Exact solution to fractional logistic equation

• The logistic equation is generalized to include memory using the fractional calculus.
• The fractional logistic equation (FLE) is solved using an infinite-order linear representation.
• The linear representation of the FLE is exact and its solution involves the inversion of an infinite-order matrix.
• The technique is applicable to a large class of polynomial nonlinear fractional rate equations.

The logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences. Herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time. The solution to the fractional logistic equation (FLE) is obtained using the Carleman embedding technique that allows the nonlinear equation to be replaced by an infinite-order set of linear equations, which we then solve exactly. The formal series expansion for the initial value solution of the FLE is shown to be expressed in terms of a series of weighted Mittag-Leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity. The numerical integration to the FLE provides an excellent fit to the analytic solution. We propose this approach as a general technique for solving a class of nonlinear fractional differential equations.