Abstract
In this article, the nonlinear partial fractional differential equation, namely the KdV equation is renewed with the help of modified Riemann- Liouville fractional derivative. The equation is transformed into the nonlinear ordinary differential equation by using the fractional complex transformation. The goal of this paper is to construct new analytical solutions of the space and time fractional nonlinear KdV equation through the extended -expansion method. The work produces abundant exact solutions in terms of hyperbolic, trigonometric, rational, exponential, and complex forms, which are new and more general than existing results in literature. The newly generated solutions show that the executed method is a well-organized and competent mathematical tool to investigate a class of nonlinear evolution fractional order equations.
Highlights
The Jumarie’s modified Riemann Liouville derivative of order α is as follows: The application of fractional order differential equations is a matter of recent decades but the theory of fractional calculus has quite a long and prominent history
During the last three decades, fractional calculus has been applied to almost every field of science, namely mathematics, engineering and technology
The results are presented by a polynomial in (G '/ G), where (G '/ G ) satisfies the second order nonlinear ordinary differential equation (ODE) as AGG′′ − BGG′ − C (G′)2 − EG2 = 0
Summary
As every nonlinear equation has its substantially noteworthy rich structure, still, significant research has to be done due to be well established the (G '/ G) -expansion method. Expansion method with the help of modified Riemann Liouville derivative by Jumarie [26,27,28] to solve nonlinear space and time fractional KdV equation. The Jumarie’s modified Riemann Liouville derivative of order α is as follows: The application of fractional order differential equations is a matter of recent decades but the theory of fractional calculus has quite a long and prominent history. A Comparative Study of Space and Time Fractional KdV Equation through Analytical Approach with Nonlinear Auxiliary Equation modified Riemann- Liouville derivatives are as [29]: Step 2.
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