scholarly journals Exact Solutions for the Integrable Sixth-Order Drinfeld-Sokolov-Satsuma-Hirota System by the Analytical Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jalil Manafian Heris ◽  
Mehrdad Lakestani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Sixth Order ◽  
Mathematical Tool ◽  
Partial Differential ◽  
Wave Solutions

We establish exact solutions including periodic wave and solitary wave solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system. We employ this system by using a generalized (G′/G)-expansion and the generalized tanh-coth methods. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

scholarly journals Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Rational Solutions ◽  
Partial Differential ◽  
Wave Solutions

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

scholarly journals A New Fractional Subequation Method and Its Applications for Space-Time Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fanwei Meng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Periodic Wave ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Liouville Derivative

A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs). The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations are obtained by using this method.

Exact Travelling Wave Solutions of two Important Nonlinear Partial Differential Equations

2014 ◽  
Vol 69 (3-4) ◽  
pp. 155-162 ◽  
Author(s):  
Hyunsoo Kim ◽  
Jae-Hyeong Bae ◽  
Rathinasamy Sakthivel
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Temporal Dynamics ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions ◽  
Kudryashov Method

Coupled nonlinear partial differential equations describing the spatio-temporal dynamics of predator-prey systems and nonlinear telegraph equations have been widely applied in many real world problems. So, finding exact solutions of such equations is very helpful in the theories and numerical studies. In this paper, the Kudryashov method is implemented to obtain exact travelling wave solutions of such physical models. Further, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviour. The results reveal that the Kudryashov method is very simple, reliable, and effective, and can be used for finding exact solution of many other nonlinear evolution equations.

scholarly journals A FAMILY OF THE SPIRAL SOLUTIONS OF THE NONLINEAR KLEIN‐GORDON EQUATION

1998 ◽  
Vol 3 (1) ◽  
pp. 98-103
Author(s):  
V. V. Gudkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Klein Gordon ◽  
Klein Gordon Equation

A family of the functions, intended for a construction the exact travelling wave solutions of nonlinear partial differential equations, is given. Exact solutions of the Klein‐Gordon equation with a special potential are obtained. The behavior of complex and hypercomplex solutions of the second order is presented.

scholarly journals Explicit Travelling Wave Solutions to Nonlinear Partial Differential Equations Arise in Mathematical Physics and Engineering

2021 ◽  
Vol 2 (01) ◽  
pp. 58-63
Author(s):  
Muktarebatul Jannah ◽  
Tarikul Islam ◽  
Armina Akter
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Travelling Wave ◽  
Research Field ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions ◽  
Non Linear

To describe the interior phenomena of the mysterious problems around the real world, non-linear partial differential equations (NLPDEs) plays a substantial role, for which construction of analytic solutions of those is most important. This paper stands for a goal to find fresh and wide-ranging solutions to some familiar NLPDEs namely the non-linear cubic Klein-Gordon (cKG) equation and the non-linear Benjamin-Ono (BO) equation. A wave variable transformation is made use to convert the mentioned equations into ordinary differential equations. To acquire the desired precise exact travelling wave solutions to the above-stated equations, the rational -expansion method is employed. Consequently, three types of equipped solutions are successfully come out in the forms of hyperbolic, trigonometric and rational functions in a compatible way. To analyse the physical problems arisen relating to nonlinear complex dynamical systems, our obtained solutions might be most helpful. So far we know, these achieved solutions are different than those in the literature. The applied method is efficient and reliable which might further be used to find different and novel solutions to many other NLPDEs successfully in research field.

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