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Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. (English) Zbl 1280.65075
The author constructs an approximate method to solve the Riccati fractional differential equation $$\sum_{k=0}^{m}P_{k}(t)\frac{d^{k\alpha}y(t)}{dt^{k\alpha}}=A(t)+B(t)+C(t)y^{2}$$ 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 $a_{k}$. Two numerical examples illustrate this method, but only in the case of the initial conditions for the differential equation.

MSC:
65L10Boundary value problems for ODE (numerical methods)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L70Error bounds (numerical methods for ODE)
34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
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