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

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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. • In order to apply the Galerkin method, the motivation of the study is to drive and use new theorems in order to facilitate the calculations of inner products between Legendre polynomials and their integer/fractional derivatives. • The Legendre-Galerkin technique is used to solve the quadratic Riccati and Bagley-Torvik and other fractional-order differential equations. • The suggested approach for the linear Bagley-Torvik differential equation error estimate is discussed and the results are computed. • To demonstrate the effectiveness of the Legendre-Galerkin method, tables and graphs were utilized to show outcomes and the estimated error.