Abstract
In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.
Highlights
Fractional differential equations are equations with derivatives of arbitrary positive orders [1,2,3,4]
Some examples of the exact analytical solutions of the nonlinear fractional differential equations are given in Section 4 of book [1]
We should note that the violation of the standard semi-group property, the violation of the standard product, and chain rules are important characteristics that should be taken into account in these equations. These non-standard properties significantly complicate the derivation of exact analytical solutions of nonlinear fractional differential equations and constructing mathematical models [17]
Summary
Fractional differential equations are equations with derivatives of arbitrary (integer and non-integer) positive orders [1,2,3,4]. We should note that the violation of the standard semi-group property, the violation of the standard product, and chain rules are important characteristics that should be taken into account in these equations These non-standard properties significantly complicate the derivation of exact analytical solutions of nonlinear fractional differential equations and constructing mathematical models [17]. We derive an exact solution of the special case of this nonlinear fractional differential equation and the conditions of the existence of solutions for this non-linear fractional differential equation. We derive the exact solution of the fractional logistic differential equation with power law coefficients as a special case of the proposed solution for the Bernoulli fractional differential equation. Examples of applications of Bernoulli fractional differential equations in physics and economics are suggested
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