Open crowlogic opened 10 months ago
This issue proposes an implementation strategy for solving fractional Riccati differential equations using the Jacobi Tau method, which leverages an operational matrix of integration.
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
.
The approach includes:
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
.
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.
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.
Fractional Derivatives: Handle fractional derivatives using an appropriate fractional calculus method, such as the Riemann-Liouville or Caputo definition.
Algebraic System Solution:
Solve the resulting system of algebraic equations for the coefficients a_n
.
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.
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.
https://www.mdpi.com/2504-3110/7/4/302
The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order
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