A mathematical analysis: From memristor to fracmemristor

The memristor is also a basic electronic component, just like resistors, capacitors and inductors. It is a nonlinear device with memory characteristics. In 2008, with HP's announcement of the discovery of the TiO₂ memristor, the new memristor system, memory capacitor (memcapacitor) and memory inductor (meminductor) were derived. Fractional-order calculus has the characteristics of non-locality, weak singularity and long term memory which traditional integer-order calculus does not have, and can accurately portray or model real-world problems better than the classic integer-order calculus. In recent years, researchers have extended the modeling method of memristor by fractional calculus, and proposed the fractional-order memristor, but its concept is not unified. This paper reviews the existing memristive elements, including integer-order memristor systems and fractional-order memristor systems. We analyze their similarities and differences, give the derivation process, circuit schematic diagrams, and an outlook on the development direction of fractional-order memristive elements.