Abstract
An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equations in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions. Some exact solutions in traveling wave forms are explicitly expressed by the proposed method for both the Korteweg-de Vries and modified Korteweg-de Vries equations.
Highlights
Where u is function of the independent variables t and z, p and q are real parameters
Different from some classical methods such as various forms of Kudryashov approach, exponential rational function technique, simple hyperbolic ansatzes, rational exponential approach [25,26,27,28,29,30,31,32], the fractional form of the Sine-Gordon equation method is implemented to both equations to derive exact solutions in traveling wave forms
Fractional traveling wave transform succeeds both reducing the time fractional KdV and modified form of the KdV (mKdV) equations to some ODEs and giving the relation between trigonometric and hyperbolic functions based on time fractional Sine-Gordon equation
Summary
Where u is function of the independent variables t and z, p and q are real parameters. The operator Dtγ represents conformal fractional derivative operator defined only for positive region of t [24]. Dtγ (k3) = 0, for all constant functions S(t) = k3 The following theorem gives the relation between the classical and conformable derivative.
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