alexa Differential Equations: Determining Partial Integrability of Wave Theory | Open Access Journals
ISSN: 2090-0902
Journal of Physical Mathematics
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

Differential Equations: Determining Partial Integrability of Wave Theory

Tim Tarver*

Department of Mathematics, Bethune-Cookman University, USA

*Corresponding Author:
Tim Tarver
Professor, Department of Mathematics
Bethune-Cookman University, USA
Tel: 386-481-2000
E-mail: [email protected]

Received date: June 01, 2016; Accepted date: July 20, 2016; Published date: July 27, 2016

Citation: Tarver T (2016) Differential Equations: Determining Partial Integrability of Wave Theory. J Phys Math 7:186. doi:10.4172/2090-0902.1000186

Copyright: © 2016 Tarver T. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Visit for more related articles at Journal of Physical Mathematics

Introduction

Substantial research shows a number of methods to integrate ordinary and partial differential equations. However, methodology towards creating a theory for the Navier-Stokes equations is not obvious. One way to possibly go about creating a theory is gathering all empirical data relating to integrability of differential equations. There exist integrability methods for linear and non-linear differential equations. In the case of Navier-Stokes, we begin to look at non-linear integrability methods for creating a possible mathematical theory towards solutions.

Explanation

One method that has been proven useful in differential systems is called Darboux’s equation for integrable systems. A common example of a Darboux integrable equation is one with an exponential component such as,

image

This method could be applied towards the Navier-Stokes equations from a classical geometric theoretical perception of differential equations developed by a number of mathematicians including Darboux [1]. The reason for using this method is to find general algorithmic approaches towards the derivation of possible solutions [1].

Another concept to study exists within the integrability of Dispersive Wave Equations. There is a method within the integrable wave equations called the Korteweg-de Vries method or KdV. There is a general equation for possible integration to yield,

image

where image [1,2]. This method of integrability is mentioned because it may have an implication of a correlation to the Navier- Stokes equations. The Korteweg-de Vries method of integrability may have a specific correlation to the next method I will discuss.

There is method for non-linear partial differential equations titled the Adomian of summation. Summation methodology is another way to integrate an equation depending on the necessity for it [3]. There could exist a two-dimensional solution for vector field image in the Navier- Stokes equation. This solution may be calculated by taking the form of

image

with a nonlinear term image by

image

Where Ai are Adomian polynomials generated for all general forms of nonlinearity [4]. Does the summation of the vector image implyimage to exist?

We now go back to the concept of Korteweg-de Vries equation and integrability analysis of it’s inverse. There is a kdV nonlinear dynamical system to yield,

image

Where image is an evolution parameter andimage The vector field being a proper subset is an implication towards the given divergent-free vector field on image in the existence of Navier- Stokes equations. The vector field image is also said to be an element of an infinite-dimensional periodic functional manifold, hence the M. Using this method, the element image can be integrated such that

image

Lastly, I want to discuss a specific concept called Kink Waves. These waves are formally defined as traveling waves which rise or descend from one asymptotic form to another [4]. It is said that one of these solutions approaches or converges to a constant. In the concept of dissipative waves, there is an equation by Burger to show,

image

Where v is the given viscosity. Since we are dealing with a viscosity and traveling waves, the Burger’s equation may have a direct correlation as to how the Navier-Stokes equations were developed. The Burger’s equation may also imply future solutions using integrability methodologies.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Article Usage

  • Total views: 8215
  • [From(publication date):
    September-2016 - Oct 23, 2017]
  • Breakdown by view type
  • HTML page views : 8111
  • PDF downloads :104
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2017-18
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri, Food, Aqua and Veterinary Science Journals

Dr. Krish

[email protected]

1-702-714-7001 Extn: 9040

Clinical and Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemical Engineering and Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001 Extn: 9040

Earth & Environmental Sciences

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

General Science and Health care Journals

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics and Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001 Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Informatics Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Material Sciences Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Mathematics and Physics Journals

Jim Willison

[email protected]

1-702-714-7001 Extn: 9042

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001 Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

John Behannon

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001 Extn: 9042

 
© 2008-2017 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
adwords