A generalized exponential transform method for solving non-linear evolution equations of physical relevance

https://doi.org/10.1016/j.amc.2009.04.074Get rights and content

Abstract

The purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied for solving non-linear partial differential equations (nPDE) especially the celebrated evolutions equations describing any time depended sequences.

The crucial step needs an auxiliary variable satisfying special class of ordinary differential equations (ODE) of first order which are introduced new in this field for the first time.

The validity and reliability of the method is tested by its application to some non-linear evolution equations leading to new class of solutions related with some new types of special functions.

Otherwise, for practical use in science and engineering the algebraic construction of new class of solutions is of fundamental interest and moreover, the proposed approach convinced by its easiness and does not need tedious steps of evaluation and can be used without studying the whole theory.

The possibility to write a symbolic software using any programming languages is given.

Further, the algorithm works efficiently, is clear structured and can be used in any applications independently from the order and the non-linearity of the underlying nPDE.

Therefore, the given novel algebraic method is suitable for a wider class of nPDE in order to augment the solution manifold by an alternative approach.

Section snippets

Introduction – general comments

Mathematical modelling of physical systems often leads to non-linear evolution equations in the general form ut = K[u], where K[u] is a locally defined function of u and its x-derivatives. The best-known example (also from historically point of view) is the Korteweg de Vries equation [1] and its varieties (cylindrical KdV, modified KdV, generalized KdV) modeling the propagation of weakly non-linear waves in dispersive media. Further, the equation represents the starting point to investigate

Description of the method

Consider a given nPDE in its two variables x and t which describes the dynamical evolution of a wave form u(x, t), u : Ω → 

in a domain Ω ⊆ 
d, d ⩾ 2 and t, so that the mapping holds for u : Ω × 
+ → 
,
+ = {t ∈ 
 : t > 0}, Ω ⊆ 
d:Pu,∂u∂x,∂u∂t,∂2u∂x2,…=0.Firstly, in our search for stationary solutions of (2.1), we introduce a travelling frame of reference ξ = x − λt (whereby λ is a constant to be determined in a later step; in general wave theory or solitary propagation λ means the velocity) to transform (2.1) into an nODE, say

Examples, calculations and results

Example 1 A non-linear evolution equation of fourth order

The scaled equation [34], [38] in (1 + 1) dimension under consideration is given by:∂2u∂t2+∂4u∂3x∂t+3∂2u∂x2∂2u∂x∂t=0,u=u(x,t),u∈C4(-∞,∞),whereby u(x, t) is a field describing a wave propagation depending on time t so that the mapping holds for u : Ω × 

+ → 
,
+ = {t ∈ 
 : t > 0}, Ω ⊆ 
d. u(x, t) should be continuous and differentiable at least four times.

We are looking for class of solutions for which u = F(x, t), F ∈ C4(D), D ⊂ 

2 holds. Performing the steps considered in Section 2 with n = 2, the following NHAS has to be

Properties of some selected class of functions

We start the analysis with the solution function (3.4b) where a graphical sketch can be seen by Fig. 1. To be a regular function, one has to claim the restriction x⧹{-1,12(-2±π)}. For the limit case t → 0 as an initial condition, one can see the behaviour by limx→±0 Re[uA,I(x, 0)] remaining a finite expression, otherwise in the case of limx→±∞Re[uA,I(x, 0)], the function is indefinite. In a considered domain the function is not stable, provided byd2dx2ReuA,I(x,0)=<0…x=0,<0…x<0,>0…x>0.Therefore, one

Conclusion remarks

In this paper, new class of solutions of some (1 + 1) non-linear evolution equations of higher order by using a new direct algebraic approach, the so called generalized exponential transform method (GETM) could derived.

It is well known that class of solutions of non-linear evolution equations can be expressed as finite series in terms of special functions, e.g. tanh-functions, Weierstrass and Jacobian functions.

All these different algebraic approaches are useful if one assume the possibility to

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