An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation

Abstract

Abstract This manuscript develops a numerical approach for approximating the solution of the fractional Riccati differential equation (FRDE): D μ u ( x ) + a ( x ) u 2 ( x ) + b ( x ) u ( x ) = g ( x ) , 0 ≤ μ ≤ 1 , 0 ≤ x ≤ t , u ( 0 ) = d , $$\begin{align*}D^{\mu}&u(x)+a(x) u^2(x)+b(x) u(x)= g(x),\quad 0\leq \mu \leq 1,\quad 0\leq x \leq t,\\&u(0)=d,\end{align*}$$ where u(x) is the unknown function, a(x), b(x) and g(x) are known continuous functions defined in [0,t] and d is a real constant. The proposed method is applied for solving the FRDE with shifted Chebyshev polynomials as basis functions. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results compared with those given using other numerical schemes.