ENGINEERING 5434:  Applied Mathematical Analysis

                                                                                                 

Instructor                   Dr. Seshu Adluri                     Teaching Assistants   TBA

E-mail                          adluri@mun.ca                      E-mail                         

Phone                         864-3800                                 Phone                        

Office Location          EN-3044                                  Office Location         

Office Hours               Monday 10:00-11:50 a.m.     Office Hours              

Website                      www.engr.mun.ca/~adluri/5434

 

 

CALENDAR ENTRY:                

 

Examines numerical and analytical solutions of applied mathematical problems in Civil Engineering, problems with higher order ordinary differential equations, stiff equations, systems of ODE, Runge-Kutta methods, boundary value problems, applications of eigen value problems (numerical solutions), Fourier analysis, elliptic, parabolic and hyperbolic partial differential equations and their numerical solutions with engineering applications.

 

COURSE DESCRIPTION:         

 

The course gives an insight into engineering problems requiring solutions that are more involved than those the students have learned thus far.  They include several types of first or higher order ordinary and partial differential equations and data analysis using Fourier methods.  In most practical cases, the problems are complicated and not easily amenable to classical closed form solutions.  Therefore the main emphasis of the course is on numerical solutions.  Several issues arise in numerical methods such as matrices, stiff equations, eigen problems, numerical stability.  These are addressed using engineering examples from mechanics, structures, heat transfer, diffusion, fluids, etc.

 

PREREQUISITES:                     ENGI 4425

 

COREQUISITES:                       A good working knowledge of spreadsheets such as EXCEL and math software such as MATLAB is required. 

 

SCHEDULE:                              LECTURE: MWF 2:00-2:50 pm            Room: EN1054

                                                TUTORIAL: Friday 4:00-4:50               Room: EN1054

 

CREDIT VALUE:                       3 credits

 

RESOURCES:                           www.engr.mun.ca/~adluri/5434

 

Text book:  No textbook assigned.  However, the book prescribed for ENG 4425 is used as supplementary reading material for part of the course. 

REFERENCES:  As discussed in class from time to time.  Consult the following: 

Chapra, S.C., & Canale, R.P.. "Numerical Methods for Engineers," McGraw-Hill.

Akai, T.J.,. "Applied Numerical Methods for Engineers," John Wiley & Sons, Inc.

Gerald, C.F., and P.O. Wheatly,. "Applied Numerical Analysis," Sixth Ed., Addison-Wesley Publishing Co.

O’Neil, P.,. "Advanced Engineering Mathematics," Third Ed., Wadsworth Publishing Co.

Epperson, J.F.,. "An Introduction to Numerical Methods and Analysis," John Wiley & Sons.

MAJOR TOPICS:                     

    1. Quick Review
    2. Euler, Heun, Runge-Kutta, etc., for single first order differential equations
    3. Systems of first order equations and Higher order differential equations
    4. Numerical solution for initial value problems
    5. Boundary Value Problems –shooting (or secant) methods.
    6. Stiff Equations and implicit methods of solution.
    7. Introduction to Finite Difference Methods, dynamic response of systems, marching solutions
    1. Eigen Value Problems –physical example, formation and analytical solution
    2. Eigen Value Problems -variations of Power method, bounding theorems, Faddeeve-Leverrier method.
    1. Introduction
    2. Theorems, series, even & odd functions, etc.
    3. Fourier expansions and applications
    4. Numerical Fourier Analysis -Harmonics, confidence of fit, data analysis
    1. Introduction - Numerical solution
    2. Elliptic Partial Differential Equations: Laplace and Poisson equations, steady state heat equation, finite difference solution, Torsion of prismatic bars, etc.
    3. Parabolic Partial Differential Equations: Thermal conduction, diffusion, and other applications. Solution techniques for systems of equations,
    4. Hyperbolic Partial Differential Equations: Wave equation, marching solutions, etc.

Any of the above may be extended with related topics and applications depending upon time and interest.

 

LEARNING OUTCOMES:

 

Upon successful completion of this course, the student will be able to:

 

1.    Understand the theoretical and practical applications of numerical methods.

2.    Find numerical solutions for Initial Value Problems.

3.    Compare and recommend different methods for numerical solution of Ordinary Differential Equations.

4.    Use numerical methods to solve Boundary Value Problems.

5.    Use implicit methods of solution to numerically solve Stiff Equation Problems.

6.    Use Finite Difference Methods for solving Differential Equations.

7.    Understand the application of Eigen Values and their use in engineering.

8.    Use numerical techniques to provide solutions to Eigen Value Problems.

9.    Use Fourier analysis to represent and analyze functions and physical data.

10.  Identify and numerically solve Elliptic, Parabolic and Hyperbolic Partial Differential Equations with applications in heat transfer, diffusion, torsion of prismatic bars, string vibrations, etc.  .

 

ASSESSMENT: 

                                                Approximate Due Dates

Assignments                  15%                            Assignments are due one week from announcement unless otherwise agreed upon.  Assignments include computer work. 

Midterm                       25%                            Feb 7

Final exam                    60%                           

Exam policy: The formula sheet is as per the announcement in the class.  All or a subset of the preannounced formula sheet will be provided in the exam.  No extra text or notes are permitted in the exam.  Electronic storage/retrieval devices are not permitted in the exam.  Please see the appropriate guidelines from the University regarding such matters. 

Comprehensive examples will be discussed primarily during tutorials. During tutorials, the students may be required to solve the problems in class to gain practice.

Please note that prewritten solutions may or may not be available for the assignments.  However, the tutorials are specifically marked for discussing the relevant solutions.  At that time, if the students ask for it, the problems can be discussed and may be partially or fully solved in class.  If the students do not raise their need for discussion of the problems, the time will be spent on solving other example problems, etc.  The same policy holds for midterm exams and quizzes, if any.

Several handouts, example solutions using EXCEL are posted at the course website.  The students are expected to make use of the various files and ask for help if needed.  

The students are expected to solve the assignment problems by themselves in order to reinforce the class instruction.  Please refer to copying policy of the University if there is any doubt.  Help with the assignment problems can be sought during contact hours and/or tutorial time. 

ACADEMIC INTEGRITY AND PROFESSIONAL CONDUCT:

 

Students are expected to conduct themselves in all aspects of the course at the highest level of academic integrity. Any student found to commit academic misconduct will be dealt with according to the Faculty and University practices. More information is available at www.engr.mun.ca/undergrad/academicintegrity.

 

Students are encouraged to consult the Faculty of Engineering and Applied Science Student Code of Conduct at http://www.engr.mun.ca/policies/codeofconduct.php and Memorial University’s Code of Student Conduct at http://www.mun.ca/student/home/conduct.php.

 

INCLUSION AND EQUITY: 

 

Students who require physical or academic accommodations are encouraged to speak privately to the instructor so that appropriate arrangements can be made to ensure your full participation in the course.  All conversations will remain confidential.

 

The university experience is enriched by the diversity of viewpoints, values, and backgrounds that each class participant possesses.  In order for this course to encourage as much insightful and comprehensive discussion among class participants as possible, there is an expectation that dialogue will be collegial and respectful across disciplinary, cultural, and personal boundaries.

 

STUDENT ASSISTANCE:  Student Affairs and Services offers help and support in a variety of areas, both academic and personal.  More information can be found at www.mun.ca/student. 

 

 

 

Assignments:

 

 

 

Marks going into final exam  

If there are any discrepancies, please talk to Dr. Adluri immediately after the final exam (before leaving the exam hall).

Please note that changes at a later date are not possible. 

 

Sample Midterm Sol:

Midterm Q1, Midterm Q2-1, Q2-2

 

 

Sample final Exam     Solution:  P2, P1, P3a, P3b, P4, P5

 

Another Sample exam, 

 

 

Formula sheet

 

Handouts and extra material for the students to help in their understanding:

 

 Intro’

Quick Review

Example for user defined functions in EXCEL -DOC file

Example for user defined functions in EXCEL - EXCEL file

Solution to cubic equations

Gauss elimination procedure

Solving simultaneous equations in EXCEL

 

Ordinary Differential Equations -First Order

Handout for various procedures

Example for Euler, Modified Euler and Runge-Kutta Methods

Comparison of relative errors

 

Systems of ODE -Initial Value Problems

Application: Predator-Prey model

Conversion of higher order ODE to a system of First order ODE

Application: Cantilever Beam

 

Systems of ODE -Boundary Value Problems

Notes on Boundary value problems

Example of Shooting using Euler Method

Example of Shooting with various methods (Macros)

Another Example of Shooting with various methods (using Macros)

Example of simply supported beam

 

 

Stiff Equations

First order stiff equation example

System of stiff equations - example

 

Eigen Values

Review of Eigen Values and Introduction to Power method

Example for Power method

Faddeeve-Leverrier Method

Example for Faddeeve-Leverrier

 

Example for Faddeeve-Leverrier- principal stresses

Example for Power method-principal stresses

Buckling of Tapered column-Faddeeve-Leverrier

 

Fourier Analysis

Fourier analysis for Discrete data

Practice problems for Fourier Analysis

PDE

Steady state heat –Laplace (Elliptic) equation

Vibration of string –Hyperbolic Eq.

 

Partial Differential equations are slightly cumbersome to be solved using spreadsheets.  Interested students may use MATLAB or other tools for this purpose.