Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials (Q371558)

The author constructs an approximate method to solve the Riccati fractional differential equation NEWLINE\[NEWLINE\sum_{k=0}^{m}P_{k}(t)\frac{d^{k\alpha}y(t)}{dt^{k\alpha}}=A(t)+B(t)+C(t)y^{2}NEWLINE\]NEWLINE on \([0,R] \), \( 0<\alpha \leq 1\), with mixed conditions. The method is based on the expansion \( y(t)= \sum_{k=0}^{\infty}a_{k}B_{k}(t)\) by a system of Bernstein polynomials. Using the truncated series and the set of collocation points \(t_{i}\subset[0,R],\) this problem is reduced to a system of nonlinear algebraic equations for the coefficients \(a_{k}\).NEWLINENEWLINETwo numerical examples illustrate this method, but only in the case of the initial conditions for the differential equation.