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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.

As the fourth fundamental circuit element, the memristor may execute computations while storing data. Fracmemristor takes advantage of the fractional calculate’s long-term memory, non-locality, weak singularity, and the memristor’s storage–computational integration. Since the physical structure of the fracmemristor is similar to the topology of the ant transfer probability flow in ACO, we propose the fractional-order memristive ant colony algorithm (FMAC), which uses the fracmemristor physical system to record the probabilistic transfer information of the nodes that the ant will crawl through in the future and pass it to the current node of the ant, so that the ant acquires the ability to predict the future transfer. After instigating the optimization capabilities with TSP, we discovered that FMAC is superior to PACO-3opt, the best integer-order ant colony algorithm currently available. FMAC operates substantially more quickly than the fractional-order memristor ant colony algorithm due to the transfer probability prediction module based on the physical fracmemristor system (FACA).

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