Degree Level and Type to be awarded for program or part of program




НазваниеDegree Level and Type to be awarded for program or part of program
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M.Sc. in Modelling and Computational Science page





Organization and Program Information


Submission Title Page


Full Legal Name of Organization: University of Ontario Institute of Technology


Operating Name of Organization: University of Ontario Institute of Technology


Common acronym of Organization: UOIT


URL for Organization Homepage: www.uoit.ca


Degree Level and Type to be awarded for program or part of program:

Master of Science in Modelling and Computational Science


Proposed Degree Titles:

Master of Science in Modelling and Computational Science


Proposed Degree Nomenclature:

M.Sc. in Modelling and Computational Science


Date of Submission: December 17, 2004


Location where program to be delivered:

University of Ontario Institute of Technology

2000 Simcoe St. N

Oshawa, ON

L1H 7K4


Contact Information:


Persons responsible for this submission:


Dr. William R. Smith

Dean, Faculty of Science

University of Ontario Institute of Technology

2000 Simcoe Street North

Oshawa, Ontario, L1H 7K4

Tel: 905-721- 3111 ext. 3235

Fax: 905-721- 3304

Email: william.smith@uoit.ca


1. Introduction


a. Brief listing of program


The program leads to the degree of Master of Science in Modelling and Computational Science. A student will have the option of obtaining their degree through a Thesis option or a Course-based option. The Thesis option combines course-based learning with the writing of a thesis that may involve original contributions to research, while the Course-based option has an increased emphasis on course-based learning, and the development of research skills is achieved through the writing of a Research Report. The expected time of completion for this degree is 20 months.


Background and rationale for program


The study of observable phenomena (physical, biological, engineering, and many others) can be approached from different perspectives and studied at various levels. Experimentalists gather data to gain an understanding of the phenomena, which may lead to a theory that explains how the underlying mechanisms function and that can predict the behaviour of the phenomena. Often, however, analysis of the observed data in and of itself is insufficient for providing a global understanding of a phenomenon or in establishing a coherent predictive theory. The construction of a theoretical model of the observed phenomenon that can be studied mathematically has become a successful and recognized additional method of gaining insight into the way nature works. Furthermore, some natural phenomena can only be studied using this approach, because they are not amenable to experimental manipulation; examples are global warming due to an increase in the concentration of carbon dioxide in the atmosphere, and the dynamics of galaxy evolution.


The use of predictions obtained using mathematical models dates back to the seventeenth century, when Isaac Newton used his theory of gravitation and differential calculus to calculate the motions of celestial bodies. A more recent example is the work of Hodgkin and Huxley in the 1950’s (Nobel prize in Physiology: 1963) who used a mathematical model involving differential equations to study the mechanisms that generate action potentials in neurons.


For mathematical models to be useful in the study of observable phenomena, it is essential to develop mathematical methods and theories for obtaining and analyzing solutions of the equations of the model. Depending on the model’s complexity, the process can involve theoretically-based analytical methods, numerical simulations, or more frequently a combination of both approaches. The resulting solutions unravel the information hidden within the mathematical models. Their analysis leads both to an understanding and interpretation of already-observed data, and to the development of testable predictions concerning behaviour of the phenomenon when parameters of the model are changed. These testable predictions determine the validity and reliability of the theoretical model as a tool for comprehending the phenomenon.


Until recent times, theoretical mathematical analysis has been the only method for reliably investigating mathematical models. In the early 1960’s, the use of computer simulations began to enter the realm of scientific investigation, as exemplified by the pioneering work of Ed Lorenz in the study of meteorological phenomena, which led to the birth of what has come to be called “Chaos Theory”. The widespread access to computers of increasing power, and the development of new computational algorithms has stimulated the growth of the emerging field of Computational Science, which is a new methodology for carrying out scientific investigation that is complementary to the traditional approaches of theory and experiment. Computational Science combines the implementation of mathematical models, computer algorithms, and knowledge in a particular area of application, in order to provide an additional tool for the study of phenomena, and in particular to facilitate the study of problems that are intractable or difficult to study using conventional approaches.


The diagram below shows schematically the various steps involved in the modelling of observable phenomena. There are three primary links in the chain of modelling: formulating the mathematical model, developing algorithms to solve the equations of the model, and performing effective computations, which often involves the use of computers.




The modelling and analysis of observable phenomena requires the development of skills and acquisition of knowledge from several disciplines (mathematics, computing science, and the area of application) for its effective implementation. The academic units initially involved in the proposed program at UOIT (the Faculty of Science and the School of Energy Systems and Nuclear Science) cut across several disciplines: the members are mathematicians, computing scientists, physicists, chemists and nuclear engineers. The proposed Master’s degree program in Modelling and Computational Science takes advantage of the interdisciplinary nature of the academic units to offer students a course of study that will introduce them to all aspects of the modelling process.


Objectives of the program:


The objectives of the program are the following:


  1. Train students in the modelling of real-world phenomena.

  2. Train students in computer skills and computer programming.

  3. Prepare students for Ph.D. studies.

  4. Develop scientific communication skills.

  5. Prepare students for the job market.


These objectives will be achieved through the following activities


  1. The training of students in the modelling of observable phenomena is achieved through course work and a research project. The three required core courses taken by all students are Mathematical Modelling, Numerical Analysis and High-Performance Computing. In the Mathematical Modelling course, the students will be exposed to various methods of formulating mathematical models of observable phenomena and learn mathematical techniques and theories for performing the analysis of the models. The Numerical Analysis course will introduce students to algorithms that can be used for obtaining approximate numerical solutions to the relevant mathematical models that typically arise in practice. The High-Performance Computing course introduces students to the efficient implementation of numerical algorithms for solving large-scale problems. The elective courses contain material that will provide the student with further exposure to the modelling process, typically in particular areas of application. Finally, in both the thesis and course-based options the students will gain first-hand experience on the three aspects of modelling through their Thesis work or their Research Report, respectively.




  1. The training of students in state-of-the-art computer skills will be provided in the Numerical Analysis and High-Performance Computing courses. In both these courses, the students will be required to implement numerical algorithms using programming languages such as Fortran, C, and MATLAB. These algorithms will include those relevant to parallel computing. Students will generally use some form of computer programming as a tool to complete their research project (whether thesis or course-based).




  1. Students completing the Master’s program in Modelling and Computational Science will be equipped to enter Ph.D. programs in Applied Mathematics, Physics, Chemistry and Engineering. See Section 1.3 below for further details.




  1. Students will develop scientific communication skills through the writing of a Thesis or a Research Report that will be evaluated by a Supervisory Committee, and an oral presentation of their work. Students in the Thesis option of the program will also be required to defend their thesis orally. The quality of their written and oral communication skills will be assessed as part of the thesis or course-based report evaluation. Moreover, all students in the program are required to participate in a seminar series, in which they will be required to give an oral presentation and will receive feedback from their Supervisor.




  1. Students completing the Master’s program in Modelling and Computational Science will be in a good position to enter the job market, by virtue of the skills aquired. See “Need and demand” part (e) of this section.


b. Relationship of proposed program to unit academic plans


From its inception, the Faculty of Science has no Departments, and has emphasized interdisciplinarity in its teaching and research. This includes team-teaching of courses cutting across disciplines. In addition, the Colloquium Series in the Faculty of Science, ongoing since Fall, 2004, emphasizes interdisciplinary research and is an ongoing focal point for research interaction among faculty and research visitors. From the beginning, Science has built a strong core of research-oriented faculty in various areas of modelling and computational science. As infrastructure support for these latter initiatives, UOIT has been a member of SHARCNET (Shared Hierarchical Academic Research Computer Network), a high-performance computing consortium of 11 universities and colleges based in South-Central Ontario. The proposed program is a natural evolution of all these developments, which are planned to continue and further evolve in the future.


The Master’s program in Modelling and Computational Science will provide an opportunity for graduate training to students from UOIT with a B.Sc. degree in Physics, Chemistry, Computing Science or Mathematics from the Faculty of Science, or with a B.Eng. degree from the School of Energy Systems and Nuclear Science or from the Faculty of Engineering and Applied Science. In particular, for students in the Faculty of Science opting for the undergraduate Computational Science specialization within the Physical Science stream, the proposed Master’s program is a natural extension of the training they receive as undergraduate students.


c. Admission Requirements


Each applicant to the Master’s program in Modelling and Computational Science must meet the following requirements.


  1. An Honours undergraduate degree in Mathematics, Science, or Engineering with a minimum average grade of B (3.0 on 4.3)




  1. Course Requirements: Normally students will have completed courses in basic numerical methods, linear algebra, differential equations, and possess some computing skills. To assist with the assessment of the application, the student should provide relevant course numbers, titles, a brief description of course content, text books used and/or chapters covered, and the grade received if applicable.




  1. Admission depends on the availability of a Research Supervisor.




  1. The student must submit an official transcript from all post-secondary institutions attended.




  1. All non-English transcripts must be accompanied by an official translation. International students whose first language is other than English must submit a TOEFL score of 220 (computer-based) or 560 (paper-based), or a CAEL score of 60.


d. In-course employment opportunities: N/A


e. Need and demand


i. A description of the general need and demand for the initiative:


The Ontario Council of Graduate Studies (OCGS) has stated that the need for graduate education in Ontario will double in the near future. One of the drivers is the so-called “Double Cohort” of students, who will graduate in 2007. The demand for graduate education is expected to be particularly strong in the area of the proposed graduate program. The need arises from both the industrial and academic sectors.


From a survey of industrial experts recently undertaken by the Canadian Advanced Technology Alliance (CATA), it was found that there is a critical need for Highly Qualified Personnel (HQP) who possess skills and knowledge in High-Performance Computing (HPC). Many companies from all sectors acknowledged this need for HQP, indicating the significant extent of the skill requirement.  "The importance of high-performance computing in sustaining Canadian competitiveness across all sectors cannot be understated," says CATA President, John Reid. The Canadian Advanced Technology Alliance (CATA) is the largest business development association dedicated to making Canadian organizations world-class producers and users of advanced technology.


The jobs in these areas are expected to be almost exclusively within interdisciplinary groups that perform a number of different tasks, and thus, problem-solving ability and the ability to communicate and work with people from a variety of disciplines will be critical. The graduates of the proposed Master’s program in Modelling and Computational Science will be in an excellent position to fill these jobs and to contribute to the province’s economy. Indeed, in addition to acquiring the necessary computational and modelling skills, the students will be exposed to a broad spectrum of mathematical analysis techniques and a broad spectrum of applications, including general physics and chemistry, fluid dynamics on scales ranging from millimeters to thousands of kilometers, medical imaging, transport phenomena, geophysics, and biology. This broad exposure will occur formally in the seminar course, as well as in other courses, and will occur informally via conversations and research interactions between the students of faculty involved in the program.


There is also a great need for graduates of the proposed program in the academic sector. Computational Science is a new, innovative and emerging interdisciplinary field of research. Technological advances have opened the door to a significant range of problems that until recently were inaccessible. Indeed, there are a great number of important scientific discoveries that could be made if there is a sufficient influx of researchers into the field. Such an influx would require the establishment of many more graduate programs such as this one.


We expect that the number of qualified applicants for the program will be greater than the number of students that the faculty will be able to support. In general, this is typically true for graduate programs in the Physical Sciences. However, we expect that there will be particularly strong demand for our proposed program. In 2007, the first undergraduate class will graduate from the Faculty of Science and from the School of Energy Systems and Nuclear Science at UOIT, and it is anticipated that many of these students will want to pursue graduate studies in those fields. As noted previously, in this same year, the students involved in the double cohort will be graduating, which will create a very significant increase in demand for graduate programs in general in the province. Furthermore, the OCGS expects that the increased demand will not be limited to the graduating year of the double cohort, but that due to the changing needs of our workforce, the demand for graduate programs will continue to increase in general. We expect that programs such as ours, in innovative and marketable fields, will be in particularly high demand.


ii. Five-year enrolment projections and estimates of demand for the program:

















YEAR

Cumulative Enrolment

Staff Requirements Projected




Full-time

Part-time

Cumulative Full-time Faculty FTE

Cumulative Part-time Faculty FTE

Technical Support Teaching Assistants, etc.

Ratio of Full-time Students/
Full-time Faculty





2006

10

0

1

0

0

10




2007

22

0

2.25

0

0

9.78




2008

27

0

2.25

0

0

12




2009

29

0

2.25

0

0

12.89




2010

29

0

2.25

0

0

12.89



The revenue and resources required are the same regardless of whether the students follow the thesis option or course-based option. Thus, in this table and the business plan, no distinction is made between the two options. Three core courses and at least 5 elective courses will be offered each year (3 in the first year of the program), and thus, 2 new faculty positions will be needed to satisfy these teaching requirements. An additional 0.25 faculty member requirement arises due to the need for a Graduate Director; this would entail a one-course teaching relief for a faculty member. Technical support will be necessary in the form of a computer systems analyst. However, the duties of the research computer systems analyst who is currently employed by the Faculty of Science will expand to accommodate the anticipated needs of the program. Secretarial duties related to the proposed program will be absorbed within existing and proposed staffing.


iii. An indication of the extent to which the proposed program prepares students for Ph.D. studies:



Both the thesis option and the course-based option of the Master’s program in Modelling and Computational Science will prepare students for Ph.D. studies in Applied Mathematics, Physics, Chemistry or Engineering. Depending on the background of the student, the proposed program is more than sufficient to ensure access to the Ph.D. programs of most universities across the country and around the world; see Appendix A for some examples of entrance requirements for Ph.D. programs.


In terms of academic background and scholarly activities, the program will prepare students for the challenges of doctoral studies. The program in general will provide the students with high-quality graduate courses, current-interest research problems that can lead to peer-reviewed publications, opportunities for scientific communication in both written and oral formats, exposure to cutting-edge research through interactions with the Faculty members at UOIT and the numerous visitors via the ongoing Science Colloquium and the Seminar Series, and the opportunity to present the results of their research at conferences. In relation to the course-based option, the thesis option will provide a student with more research experience. However, the increased breadth that a student will obtain in the course-based option will be a significant asset in the pursuit of a doctorate degree.


iv. The existence of similar or complementary programs elsewhere in Ontario/Canada:


Currently, the only similar program in Ontario is the Graduate Program in Computational Science at the University of Western Ontario; the only other similar program in Canada is at Memorial University in Saint John’s, Newfoundland. The graduate programs offered at Western Ontario and Memorial are superstructures built on top of existing Departments. At Western Ontario, the students register for a M.Sc. or Ph.D. in their respective disciplines and upon completion of the program receive a mention of “Scientific Computing” in their degree. Only Memorial University has a bona fide and standalone M.Sc. program in Computational Science.


The proposed Master’s program in Modelling and Computational Science is different from the aforementioned programs in two significant aspects. First, it embodies a broader scope; that is, it includes both the modelling and the computational aspects of scientific investigation. Our program emphasizes the development of mathematical models themselves, as well as the numerical computation of solutions intrinsic to Computational Science. Second, because of the structure of the academic units at UOIT, the proposed program is fundamentally interdisciplinary and does not require the coordination of several Departments. This is a cooperative program initially involving faculty of the Faculty of Science and the School of Energy Systems and Nuclear Science. However, any faculty member at UOIT can potentially be a member of the program.


2. Degree Requirements


Thesis option:


  1. Successful completion of 30 credits, including 6 three-credit courses and a 12-credit thesis. The 6 three-credit courses must include 3 core courses and 3 elective courses, and a minimum grade of B- must be achieved in each course. With approval from the student’s Supervisory Committee, at most one elective course may be a fourth-year undergraduate course not included in the list of graduate course electives. At most two of the electives may be courses either listed or cross-listed as undergraduate courses.

  2. Successful completion of the non-credit Graduate Seminar in Modelling and Computational Science course.

  3. Before the start of their second year of the program, students in the Thesis option will be required to submit to their Supervisory Committee a document describing the research project they will be investigating and a plan for completion. Moreover, they will be required to give a short oral presentation to the Supervisory Committee describing this information.

  4. Successful completion of a 12-credit Thesis and an oral thesis defense that will include a 30-minute oral presentation. Evaluation will be carried out by the Supervisory Committee that will not only assess the scientific content of the thesis and the oral defense, but also the written and oral communication skills of the student.


Course-based option:



  1. Successful completion of 30 credits, including 8 three-credit courses and a 6-credit research project. The 8 three-credit courses must include 3 core courses and 5 elective courses, and a minimum grade of B- must be achieved in each course. With approval from the student’s Supervisory Committee, at most two of the elective courses may be fourth-year undergraduate courses not included in the list of graduate course electives. At most two of the electives may be courses either listed or cross-listed as undergraduate courses.

  2. Successful completion of the non-credit Graduate Seminar in Modelling and Computational Science course.

  3. Before the beginning of their second year, students in the course-based option will be required to choose, in consultation with their supervisor, a research project.

  4. Successful completion of a 6-credit Research Report. Evaluation will be carried out by the Supervisory Committee that will not only assess the scientific content of the thesis, but also the written communication skills of the student.





a. Program maps and course descriptions:


i. Courses currently offered;

MCSC 6060G Advanced Statistical Mechanics will be cross-listed with the undergraduate course PHY 4010U Statistical Physics II.

MCSC 6070G Advanced Quantum Mechanics will be cross-listed with the undergraduate course PHY 4020U Quantum Mechanics II.

MCSC 6080G Advanced Optimization will be cross-listed with the undergraduate course MAT4060U Optimization (formerly Operations Research II).



ii. New courses:



Core Courses


MCSC 6010G Mathematical Modelling

MCSC 6020G Numerical Analysis

MCSC 6030G High-Performance Computing


Elective Courses


MCSC 6120G Numerical Methods for Ordinary Differential Equations

MCSC 6125G Numerical Methods for Partial Differential Equations

MCSC 6140G Dynamical Systems and Bifurcations

MCSC 6150G Fluid Dynamics

MCSC 6160G Transport Theory

MCSC 6165G Monte Carlo Methods

MCSC 6170G Computational Chemistry

MCSC 6180G Computational Physics

MCSC 6210G Advanced Topics in Mathematical Modelling

MCSC 6220G Advanced Topics in Numerical Analysis

MCSC 6230G Advanced Topics in High-Performance Computing

MCSC 6240G Advanced Topics in Dynamical Systems

MCSC 6270G Advanced Topics in Computational Science


Required Non-Credit Course


MCSC 6000G Graduate Seminar in Modelling and Computational Science



Expected student progress through the MSc Thesis option

M.Sc. – Year 1 – Fall

3 three-credit courses (2 core + 1 elective)

M.Sc. – Year 1 – Winter

3 three-credit courses (1 core + 2 electives)

M.Sc. – Year 1 – Summer

Begin Thesis research

M.Sc. – Year 2 – Fall

Thesis research

M.Sc. – Year 2 – Winter

Complete Thesis




Expected student progress through the MSc Course-based option

M.Sc. – Year 1 – Fall

3 three-credit courses (2 core + 1 elective)

M.Sc. – Year 1 – Winter

3 three-credit courses (1 core + 2 electives)

M.Sc. – Year 1 – Summer

Begin research for Research Report

M.Sc. – Year 2 – Fall

2 three-credit courses (2 electives) + research report

M.Sc. – Year 2 – Winter

Complete Research Report

Program Map for Thesis option:



Year- Semester

Course

Course

Course

1-1

Mathematical Modelling

MCSC 6010G

Numerical Analysis

MCSC 6020G

Elective

1-2

High-Performance Computing

MCSC 6030G

Elective

Elective

1-3

Thesis







2-1

Thesis







2-2

Thesis










Program Map for Course-based option:



Year- Semester

Course

Course

Course

1-1

Mathematical Modelling

MCSC 6010G

Numerical Analysis

MCSC 6020G

Elective

1-2

High-Performance Computing

MCSC 6030G

Elective

Elective

1-3

Research Project







2-1

Elective

Elective

Research Project

2-2

Research Project









Course Offerings and Frequency:


The following courses will be offered each year:


  1. MCSC 6010G Mathematical Modelling

  2. MCSC 6020G Numerical Analysis

  3. MCSC 6030G High-Performance Computing

  4. MCSC 6060G Advanced Statistical Mechanics (cross-listed with undergraduate course PHY4010U)

  5. MCSC 6070G Advanced Quantum Mechanics (cross-listed with undergraduate course PHY4020U)

  6. MCSC 6080G Advanced Optimization (cross-listed with undergraduate course MATH4060U)

  7. MCSC 6000G (non-credit) Graduate Seminar in Modelling and Computational Science.


At least 5 of the following courses will be offered each year (at least 3 in the first year the program is offered), subject to demand and resource requirements:


  1. MCSC 6120G Numerical Methods for Ordinary Differential Equations

  2. MCSC 6125G Numerical Methods for Partial Differential Equations

  3. MCSC 6140G Dynamical Systems and Bifurcations

  4. MCSC 6150G Fluid Dynamics

  5. MCSC 6160G Transport Theory

  6. MCSC 6165G Monte Carlo Methods

  7. MCSC 6170G Computational Chemistry

  8. MCSC 6180G Computational Physics

  9. MCSC 6210G Advanced Topics in Mathematical Modelling

  10. MCSC 6220G Advanced Topics in Numerical Analysis

  11. MCSC 6230G Advanced Topics in High-Performance Computing

  12. MCSC 6240G Advanced Topics in Dynamical Systems

  13. MCSC 6270G Advanced Topics in Computational Science


iii. Required courses mounted by other units;


None


3. Calendar Details



Core Courses:


MCSC 6010G Mathematical Modelling. This is a core course and forms an essential part of the program. The student will become familiar with the fundamental principles and techniques of mathematical modelling, showcased through the use of classical and advanced mathematical models in physics, biology and chemistry. An array of analytical techniques will be introduced through the study of the mathematical models. Lect: 3 hrs. Prerequisites: Admission to the program.


MCSC 6020G Numerical Analysis. Numerical analysis is the study of computer algorithms developed to solve the problems of continuous mathematics. Students taking this course gain a foundation in approximation theory, functional analysis, and numerical linear algebra from which the practical algorithms of scientific computing are derived. A major goal of this course is to develop skills in analyzing numerical algorithms in terms of their accuracy, stability, and computational complexity. Topics include best approximations, least squares problems (continuous, discrete, and weighted), eigenvalue problems, and iterative methods for systems of linear and nonlinear equations. Demonstrations and programming assignments are used to illustrate the use of available software tools for the solution of modelling problems that arise in physical, biological, economic, and engineering applications. Lect: 3 hrs. Prerequisites: Admission to the program.


MCSC 6030G High-Performance Computing. The goal of this course is to introduce students to the tools and methods of high-performance computing (HPC) using state-of-the-art technologies. The course includes an overview of high-performance scientific computing architectures (interconnection networks, processor arrays, multiprocessors, shared and distributed memory, etc.) and examples of applications that require HPC. The emphasis is on giving students practical skills needed to exploit distributed and parallel computing hardware for maximizing efficiency and performance. Building on MCSC 6020G, students will implement numerical algorithms that can be scaled up for large systems of linear or nonlinear equations. Lect: 3 hrs. Prerequisites: Admission to the program.


Elective Courses


MCSC 6120G Numerical Methods for Ordinary Differential Equations.

Differential equations are an indispensable tool for the modelling of physical and biological phenomena. However, most often in practice, analytical solutions to model equations cannot be found, and numerical approximations must be implemented. In this course, practical computational techniques for the numerical solution of ordinary differential equations will be covered, with an emphasis on their implementation and the fundamental concepts in their analysis. By the end of the course, when presented with an ordinary differential equation requiring a numerical solution, the student will be able to choose and implement the appropriate numerical technique, and will understand the limitations of the approximation. Lect: 3 hrs. Prerequisites: Numerical Analysis MCSC 6010G


MCSC 6125G Numerical Methods for Partial Differential Equations.

This course is an introduction to the mathematical concepts needed to develop accurate, reliable, and efficient numerical methods for the approximate solution of partial differential equations (PDEs). Partial differential equations constitute a vital modelling tool in science and a rich field of applied mathematical research. The basic problems of elliptic, parabolic, and hyperbolic type are examined with corresponding numerical approximation techniques. This course includes a study of various discretization frameworks: finite-difference methods, finite-element methods, finite-volume methods, and spectral collocation methods. Approximation schemes are compared and contrasted with an emphasis on error estimation, consistency, stability, and convergence as well as the availability and convenience of existing software. There will also be discussion of elements of iterative methods used for solving linear algebraic and nonlinear systems that arise from discretization of PDE problems. Lect: 3 hrs. Prerequisites: Numerical Analysis MCSC 6010G


MCSC 6140G Dynamical Systems and Bifurcations. This course provides an introduction to the modern theory of dynamical systems and bifurcation theory, including chaos theory. Dynamical systems theory is an important tool in the modelling of many physical systems, but it is also a rich field of mathematical research in itself. By the end of this course, the student will have acquired a large toolkit of techniques to analyze the dynamical features of both ordinary differential equations and discrete dynamical systems. Topics include: invariant manifolds, reduction methods, local bifurcations of vector fields and maps, chaotic dynamics, homoclinic bifurcations, applications. Lect: 3 hrs. Prerequisites: Undergraduate-level modern theory of ordinary differential equations.


MCSC 6150G Fluid Dynamics. The dynamics of fluids like oil, water, or air can be described by a set of equations known as the Navier-Stokes equations. Much can be learned about the behaviour of fluids by studying solutions to these equations. However, finding solutions is a mathematical problem of almost infinite variability and often staggering complexity. Thus, solutions to physically relevant problems generally involve approximations that are motivated by physical insight, and based on the identification of the key parameters that determine the behaviour of the fluid. Unavoidably, the field of fluid mechanics has broken up into a great number of subfields.  However, this course will try to give a more unified view by emphasizing mathematical structures that reappear in different guises in almost all those sub-specialties. Topics include: elementary viscous flow, waves: surface waves, internal gravity waves, sound waves, shocks, solitons, vortex motion, classical aerofoil theory, Navier-Stokes equations, non-Newtonian fluids, boundary layers, instability: Reynolds number, bifurcations, Kelvin-Helmholtz, jets, Rayleigh-Benard convection, centrifugal instability, parallel flows. Lect: 3hrs. Prerequisites: Admission to the program.

MCSC 6160G Transport Theory. The course is a general introduction to transport theory. Continuous-medium transport and discrete-particle transport are presented in a unified manner through the use of the probability distribution function. Various types of transport problems are presented together with analytic solutions for the simpler problems that allow them. Approximate and numerical methods are also covered. Lect: 3 hrs. Prerequisites: Undergraduate linear algebra, theory of differential equations, and vector calculus.
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