| Citation: |
Masakazu Onitsuka, Iz-iddine EL-Fassi. GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 964-975. doi: 10.11948/20230221
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GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DIFFERENTIAL EQUATION
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Abstract
In this paper, a generalization of the Caputo-Fabrizio fractional derivative is proposed. The purpose of this study is to derive a solution formula for ordinary differential equations with the generalized Caputo-Fabrizio fractional derivative. The main result can be applied to solve the Caputo-Fabrizio fractional differential equation $\mathcal{D}^\alpha y=f(y)$. That is, a new result even for common Caputo-Fabrizio fractional differential equations is obtained. The strength of the results obtained in this study is that the solution to the differential equation can be given using only the kernel included in the derivative and the right-hand side f of the equation. In other words, rather than providing a method to solve the solution, this study provides a formula for the solution. This study is proposed as a tool for solving many nonlinear equations, including the logistic type fractional differential equations.
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References
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Masakazu Onitsuka, Iz-iddine EL-Fassi. GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 964-975. doi: 10.11948/20230221
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Figure 1.
Solution curves of classical logistic equation (black), Caputo-Fabrizio fractional logistic equation (1.3) (blue), and generalized Caputo-Fabrizio fractional logistic equation (4.3) with (4.4) (red), with
$ y(0)=0.1 $ $ \alpha=0.1 $ -
Figure 2.
Solution curves of classical logistic equation (black), Caputo-Fabrizio fractional logistic equation (1.3) (blue), and generalized Caputo-Fabrizio fractional logistic equation (4.3) with (4.4) (red), with
$ y(0)=0.1 $ $ \alpha=0.5 $ -
Figure 3.
Solution curves of classical logistic equation (black), Caputo-Fabrizio fractional logistic equation (1.3) (blue), and generalized Caputo-Fabrizio fractional logistic equation (4.3) with (4.4) (red), with
$ y(0)=0.1 $ $ \alpha=0.9 $