Abstract
We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.
Highlights
Fractional calculus has been become a hot field and applied to physics, mechanics, chemistry, biology, automatic control and so forth [1,2,3,4,5,6]
In the paper, based on symmetry, we reduce the time-fractional fifth order KdV equation to a less order equation, and we use Liu’s methods to obtain its abundant exact solutions
Fractional fifth order KdV equation is a typical high order rank homogenous partial differential equation, which in general cannot be directly transformed into the integral form
Summary
Fractional calculus has been become a hot field and applied to physics, mechanics, chemistry, biology, automatic control and so forth [1,2,3,4,5,6]. We study a conformal time-fractional fifth order KdV equation. We use a powerful mathematical tool namely trial equation method to find an integrable factor of nonlinear fifth order KdV differential operator. By using Liu’s method, we need not assume the forms of solutions, while we can directly derive out all solutions of the integral and give the complete classification of these solutions according to the parameters Up to now, these two methods have become the routine tools of giving the exact solutions to nonlinear partial differential equations. In the paper, based on symmetry, we reduce the time-fractional fifth order KdV equation to a less order equation, and we use Liu’s methods to obtain its abundant exact solutions.
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