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Huber A.,Prottesweg 2a
Journal of Computational Methods in Sciences and Engineering | Year: 2011

The purpose of the present paper is to cover two intentions: Firstly we introduce a new computational algebraic procedure that can be applied to derive classes of solutions of nonlinear partial differential equations especially of higher order important in scientific and technical applications. The crucial step needs an auxiliary variable satisfying some ordinary differential equations of the first order containing sine, cosine and their hyperbolic varieties introducing for the first time connected by a special functionally dependence. Stated in most general form the solution manifold of these ordinary differential equations admits elliptic integrals and functions, respectively, as analytical solutions. Secondly the validity and reliability of the method is tested by its application to a rarely studied nonlinear evolution equation of the fourth order and leads to new classes of solutions. Nevertheless it should be emphasised that this technique does not need the solution of complicated nonlinear ordinary differential equations as caused the case of similarity reduction techniques. Further the algorithm works efficiently, is clearly structured and can be used in applications independently of the order of the equation. For computational purposes the method is appropriate to be written in any computer language. Therefore the given novel algebraic approach is suitable for wider classes of nonlinear partial differential equations in order to augment the solution manifold by a straightforward alternative approach. © 2011 - IOS Press and the authors. All rights reserved. Source


In this paper we introduce a nonlinear partial differential equation (nPDE) of the third order to the first time. This new model equation allows the extension of the Debye-Hückel Theory (DHT) considering time dependence explicitly. This also leads to a new formulation of the meaning of the nonlinear Poisson-Boltzmann Equation (PBE) and therefore we call it the modified Poisson-Boltzmann Equation (mPBE). In the present first part of this extensive study we derive the equation from the electromagnetics from a quasistatic perspective, or more precisely the electroquasistatic approximation (EQS). Our main focus will be the analysis via the Lie group formalism and since that up to now no symmetry calculation is available we believe that it seems indispensable to apply this method yielding a deeper insight into the behaviour of the solution manifold of this new equation following electrochemical considerations. We determine the classical Lie point symmetries including algebraic properties. Similarity solutions in a most general form and suitable nonlinear transformations are obtained. In addition, a note relating to potential and generalized symmetries is drawn. Moreover we show how the equation leads to approximate symmetries and we apply the method to the first time. The second part appearing shortly after will deal with algebraic solution methods and we shall show that closed-form solutions can be calculated without any numerical methods. Finally the third part will consider appropriate electrochemical experiments proving the model under consideration. © 2010 Springer Science+Business Media, LLC. Source


Huber A.,Prottesweg 2a
Journal of Computational Methods in Sciences and Engineering | Year: 2010

The classical Lie group formalism as well as the nonclassical procedure is applied to study a nonlinear partial differential equation of the second order. It is important to stress that until now no symmetry calculation is available. Therefore it seems indispensable to apply this method yielding a deeper insight into the behaviour of the solution manifold. Firstly we determine the classical Lie point symmetries including algebraic properties. Similarity solutions in a most general form and nonlinear transformations are obtained. Also a statement relating to potential symmetries is performed. Then we show how the equation leads to approximate symmetries and we apply the method for the first time. Secondly some important hints relating to different alge braic solution techniques are given in order to construct further closed-form solutions. We finally show how this less-studied equation admits solitary and peakon classes of solutions of practical relevance. © 2010 IOS Press and the authors. All rights reserved. Source


Huber A.,Prottesweg 2a
Journal of Computational Methods in Sciences and Engineering | Year: 2010

The classical Lie group formalism is applied to deduce classes of solutions of a special nonlinear partial differential equation, the so called Short-Pulse-Equation important in physical applications. We determine the Lie point symmetries and their algebraic properties. Similarity solutions are given as well as nonlinear transformations. In addition we discuss approximate symmetries for the first time. This analysis allows one to deduce wider classes of new unknown solutions either of practical or theoretical use. © 2010 IOS Press and the authors. All rights reserved. Source


Huber A.,Prottesweg 2a
Journal of Computational Methods in Sciences and Engineering | Year: 2011

A computational approach is used to find new solitary solutions of some nonlinear partial differential equations of higher order. The usual starting point is a special transformation converting the equation under consideration in its two variables x and t into a nonlinear ordinary differential equation in the single variable ξ. When we consider the method of hyperbolic functions, new distinctive classes of solutions of physical relevance result. The new feature of this paper is the fact that we are able to calculate distinctive classes of solitary solutions which cannot be found in the literature. In other words using this method the solution manifold is augmented by further classes of solution functions. Simultaneously we stress the necessity of such sophisticated methods since a general theory of nonlinear partial differential equations does not exist. Otherwise this paper is a natural completion of our recent results using alternative approaches for calculating soliton-like solutions. © 2011 - IOS Press and the authors. All rights reserved. Source

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