585.615.81 - Mathematical Methods

Course Structure

Course Topics

• Fourier Series
• Fourier Integral Transforms
• Generalized functions that include Dirac delta and Step function
• Convolution
• Laplace Integral Transforms and their application to solution of Differential Equations with initial conditions
  
• Legendre and Euler Linear Differential Equations with variable coefficients
• Non-homogenous Differential Equations solutions via methods of Variation of Parameters and Green’s Functions
• Series Solutions to second-order Ordinary Differential Equations
  
• Adjoint and Hermitian Operators
• Hilbert Space
  
• Strum-Liouville Equations
• Special Functions to include Legendre and Bessel Functions
• Gamma and related functions
• Introduction to Partial Differential Equations (PDEs)
• Separation of Variables as applied to wave and heat PDEs in one and two spatial dimensions and/or time
• Introduction to Complex Variables
• Derivatives of functions of a complex variable and the Cauchy-Riemann relation for analytic functions
  
• Complex Integrals and Cauchy’s theorems
  
• Taylor and Laurent Series for functions of a complex variable
• Theory of Residues.
• Inverse Laplace Transform via Bromwich Integral 
  

Course Goals

The goal of this course is to provide a general background for all biomedical engineers in advanced mathematical techniques. This course will provide mathematical techniques that will be seen in many of the courses in the Biomedical Engineering program. More importantly the mathematical techniques covered in this class provide a substantial framework to read and understand many of the journals and technical descriptions that students will 
see in their professional careers.

Course Learning Outcomes (CLOs)

• Represent functions in terms of Fourier series and perform Fourier integral transforms.
• Apply methods of Fourier series and Fourier or Laplace integral transforms in the solution of engineering and/or biomedical engineering problems.
• Solve ordinary differential equations or system of equations with initial values using Laplace integral transform and apply this technique to solve engineering and/or biomedical engineering problems.
• Solve non-homogenous differential equations using he methods of variation of parameters and Green’s functions.
• Find series solutions to second-order ordinary differential equations.
• Derive and solve some important partial differential equations (PDEs) (especially that for the wave and heat equation in one and two spatial dimensions and time) that arise in engineering and/or biomedical engineering problems.
• Perform computations and graph functions of a complex variable that involve powers, nth roots and circular functions such as exponentials and trigonometric functions.
• Compute derivatives of functions of a complex variable and apply the Cauchy-Riemann equations to check functions for analyticity.
• Compute complex line integrals a number of ways, including the following methods: antiderivatives, parametrization of the path of integration, application of Cauchy’s theorems and formulas and application of the theory of residues.

Textbooks

Required

Riley, K. F., Hobson M. P. (2011) Essential Mathematical Methods for the Physical Sciences, Cambridge University Press

ISBN-10: 1-139-63659-6
ISBN-13:  978-1-139-63659-9   

ISBN-10: 1-107-48432-4 
ISBN-13:  978-1-107-48432-0 

Optional

Kreyszig, E. (2011) Advanced Engineering Mathematics, 10th ed., Wiley ISBN-10: 9780470458365

ISBN-13: 978-0470458365

Additionally, any of the following texts or other texts that you may have from previous courses may be useful for this course if you find yourself struggling with specific skills:

• Tenenbaum, M., Pollard, H. (1985) Ordinary Differential Equations, Revised ed., Dover Books
• Strang, G. (2009) Introduction to Linear Algebra, Fourth Edition 4th ed., Wellesley Cambridge Press

Required Software

MATLAB

You will need access to a recent version of MATLAB. The MATLAB Total Academic Headcount (TAH) license is now in effect. This license is provided at no cost to you. Send an email to software@jhu.edu to request your license file/code. Please indicate that you need a standalone file/code. You will need to provide your first and last name, as well as your Hopkins email address. You will receive an email from Mathworks with instructions to create a Mathworks account. The MATLAB software will be available for download from the Mathworks site.

Student Coursework Requirements

1. Interactive Assignments (12% of Final Grade Calculation) 

Participation in interactive assignment discussions is an essential part of your grade for this course. Most weeks a set of homework problems will be assigned to the entire class. A portion of these problems will also be chosen and assigned to some students (chosen on a weekly basis) so that they can post their initial attempt at a solution. If you are the initial presenter then it is expected that you have worked on the solution to the problem you have been assigned and made it presentable to the class. Posting your initial answer in the interactive assignment discussion by day 5 (at the latest, i.e. Timeliness) for that module week.

Another part of your grade for module discussion is your interaction (i.e., responding to classmate postings with thoughtful responses, i.e., Critical Thinking). Just posting your response to a discussion question is not sufficient; we want you to interact with your classmates. Be detailed in your postings and in your responses to your classmates' postings. Feel free to agree or disagree with your classmates. Please ensure that your postings are civil and constructive.

The TA and myself will monitor module discussions and will respond in the discussions in some instances,  My corrected answers will ultimately be posted. Evaluation of preparation and participation is based on contribution to discussions. Preparation and participation in homework discussions is evaluated by the following grading elements: 

1. Timeliness (50%)
2. Critical Thinking (30%)
3. Correctness of posted analysis either in initial answer or follow up discussion (20%) Preparation and participation is graded as follows:
   

100 - 90 A—Timeliness [regularly participates; all required postings; early in discussion; throughout the discussion]; Critical Thinking [correct technique; insightful critic or added value]; Correctness
89–80 = B—Timeliness [frequently participates; all required postings; some not in time for others to read and respond]; Critical Thinking [substantially correct technique; some relevant critic]; Correctness.
79–70 = C—Timeliness [infrequently participates; all required postings; most at the last minute without allowing for response time]; Critical Thinking [some initial attempt; some critic]; Correctness.

2. Submission of completed Homework Assignments (8% of Final Grade Calculation) Submission of completed homework (twice during the semester):
   1. Timeliness of completed homework (50%)
   2. General quality of completed homework (50%)

Completed Homework Assignments should consist of all homework problems assigned for that portion of the semester. Homework submitted should be reasonably detailed and show all appropriate steps. In the case where homework problems have already been presented and discussed online make sure that you show all your own details and steps in solving them since it is expected that your answer will be correct. Remember that while problems often require the same steps to solve that solution styles and details can differ significantly from person to person! I also recognize the fact that for some problem there is little latitude in the technique for working out the problem - just make an honest attempt to provide your own work. Also include your name and a page number indicator (i.e., page x of y) on each page of your submissions.

Each problem should be identified by book chapter and number of the problem in the book chapter. All Figures and Tables if appropriate should be captioned and labeled appropriately. Completed homework (twice during the semester) is graded as follows:

100–90 = A—Timeliness [All homework was turned in when due]; General quality [neat, shows all steps with attention to details when necessary and answers are almost all correct].
89–80 = B—Timeliness [Most homework was turned in when due]; General quality [mostly neat, shows all steps and has mostly correct answers].
79–70 = C—Timeliness [Some homework not turned in when due]; General quality [not very neat, steps missing and many answers not supported by work or not correct ].

3. Course Individual Projects (30% of Final Grade Calculation) Three course project will be assigned.

The course projects will be evaluated by the following grading elements:

1. General organization of project answers (10%)
2. Mathematical setup and assumptions (if necessary) are correctly and clearly presented for project questions(20%)
3. Each part of questions is answered (20%)
4. Appropriateness of mathematical techniques, including major and intermediate steps are included (30%)
5. Correctness of answers and associated mathematics (20%)

Course Projects are graded as follows: 

100–90 = A— General organization [Clear, neat and well organized work]; Mathematical setup and assumptions [Correct setup and assumptions presented]; Appropriateness of mathematical techniques [Appropriate, detailed and correct techniques are used]; Correctness.
89–80 = B— General organization [Mostly Clear, neat and well organized work]; Mathematical setup and assumptions [Generally correct setup and assumptions presented]; Appropriateness of mathematical techniques [Mostly appropriate, some detailed and techniques missing or not correct]; Correctness.
79–70 = C— General organization [Not very clear or neat and organized poorly]; Mathematical setup and assumptions [Some incorrect setup and assumptions presented]; Appropriateness of mathematical techniques [Many details missing and some major techniques inappropriate]; Correctness.

4. Exam[s] (25% of Final Grade Calculation, combined from 25% for Midterm and 25% for Final)

The midterm exam will be available in Module 8 and the final exam will be available in the final Module (14). You will have one week to complete the exams and they will be due by 5PM exactly one week from their release. You may use all course materials to complete the exams.

The exams are evaluated by the following grading elements:

1. General organization or presentation of answers (20%)
2. Each part of questions is answered (20%)
3. Appropriateness of mathematical techniques, including major and intermediate steps are included (30%)
4. Correctness of answers and associated mathematics (30%) Exams are graded asfollows:

100–90 = A— General organization or presentation of answers; Writing Quality/ Rationale/ Examples/ Outside References [rich in content; full of thought, insight, and analysis].
89–80 = B—All parts of the question are addressed; Writing Quality/ Rationale/ Examples/ Outside References [substantial information; thought, insight, and analysis has taken place].
79–70 = C—Majority of parts of the question are addressed; Writing Quality/ Rationale/ Examples/ Outside References [generally competent; information is thin and commonplace].
 

Grading Policy

Course grading

Assignments are due according to the dates posted in your Blackboard course site. You may check these due dates in the Course Calendar or the Assignments in the corresponding modules. I will post grades one week after assignment due dates. A grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussion in every week. A grade of B indicates work that meets all course requirements on a level appropriate for graduate academic work. Course grade ranges may be adjusted if necessary. These criteria apply to both undergraduates and graduate students taking the course.

Final grades will be determined by the following weighting:

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.