Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative

Abstract

In this paper, a modified reproducing kernel algorithm is proposed to solve a class of quadratic and cubic logistic equations with Caputo-Fabrizio fractional derivative in Hilbert space. These equations are the generalization of Verhulst’s model which describes population growth taking in account that individuals will compete for limited resources. A novel reproducing kernel function is constructed to create an orthogonal system and to calculate the analytical and approximate solutions in the desirable Sobolev space. The stability, convergence, and complexity of the proposed approach are discussed. Furthermore, the effects of the Caputo-Fabrizio fractional derivatives are studied in solving the population growth model comparing with those of the classical Caputo derivatives. The main motivation for using the proposed technique is high accuracy and low computational cost compared to other existing methods especially when involving fractional differentiation operators. In this orientation, the effectiveness, applicability, and feasibility of this technique are verified by numerical examples. In a numerical viewpoint, the obtained results indicate that the suggested intelligent method has many advantages in accuracy and stability using the new Caputo-Fabrizio derivative.

Numerical solution of the fractional‐order logistic equation via the first‐kind Dickson polynomials and spectral tau method

“…The unknown expansion coefficients a i of dimension (N + 1) are constructed as a system of algebraic equations by using Equation (34) together with (37). The Gaussian elimination procedure will be used for solving the obtained algebraic system.…”

“…Table 1 contains the absolute error results for 𝜎 = 0.25, 𝛽 = 1, with N = 5, 7, respectively, besides the results of the same errors of the publication. 34 Also, the results of numerical and analytical solutions for Example 6.1 are plotted in Figure 1. Moreover, the absolute errors plotted according to N = 5, 7 with 𝜎 = 0.25, 𝛽 = 1 are illustrated in Figure 2.…”

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and SomeError Analysis

“…Among the effective numerical methods that used to handle several types of ordinary differential equations is the reproducing kernel Hilbert space (RKHS) method. In this direction, to see more information, definitions, kinds of operators, and applications of RKHS method in solving differential and integrodifferential equations of different types and orders, the reader is asked to refer to [17][18][19][20][21][22][23][24]. By and large, when trying to apply numerical and analytical methods to solve stiff system, it fails.…”

An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

“…Caputo and Fabrizio [10] introduced a novel concept of the fractional derivative, known as the Caputo-Fabrizio fractional derivative, which contains a nonsingular exponential kernel that leads to better modeling of real-world problems. Since then, the Caputo-Fabrizio and Atangana-Baleanu-Caputo (ABC) concepts have witnessed an increasing demand from scholars to conduct numerous analytical and applied studies in various scopes of engineering, physics and pure mathematics [11][12][13][14][15][16][17][18].…”