Chaos theory Applications to PDEs (geometry design) - Essay Example

Comments (0) Cite this document
Summary
Application of Chaos Theory in the Differential Equations Name Institution Application of Chaos Theory in the Differential Equations Introduction The chaos theory in the finite dimensional dynamical systems is known to exist and it includes or leads to the development of discrete systems and maps of the ordinary differential equations…
Download full paperFile format: .doc, available for editing
GRAB THE BEST PAPER91% of users find it useful
Chaos theory Applications to PDEs (geometry design)
Read TextPreview

Extract of sample "Chaos theory Applications to PDEs (geometry design)"

Download file to see previous pages 55). Therefore, there has been a growing demand for the development for a much stronger theory than for the finite dimensional systems. In mathematics, there are significant challenges in the studies on the infinite dimensional systems (Taylor, 1996; p. 88). For instance, as phase spaces, the Banach spaces have many structures than in Euclidean spaces. In application, the most vital natural phenomena are explained by the partial differential equations, most of important natural phenomena are described by the Yang-Mills equations, partial differential equations, nonlinear wave equations, and Navier-Stokes equations among others. Problem Statement Chaos theory has led to profound mathematical equations and theorems that have numerous applications in different fields including chemistry, biology, physics, and engineering among other fields or professions. Problem Definition The nonlinear wave equations are usually significant class of equations especially natural sciences (Cyganowski, Kloeden, and Ombach, 2002; p. 33). They usually describe a wide spectrum of phenomena including water waves, motion of plasma, vortex motion, and nonlinear optics (laser) among others (Wasow, 2002). Notably, these types of equations often describe differences and varied phenomena; particularly, similar soliton equation that describes several different situations. These types of equations can be described by the nonlinear Schrodinger equation 1 The equation 1 above has a soliton solution 2 Where the variable This leads to 3 The equation leads to the development of the soliton equations whose Cauchy problems that are solved completely through the scattering transformations. The soliton equations are similar to the integrable Hamiltonian equations that are naturally counterparts of the finite dimensionalintegrable differential systems. Setting up the systematic study of the chaos theory in the partial differential equations, there is a need to start with the perturbed soliton equations (Wasow, 2002). The perturbed soliton equations can be classified into three main categories including: 1. Perturbed (1=1) dimensional soliton equations 2. Perturbed soliton lattices 3. Perturbed (1 + n) dimensional soliton equations (n? 2). For each of the above categories, to analyze the chaos theory in the partial differential equations, there is needed to choose a candidate for study. The integrable theories are often parallel for every member within the same category (Taylor, 1996; p. 102). Moreover, members of different categories are often different substantial. Therefore, the theorem that describes the existence of chaos on each candidate can be generalized parallely to other members under the same category (Wasow, 2002). For instance; The candidate in the first category is often described by a perturbed cubic that often focuses on the nonlinear Schrodinger equation 4 Under even and periodic boundary conditions q (x+1) = q (x) and q (x) =q (x), and is a real constant. The candidates in category 2 are often considered as the perturbed discrete cubic that often focus on the nonlinear Schrodinger equation + Perturbations, 5 The above equation is only valid under even and periodic boundary conditions described by +N = The candidates falling under category 3 are perturbed Davey-Stewartson II equations 6 The equation is only satisfied under the even and periodic ...Download file to see next pagesRead More
Cite this document
  • APA
  • MLA
  • CHICAGO
(“Chaos theory Applications to PDEs (geometry design) Essay”, n.d.)
Chaos theory Applications to PDEs (geometry design) Essay. Retrieved from https://studentshare.org/mathematics/1466530-chaos-theory-applications-to-pdes-geometry-design
(Chaos Theory Applications to PDEs (geometry Design) Essay)
Chaos Theory Applications to PDEs (geometry Design) Essay. https://studentshare.org/mathematics/1466530-chaos-theory-applications-to-pdes-geometry-design.
“Chaos Theory Applications to PDEs (geometry Design) Essay”, n.d. https://studentshare.org/mathematics/1466530-chaos-theory-applications-to-pdes-geometry-design.
  • Cited: 0 times
Comments (0)
Click to create a comment or rate a document

CHECK THESE SAMPLES OF Chaos theory Applications to PDEs (geometry design)

College Geometry - neutral geometry and Euclidean geometry

...? GEOMETRY Introduction Geometry is a branch of measure mathematics that revolves in and around size, shape, relative figure’s position, together with properties of space. It is one of the oldest branches of mathematics that originated from the ancient Greek among the mathematics scientists. Initially, it dealt with areas, volumes, and lengths but in the 3 BC, geometry took a new path as a result of efforts from Mathematicians such as Euclid with his Euclidean geometry. He laid down the foundation of today's geometry; absolute geometry is type of geometry that has developed in the recent years by...
3 Pages(750 words)Essay

Economics Theory Applications

...Economics Theory Applications It is worth noting that just like economies that have developed time so are the economic theories. Even though the word “economy” may be old, modern economic thought dates back in time to the 1600s and 1700s when the world economies began their transformations from the agrarian to industrial societies (Samuelson, 1986, 2). Economic theories are of two types and these are microeconomic and macroeconomic theories. Microeconomics is defined as a social science that functions to resolve the problem of allocation of limited resources in a way that should meet the human wants that are normally unlimited. According to Mandal (2007,...
8 Pages(2000 words)Research Paper

Chaos Theory on Financial Forecasting

...that surrounds us we need to focus more on Complexity here describes the way of acting on an occasion, not the result obtained. Handling complexity of economic particles, rather than assuming them as their act is based on an identical behavior such as strict rationality. This outcome brings us to the fact that linear way of thinking may be a clear path to think but its progress is too straight to analyse systems that involve complexity. This handicap might cancel out the significant factors that affect the accuracy related to the prediction of destination point of the system. Chaos Theory applications in economics It has now been almost twenty years since economists first began...
8 Pages(2000 words)Essay

Fractal Geometry

...Fractal Geometry Fractal geometry is rather new mathematical theory which is completely different from the traditional concepts of Euclidean Geometry. Fractal geometry describes elf-similar or scale symmetric objects. It means that if to magnify these objects, their parts will bear an exact resemblance to the whole object. The word "fractal" was created by Benoit Mandelbrot and it means "to break", whereas the form of adjective "fractus" means "fragmentated". (Brandt 24) It is a matter of fact that the word "fractal" has two definite meanings: the first meaning refer to colloquial use and the second is connected with geometry. In...
2 Pages(500 words)Essay

Learning Theory Applications

...Learning Theory Application Key Ideas Application Learning Theory: Culture and Learning For teachers to understand who the are and where they come from, they must know how to use families/home as resource. Kathleen Hayes introduced the “Memoirs” project to help her gain an understanding of her students’ family and cultural background which are vital information for the year. Parents were invited to talk in class which motivated the students’ learning. Suggestion: Allow time for students to share their won memoirs to classmates in paired/group work. 2. A culturally relevant classroom creates a sense of belongingness to the school community helping them to transcend...
3 Pages(750 words)Research Paper

Design Principles and Applications

...Client Brief Requirement Initial stages of a project that a client engages a professional contractor or designer in require specifications and reservations that the client has over the project. Throughout the design process, the ideas of the designer and the client are brought together by the client brief. Clients submit their own ideas to the designers who assist them in understanding the entire process through a well outlined explanation on the same. Designers encourage clients to forward their reservations since it is the first step towards the understanding of the needs of the client; because a designer works to satisfy the needs of the clients. A client brief is therefore a rough draft or sketch of the project design... that the...
5 Pages(1250 words)Assignment

Geometry

...Quantitative Literacy Steve Beckman WGU QLT1 Task September29, A. Complete the following graphs: 1. Graph the following values on a single number line • Value 1: 1 • Value 2: 0 • Value 3: –6 • Value 4: 3/4 • Value 5: –1.7 2. Graph the following points on a single coordinate plane. Make sure to indicate labels for each quadrant of the coordinate plane. • Point 1: (3, –2) • Point 2: (0, 0) • Point 3: (–1, 7) • Point 4: (3, 5) • Point 5: (–4, –5) (3) Graph the following functions on a separate coordinate planes. Function 1: y = 2x - 1 y Function 2: y = (-3/4)*x + 5 y Function 3: y = x2 - 4... Literacy Steve Beckman...
7 Pages(1750 words)Speech or Presentation

Geometry

...Geometry Geometry is a branch of mathematics that deals or attempts to explain and describe the shape, size, or relative position of an object in space. The study of geometry dates back to the time of ancient Greek. The Euclidian geometry for example originated in the 6th century and it laid the foundation for modern geometry. Currently, the study has advanced to include other branches of mathematics such as number theory, algebra, and topology. Riemannian geometry for example is among the latest braches of geometry. Each of the above fields of specialization has a different approach to the study of geometry (Mlodinow, 2002). Indeed, a good mathematician will view the different branches of geometry as mere tools designed to enable... ...
2 Pages(500 words)Admission/Application Essay

Geometry

...Step Draw a line AB that we will be divided into 3 (in this case) equal parts. From point A, draw a line segment at an angle to the given line, and about the same length. The exact length is not important. Set the compasses on A, and set its width to a bit less than one third of the length of the new line. Step the compasses along the line, marking off 3 arcs. Label the last one C. With the compasses width set to CB, draw an arc from A just below it. With the compasses width set to AC, draw an arc from B crossing the one drawn in step 4. This intersection is point D. Draw a line from D to B. Using the same compasses width as used to step along AC, step the compasses from D along DB making 3 new arcs across the line Draw lines... Draw a line...
1 Pages(250 words)Speech or Presentation

Applications of Graph Theory

...that connected two islands to the mainland in Konigsberg. People wondered whether one could walk through the seven bridges once. Euler solved the problem by using the graph theory to prove that one could not walk through all the seven bridges once (Shirinivas et al., 2010). Concepts of graph theory are extensively applied in models and study applications in different fields. For example, graph theory concepts are used to study molecules, atoms and bond construction in chemistry. Graph theory concepts are also applied to explore diffusion mechanisms and to measure actors’ prestige in sociology. In biology, graph theory concepts such as...
4 Pages(1000 words)Assignment
sponsored ads
We use cookies to create the best experience for you. Keep on browsing if you are OK with that, or find out how to manage cookies.

Let us find you another Essay on topic Chaos theory Applications to PDEs (geometry design) for FREE!

Contact Us