The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order

Abstract

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.

Analytical Solution of One-Dimensional Keller-Segel Equations via New Homotopy Perturbation Method

“…There are many numerical methods proposed for solving PDE phenomena, for example, the finite volume method (FVM) [1], the finite different method (FDM) [2][3][4], the variational iteration method (VIM) [5][6][7][8]. There are also many works in numerical methods for solving PDE phenomena of all kinds; we mention some of them [9][10][11][12][13][14], and the homotopy perturbation method (HPM) [15][16][17][18][19][20][21][22][23][24][25][26]. This is the last one (HPM), which was established by He [5,27].…”