scholarly journals Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Huitzilin Yépez-Martínez ◽  
Ivan O. Sosa ◽  
Juan M. Reyes
Keyword(s):  
Integral Method ◽  
First Integral ◽  
Space Time ◽  
First Integrals ◽  
First Integral Method ◽  
Fractional Equations ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Generalized Fisher Equation ◽  
Manageable Method

The fractional derivatives in the sense of the modified Riemann-Liouville derivative and Feng’s first integral method are employed to obtain the exact solutions of the nonlinear space-time fractional ZKBBM equation and the nonlinear space-time fractional generalized Fisher equation. The power of this manageable method is presented by applying it to the above equations. Our approach provides first integrals in polynomial form with high accuracy. Exact analytical solutions are obtained through establishing first integrals. The present method is efficient and reliable, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner ◽  
Ömer Ünsal
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Integral Method ◽  
First Integral ◽  
Space Time ◽  
First Integral Method ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Complex Transform ◽  
Fractional Differential

In this paper, we establish exact solutions for some nonlinear fractional differential equations (FDEs). The first integral method with help of the fractional complex transform (FCT) is used to obtain exact solutions for the time fractional modified Korteweg–de Vries (fmKdV) equation and the space–time fractional modified Benjamin–Bona–Mahony (fmBBM) equation. This method is efficient and powerful in solving kind of other nonlinear FDEs.

Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method

2019 ◽  
Vol 20 (5) ◽  
pp. 511-516
Author(s):  
Esin Aksoy ◽  
Ahmet Bekir ◽  
Adem C Çevikel
Keyword(s):  
Differential Equations ◽  
Kdv Equation ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Fractional Equations ◽  
Complex Transformation ◽  
Liouville Derivative ◽  
Kudryashov Method ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

AbstractIn this work, the Kudryashov method is handled to find exact solutions of nonlinear fractional partial differential equations in the sense of the modified Riemann–Liouville derivative as given by Guy Jumarie. Firstly, these fractional equations can be turned into another nonlinear ordinary differential equations by fractional complex transformation. Then, the method is applied to solve the space-time fractional Symmetric Regularized Long Wave equation and the space-time fractional generalized Hirota–Satsuma coupled KdV equation. The obtained solutions include generalized hyperbolic functions solutions.

scholarly journals Travelling Wave Solutions to the Generalized Pochhammer-Chree (PC) Equations Using the First Integral Method

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Shoukry Ibrahim Atia El-Ganaini
Keyword(s):  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
Periodic Wave ◽  
Travelling Wave ◽  
First Integral Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Manageable Method ◽  
Complex Exponential

By using the first integral method, the traveling wave solutions for the generalized Pochhammer-Chree (PC) equations are constructed. The obtained results include complex exponential function solutions, complex traveling solitary wave solutions, complex periodic wave solutions, and complex rational function solutions. The power of this manageable method is confirmed.

scholarly journals New Exact Solutions of Some Nonlinear Systems of Partial Differential Equations Using the First Integral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shoukry Ibrahim Atia El-Ganaini
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
First Integrals ◽  
Partial Differential ◽  
First Integral Method ◽  
New Exact Solutions

The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE), (2 + 1)-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.

Exact solutions for the fractional differential equations by using the first integral method

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Hossein Aminikhah ◽  
A. Refahi Sheikhani ◽  
Hadi Rezazadeh
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Commutative Algebra ◽  
Integral Method ◽  
First Integral ◽  
Ring Theory ◽  
First Integral Method ◽  
Fractional Differential ◽  
Method Show

AbstractIn this paper, we apply the first integral method to study the solutions of the nonlinear fractional modified Benjamin-Bona-Mahony equation, the nonlinear fractional modified Zakharov-Kuznetsov equation and the nonlinear fractional Whitham-Broer-Kaup-Like systems. This method is based on the ring theory of commutative algebra. The results obtained by the proposed method show that the approach is effective and general. This approach can also be applied to other nonlinear fractional differential equations, which are arising in the theory of solitons and other areas.

scholarly journals The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Shoukry Ibrahim Atia El-Ganaini
Keyword(s):  
Optical Fibers ◽  
Integral Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
Higher Dimensions ◽  
First Integrals ◽  
First Integral Method ◽  
Nonlinear Schrödinger ◽  
Planar Systems

The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

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