Solution of fractional autonomous ordinary differential equations

: Autonomous differential equations of fractional order and non-singular kernel are 1 solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we 2 seek an implicit analytical solution. 3


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Fractional calculus has resurfaced and gained momentum due to its potential in 8 engineering systems, multidisciplinary fields, biology, medicine, and applied sciences. 14 The autonomous ordinary differential equations dy dt = F(y(t)) (1.1) play a major role in Engineering, Physics, Biology, and other fields like economics and medicine. The solution to (1.1), for the given initial condition y(t 0 ) = y 0 , is The logistic differential equation y ′ (t) = y(t)(1 − y(t)) represents a special case of (1.1). It's an autonomous ordinary differential equation with a wide range of engineering applications.With an initial condition y(0) = 1/2, the logistic differential equation yields the solution (1.2) tive, the Cupato fractional derivative is more suitable to applied problems. The initial conditions are properly defined. The Caputo fractional derivative is defined as where n − 1 < α ≤ n.

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In this paper, we proceed to study the following fractional version of the autonomous differential equation (1.1): where CF D α is the Caputo-Fabrizio fractional derivative. We provide an example of a 16 differential fractional version of the exponential growth function.
According to this definition, the following are the Caputo-Fabrizio fractional deriva-20 tives for some elementary functions.
The following proposition gives the Laplace transform of the Caputo-Fabrizio 25 fractional derivative. The following proposition defines the "antiderivative" of the 26 Caputo-Fabrizio fractional derivative. .

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2021
Differentiate both sides to get that, Integrating both sides and using g(0) = 0 give the desired result.
Differentiating both sides of (2.2) and simplify to get: Taking Laplace transform for both sides implies: Now, solving (3.2) for G(s), the result follows.

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This result is read as where L −1 is the inverse Laplace transform.

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Example 2.7. In this example, we solve a fractional version of the exponential growth (decay) differential equation y ′ = ky. Consider the differential equation ( CF D α y)(t) = ky(t). (2.5) To solve this equation, take the Laplace transform for both sides of this equation to get:

Solving this equation for Y(s) implies that
Therefore, the solution of (2.5) is y(t) = Ce  is given implicitly as Proof. Let y(t) be a solution of (3.1). Then the definition (2.1) gives: Differentiate both sides and simplify to get: Rearrange the terms as: Integrate both sides from t 0 to t to get: Use integration by substitution for the integrals in the left side to get the desired re-38 sult.
This implies the solution of (2.5) is It yields same result as in Example 2.7.

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Example 3.3. We construct the solution of the fractional logistic differential equation and plot for different values of α ( CF D α y)(t) = y(t)(1 − y(t)). (3.8) According to (3.2), the solution is implicitely given as After simplifications, we will get: where C is a constant which depends on α and the initial conditions t 0 and y(t 0 ). If we assume that y(0) = 1/2 then the resluting solution is The same solution of the logistic fractional differential equation was obtained previously 43 in [8].

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2021
The set of solutions, for different values of α, along with the solution of the differential equation 45 y ′ = 1 + y 2 are represented by the figure 2.