Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

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.

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This paper deals with the generalized Bagley–Torvik equation based on the concept of the Caputo–Fabrizio fractional derivative using a modified reproducing kernel Hilbert space treatment. The generalized Bagley–Torvik equation is studied along with initial and boundary conditions to investigate numerical solution in the Caputo–Fabrizio sense. Regarding the generalized Bagley–Torvik equation with initial conditions, in order to have a better approach and lower cost, we reformulate the issue as a system of fractional differential equations while preserving the second type of these equations. Reproducing kernel functions are established to construct an orthogonal system used to formulate the analytical and approximate solutions of both equations in the appropriate Hilbert spaces. The feasibility of the proposed method and the effect of the novel derivative with the nonsingular kernel were verified by listing and treating several numerical examples with the required accuracy and speed. From a numerical point of view, the results obtained indicate the accuracy, efficiency, and reliability of the proposed method in solving various real life problems.

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