MODIFIED METHOD OF SIMPLEST EQUATION FOR OBTAINING EXACT SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS: HISTORY, RECENT DEVELOPMENTS OF THE METHODOLOGY AND STUDIED CLASSES OF EQUATIONS

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.

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New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

Journal of Mathematics ◽

10.1155/2013/201276 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

Yusuf Pandir ◽

Halime Ulusoy

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Wave Equations ◽

Kdv Equation ◽

Nonlinear Partial Differential Equations ◽

Hyperbolic Function ◽

Hyperbolic Functions ◽

Partial Differential ◽

New Exact Solutions

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

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B-determining equations: applications to nonlinear partial differential equations

European Journal of Applied Mathematics ◽

10.1017/s0956792500001832 ◽

1995 ◽

Vol 6 (3) ◽

pp. 265-286 ◽

Cited By ~ 12

Author(s):

O. V. Kaptsov

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fluid Mechanics ◽

Exact Solutions ◽

Nonlinear Partial Differential Equations ◽

Symmetry Groups ◽

Partial Differential ◽

Physical Importance

We introduce the concept of B-determining equations of a system of partial differential equations that generalize the defining equations of the symmetry groups. We show how this concept may be applied to obtain exact solutions of partial differential equations. The exposition is reasonable self-contained, and supplemented by examples of direct physical importance, chosen from fluid mechanics.

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Classification of Exact Solutions for Generalized Form of Equation

Abstract and Applied Analysis ◽

10.1155/2013/742643 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Hasan Bulut

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Nonlinear Partial Differential Equations ◽

Generalized Equation ◽

Polynomial Method ◽

Partial Differential ◽

Elliptic Solutions ◽

Complete Discrimination System

The classification of exact solutions, including solitons and elliptic solutions, to the generalized equation by the complete discrimination system for polynomial method has been obtained. From here, we find some interesting results for nonlinear partial differential equations with generalized evolution.

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