Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations

Abstract

The Galerkin method is presented and applied for getting semi-analytical solutions of quadratic Riccati and Bagley-Torvik differential equations in fractional order. New theorems are proved to minimize the generated residual after invoking the Legendre polynomials as a basis in the Galerkin method. The proposed method is compared with other methods by solving some initial value problems of different fractional orders. The comparisons and results are illustrated via tables and figures. It can be concluded that the Legendre-Galerkin method is convenient for these problems due to its efficiency and reliability.