We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.
In this paper, we study a class of second-order delay fractional differential equations with a variable-order Caputo derivative. This type of equation is an extension to ordinary delay equations which are used in the modeling of several biological systems such as population dynamics, epidemiology, and immunology. Usually, fractional differential equations are difficult to solve analytically, and with fractional derivatives of variable-order, they become more challenging. Therefore, the need for reliable numerical techniques is worth investigating. To solve this type of equation, we derive a new approach based on the operational matrix. We use the shifted Chebyshev polynomials of the second kind as the basis for the approximate solutions. A convergence analysis is discussed and the uniform convergence of the approximate solutions is proven. Several examples are discussed to illustrate the efficiency of the presented approach. The computed errors, figures, and tables show that the approximate solutions converge to the exact ones by considering only a few terms in the expansion, and illustrate the novelty of the presented approach.
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.
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