Abstract
In this paper, we firstly give a counterexample to indicate that the chain rule is lack of accuracy. After that, we put forward the fractional Riccati expansion method. No need to use the chain rule, we apply this method to fractional KdV-type and fractional Telegraph equations and obtain the tangent and cotangent functions solutions of these fractional equations for the first time.
Highlights
As we know that the global quasi-operator fractional-order derivative owns the properties of depending on history, and posses more advantages than the local operator integral-order in describing the memory and hereditary characteristic of different substances, the fractional-order derivative is usually used by simulating the dynamic behavior of soft material, which is a kind of material between solid and fluid
We present the new fractional Riccati expansion method
(2) when α = 1, (44) and (45) are the new solitary wave solutions of Telegraph equation not be obtained by wang [25]
Summary
As we know that the global quasi-operator fractional-order derivative owns the properties of depending on history, and posses more advantages than the local operator integral-order in describing the memory and hereditary characteristic of different substances, the fractional-order derivative is usually used by simulating the dynamic behavior of soft material, which is a kind of material between solid and fluid. Many authors [9] [10] studied the exact traveling wave solutions of space-time fractional equations by using of formulae (4) and (5)and G’/G-expansion method, improved F-expansion, first integral method etc. We must point out that the constants σ1,σ 2 are lack of accuracy, if the fractional transformation U (ξ ) contains only one term, formula (5) is correct. We present the new fractional Riccati expansion method By this method, we firstly transform fractional partial differential equations into fractional ordinary equations with the same order by using traveling wave transformation. The properties and definition of conformable fractional derivative are listed, steps of the fractional Riccati expansion method are presented. It is obviously that formula (5) is right for the compound functions containing one term, such as f= ( g ( x)) g= 2 , g ( x) x2
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