Create Fractional Riccati Differential Equation Solver using Jacobi Tau Method

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Issue Title

Create Fractional Riccati Differential Equation Solver using Jacobi Tau Method

Issue Description

Overview

Develop a solver for fractional Riccati differential equations (FREs) based on the Jacobi Tau method.

Details

• Implement a solution algorithm based on the Jacobi Tau method to handle FREs.
• Consider integrating Jacobi polynomials as part of the method, as they are relevant to the Jacobi Tau method.
• Include necessary tests to validate the solver's accuracy and functionality.
• Provide clear documentation on how to use the solver, including any required dependencies

[crowlogic]

Fractional Riccati Differential Equation Solver Implementation with Jacobi Tau Method

Overview

This issue proposes an implementation strategy for solving fractional Riccati differential equations using the Jacobi Tau method, which leverages an operational matrix of integration.

Background

The Riccati differential equation is nonlinear and characterized by:

y'(x) = q_0(x) + q_1(x)y(x) + q_2(x)y^2(x)

For the fractional case, the derivative is of non-integer order, adding complexity to the solution process.

The Jacobi Tau method employs Jacobi polynomials P_n^{(α, β)}(x) as basis functions, which are orthogonal on the interval [-1, 1] with respect to the weight (1-x)^α(1+x)^β.

Method

The approach includes:

1. Jacobi Polynomial Expansion:
   Express y(x) as:

   y(x) ≈ ∑_{n=0}^N a_n P_n^{(α, β)}(x)

   aiming to determine the coefficients a_n.

2. Operational Matrix of Integration:
   Utilize the operational matrix of integration associated with Jacobi polynomials to transform the fractional differential equation into a system of algebraic equations.

3. Tau Method:
   Apply the Tau method, which involves truncating the infinite series to a finite number of terms and enforcing the boundary conditions to solve for the unknown coefficients.

4. Fractional Derivatives:
   Handle fractional derivatives using an appropriate fractional calculus method, such as the Riemann-Liouville or Caputo definition.

5. Algebraic System Solution:
   Solve the resulting system of algebraic equations for the coefficients a_n.

Implementation Challenges

• Deriving the operational matrix of integration for fractional derivatives of Jacobi polynomials.
• Ensuring that the nonlinearity of the Riccati equation is accurately captured within the algebraic system.
• Addressing the boundary conditions in a way that leverages the Tau method's advantages.

Potential for Exact Solutions

If the system of algebraic equations derived from the method allows, exact solutions for the coefficients a_n may be found, leading to an exact or highly accurate solution of the original differential equation.

Implementation Notes

• The solution accuracy depends on the number of Jacobi polynomials used in the expansion.
• A detailed examination of the operational matrix of integration for Jacobi polynomials in the context of fractional derivatives is required.
• Numerical methods for solving the algebraic system need to be robust to handle potential nonlinearities.

Questions and Discussion

Input is sought on the construction of the operational matrix of integration for Jacobi polynomials and experiences with the Tau method for solving nonlinear differential equations.

[crowlogic]

https://www.mdpi.com/2504-3110/7/4/302

The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order