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We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approximate solution method, called multistage Bernstein collocation method (MBCM), to solve strongly nonlinear damped systems. The method is given with an error analysis. The systems investigated include step function external excitation and periodical function external excitation. We found that the MBCM gives more accurate approximate solutions than the standard collocation method. Since MBCM splits the domain of the problem [a,b] into n parts, one can overcome oscillations arising from the standard collocation method by increasing n. The results obtained from MBCM compare favorably with that of the fourth-order Runge–Kutta method (RK4). [ABSTRACT FROM AUTHOR]