Innate Character of Conformable Fractional Derivative and Its Effects on Solutions of Differential Equations

In this paper, the innate character of the conformable fractional derivative is studied and a new type of soliton named semidomain soliton has been discovered in the conformable fractional model. The investigation shows that there is a large difference between the conformable fractional derivative and the classical Riemann–Liouville fractional derivative. The conformable fractional differential operator no longer have memory function as Riemann–Liouville fractional differential operator. The further investigation shows that the conformable fractional derivative is essentially a kind of cognate derivative of the integer-order derivative. It is found that the solutions between the conformable fractional differential equations and the integer-order differential equations can be transformed directly through the variable replacements. Several theorems for solution replacements between the conformable fractional differential equation and the integer-order differential equation are given and proved. It is found that there are some differences on dynamical properties of solutions between the conformable fractional differential equations and the classical integer-order differential equations. The finding of the above interesting properties implies that the conformable fractional differential operator will have good application prospects on establishing mathematical models in the future.