Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

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.

Keywords

• Bernoulli differential equation
• nonlinear fractional differential equation
• fractional logistic equation
• fractional dynamics
• economic models with memory
• fractional oscillator with memory