Approximate Analytic Solution of Riccati Equation with Fractional Order of Multi-Parameters

Abstract

In this paper, we present an approximate analytic solution of the Riccati equation with fractional order of multi-parameters. The fractional order of Caputo types with generalized Mittag–Leffler kernel is adaptive, this kind of fractional derivative has three fractional parameters. Several properties of the fractional derivative and integral are studied. We use the homotopy analysis method to generate the approximate analytic solution to the problem. The effect of the fractional parameters on the behavior of the solution is studied, each parameter of the fractional derivative can change not only the solution behaviors but also the existence of the solution. Two examples are presented to demonstrate the efficiency of the method. Comparisons of the exact solution and the approximate solution in the case of the standard derivative are made. For the fractional case, we calculate the residual error of the approximate solution. In all cases, the solution is accurate and simply applies.