FractionalRiccatiEquation

Based on the gathered information, the approach to solving fractional Riccati equations using shifted Jacobi polynomials can be outlined as follows:

1. Representation of Unknown Function: The unknown function $y(x)$ in the fractional Riccati equation is expressed as a series expansion using shifted Jacobi polynomials:

$$y(x) = \sum_{n=0}^N a_n P_n^{(\alpha, \beta)}(x)$$

where $a_n$ are the coefficients to be determined, $P_n^{(\alpha, \beta)}(x)$ are the shifted Jacobi polynomials, and $N$ is the truncation number (source).

2. Operational Matrix: An operational matrix, which could be an integral or derivative operational matrix, is defined for the shifted Jacobi polynomials. This matrix facilitates the handling of fractional derivatives and integrals in the Riccati equation:

$$\text{Operational Matrix} = \left[ \begin{array}{cccc} \ldots & \ldots & \ldots & \ldots \\\ \ldots & \ldots & \ldots & \ldots \\\ \end{array} \right]$$

Specific elements of this matrix are derived based on the properties of the shifted Jacobi polynomials (source).

3. Transformation to Algebraic System: The fractional Riccati equation is transformed into a system of algebraic equations using the operational matrix and the series expansion of the unknown function. This transformation leverages the properties of the operational matrix to handle the fractional derivatives and integrals involved in the Riccati equation (source).

4. Solution of Algebraic System: The system of algebraic equations is then solved to obtain the coefficients $a_n$ which are used to construct the approximate solution to the fractional Riccati equation:

$$y(x) = \sum_{n=0}^N a_n P_n^{(\alpha, \beta)}(x)$$

5. Fractional Derivatives of Shifted Jacobi Polynomials: In some approaches, formulas for the fractional derivatives of shifted Jacobi polynomials are derived and utilized in the solution process. For instance, a formula of Caputo fractional-order derivatives of shifted Jacobi polynomials in terms of shifted Jacobi polynomials themselves is proved, aiding in the direct solution of fractional differential equations (source) (source).

This methodology enables the transformation of the fractional Riccati equation into a more manageable algebraic system, which can then be solved to obtain an approximate solution. The accuracy of the solution can potentially be enhanced by increasing the truncation number $N$, thereby including more terms in the series expansion.