Jacobi Spectral Tau Method for Fractional Riccati Equation

The general form of a Riccati equation is:

$$ D^\alpha y(x) = q(x) + p(x)y(x) + r(x)y(x)^2, $$

where $D^\alpha y(x)$ denotes a fractional derivative of order $\alpha$, and $p(x)$, $q(x)$, and $r(x)$ are known functions.

Solution via Jacobi Spectral Tau Method

1. Series Expansion: Express the solution $y(x)$ as a sum of Jacobi polynomials $P_n^{(\alpha, \beta)}(x)$:

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

where $a_n$ are the coefficients to be determined, and $N$ is the truncation limit of the series.

2. OperationalMatrix: Compute the operational matrix of differentiation $D$ for Jacobi polynomials, which allows representing the derivative of any function expanded in this basis as a linear combination of Jacobi polynomials. For fractional derivatives, appropriate fractional operational matrices are used.

3. Projection: Project the differential equation onto the same polynomial basis. This involves multiplying both sides of the equation by $P_m^{(\alpha, \beta)}(x)$ and integrating over the domain, usually $[-1, 1]$, to form a system of equations for the coefficients $a_n$.

4. Incorporating the Functions $p(x)$, $q(x)$, and $r(x)$: The functions $p(x)$, $q(x)$, and $r(x)$ are also expanded in terms of the Jacobi polynomials or directly incorporated into the system of equations. For nonlinear terms like $r(x)y(x)^2$, products of the series expansions are used.

5. Tau Method: Apply the tau method by adding boundary conditions or other constraints as additional equations or modifications to the system, effectively truncating the series and solving for the coefficients $a_n$.

The resulting system of equations (linear or nonlinear, depending on $p(x)$, $q(x)$, and $r(x)$ for the coefficients $a_n$ yields the solution to the Riccati equation.