Interconnection between mathematical models described by integer- and fractional-order differential equations

Abstract

 A heuristic solution-equivalence principle is proposed for mathematical models described by ordinary differential equations of integer and fractional order. In accordance with this principle, an integer-order ordinary differential equation exists for any ordinary fractional differential equation such that both equations have the same solution and are connected by a nonlocal transformation. The proposed principle is verified in the paper by several examples. Some qualitative properties for solution-equivalent equations and corresponding initial conditions are also discussed. Furthermore, it is shown that the solution-equivalence principle can be enhanced to partial differential equations of integer and fractional orders.