Instructor Information

Tony Johnson

Cell Phone: 240-636-2708

Anthony (Tony) Johnson is a professional staff member and senior research scientist at the Johns Hopkins University Applied Physics Laboratory.
Tony is a former US Army Officer and was an Academy Professor in the Department of Mathematical Sciences at the United States Military Academy. He is a graduate of the Naval Postgraduate Applied Mathematics Doctoral program and has over 15 years of experience in computational analysis. He served as the Program Director and Chief Analyst of the Mathematical Sciences Research Program at the United States Military Academy. In addition, he held positions as the Assistant Dean for Research as well as the Director and Chief Analyst of both the Mathematical Sciences Center and the Network Science Center at West Point prior to joining the Johns Hopkins University Applied Physics Laboratory.
He specializes in finite element modeling, has done extensive research using the complex boundary element method to model fluid flow problems, and developed complex variable applications to network science problems.

Course Information

Course Description

This course is a companion to EN.625.250. Topics include ordinary differential equations, Fourier series and integrals, the Laplace transformation, Bessel functions and Legendre polynomials, and an introduction to partial differential equations. Prerequisite(s): Differential and integral calculus. Students with no experience in linear algebra may find it helpful to take EN.625.250 Multivariable and Complex Analysis first. Course Note(s): Not for graduate credit.

Course Goal

To provide a comprehensive and thorough treatment of engineering mathematics by introducing students of engineering, physics, mathematics, computer science and related fields to those areas of applied mathematics that are the most relevant for solving practical problems.

Course Objectives

  • (Ordinary Differential Equations) By the end of the course, students should be able to:
    Classify differential equations by order, linearity, and homogeneity.
    Solve first order linear differential equations.
    Solve exact differential equations.
    Use the method of undetermined coefficients to solve differential equations
    Solve higher order linear equations with constant coefficients.
    Determine whether a system of functions is linearly independent using the Wronksian.
    Use separation of variables to solve differential equations.
    Model real-life applications using differential equations.
    Use power series to solve differential equations.
    Use Laplace transforms and their inverses to solve differential equations.
  • (Partial Differential Equations) By the end of the course, students should be able to:
    State the heat, wave, Laplace, and Poisson equations and explain their physical origins.
    Identify and classify linear PDEs.
    Identify homogeneous PDEs and evolution equations.
    Model vibrating using the wave equation.
    Solve the wave equation using d’Alembert’s formula.
    Solve wave equation by separating variables and Fourier series.
    Model heat flow from a body in space.
    Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions.
    Solve PDEs using Fourier integrals and transforms.
    Solve PDEs by Laplace Transforms.
  • (Numerical Methods) By the end of the course, students should be able to:
    Use numerical methods to solve first order differential equations.
    Apply Adams-Bashford and Adams-Moulton multi-step methods to solve PDEs.
    Use numerical methods to solve systems and higher order ODEs.
    Apply methods for elliptic, parabolic, and hyperbolic PDEs.
    Use difference equations to solve Laplace and Poisson equations.
    Demonstrate understanding of Dirichlet, Neumann, and mixed problems
    Solve irregular boundary problems.

When This Course is Typically Offered

This course is typically offered in the spring term at APL


  • First-Order ODEs
  • Second-Order Linear ODEs
  • Variation of Parameters
  • Higher Order Linear ODEs
  • Systems of ODEs
  • Series Solutions
  • Legendre's Equation and Polynomial
  • Bessel Functions
  • Laplace Transforms
  • Fourier Series
  • Sturm-Liouville Problems
  • Fourier Integrals
  • Methods for First-Order ODEs
  • Multistep Methods

Student Assessment Criteria

Class Preparation and Participation 10%
Quizzes 30%
Midterm Exam 30%
Final Exam 30%

Timely feedback on students' performance is an established learning tool. I will endeavor to return all graded assignments back to you, as quickly as possible.

Class preparation and participation involves demonstration that one has completed the homework problems and can articulate problem solving strategy when called upon in class.

Quizzes will normally be graded and returned to you before the start of the next graded assignment. Homework is your responsibility and will be address only to the extent that is required to prepare for quizzes and exams.

A final grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussions every week.

A final grade of B indicates work that meets all course requirements on a level appropriate for graduate academic work. These criteria apply to both undergraduates and graduate students taking the course.

Final grades will be determined by the following weighting:

100-90 = A

89-80 = B

79-70 = C

<70 = F

The policy on exams and quizzes is that the work will be your own.  However, collaboration for homework is encouraged, but with caution. Be certain that you are learning the concepts and not simply copying the answers.

Computer and Technical Requirements

Mathematica and Matlab will be used to illustrate some principles used in this course.  Students may use Mathematica, Matlab or another type of mathematical software package or other software language, such as Java, C++ etc. Links to other mathematical packages (SciLab, Octave) will be provided as an alternative to Matlab. Computer software will not be used on any graded assignment.

Participation Expectations

Blackboard will be used for this course and students are expected to participate in any discussion posted on the course website.  Students will also be expected to participate in class discussion of concepts and example problems.

Homework will be given at the end of each class. Quizzes based on the material covered in the last class will be given at the beginning of class and should take no more that 30 minutes.


Textbook information for this course is available online through the MBS Direct Virtual Bookstore.

Course Notes

There are no notes for this course.

Final Words from the Instructor

Advanced Engineering Mathematics 10th (Tenth) Edition by Kreyszig 10th (Tenth) Edition 

(Last Modified: 09/03/2021 10:00:22 AM)