A multistage successive approximation method for Riccati differential equations

Riccati differential equations have played important roles in the theory and practice of control systems engineering. Our goal in this paper is to propose a new multistage successive approximation method for solving Riccati differential equations. The multistage successive approximation method is derived from an existing piecewise variational iteration method for solving Riccati differential equations. The multistage successive approximation method is simpler in terms of computing implementation in comparison with the existing piecewise variational iteration method. Computational tests show that the order of accuracy of the multistage successive approximation method can be made higher by simply taking more number of successive iterations in the multistage evolution. Furthermore, taking small size of each subinterval and taking large number of iterations in the multistage evolution lead that our proposed method produces small error and becomes high order accurate.

Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations

In the present paper, fractional-order generalized Legendre wavelets (FOGLWs) are introduced. We apply the FOGLWs for solving fractional Riccati differential equation. By using the hypergeometric function, we obtain an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FOGLWs. By using this exact formula and the properties of the FOGLWs, we reduce the solution of the fractional Riccati differential equation to the solution of an algebraic system. This algebraic system can be solved effectively. This method gives very accurate results. The given numerical examples support this claim.