy to apply and is not only enough accurate but also quite stable.

Highlights

Nonlinear phenomena occur in a wide variety of scientific applications such as plasma physics, solid state physics, fluid dynamics and chemical kinetics [2]
As well-known, a one-dimensional static Schrodinger equation is closely related to Riccati differential equation
We introduce the Haar wavelet operational matrix FHα of integration of the fractional order α is given by f1 for x ∈ [ k, k+0.5)

Summary

Introduction

Nonlinear phenomena occur in a wide variety of scientific applications such as plasma physics, solid state physics, fluid dynamics and chemical kinetics [2]. Lepik [17,18] presented solution for differential and integral by the Haar wavelet method. Bujurke et al [5,6,7] used the Haar wavelet method to establish the solution of Nonlinear Oscillator Equations, Stiff systems, regular SturmLiouville problems etc. Dhawan et al [10] applied the Haar wavelet scheme for the solution of differential equations. Siraj-ul-Islam et al [15] presented numerical solution of second-order boundary-value problems by the Haar wavelets. [14]) much more general values of the order α appear in the equations and one needs to consider numerical and analytical methods to solve differential equations of arbitrary order. Decomposition method is presented for solving fractional Riccati differential equations in [20].

Haar wavelet

Operational matrix of the fractional order integration of the Haar wavelet

First order Riccati differential equations

Fractional order Riccati differential equations

Numerical examples

Conclusions