MA530 Advanced Engineering Mathematics II

Instructor: Nikolay S. Strigul

E-mails: nstrigul@stevens.edu, nstrigul@princeton.edu

Office Hours: by appointment

Lectures: Wednesdays 06:30-09:00 pm, E. A. Stevens, 229A

Homeworks: There will be weekly homework assignments, due every Wednesday, when the correct solutions will be given out.

Exams: There will be Midterm and Final Exams.

Quizzes: Every class will start with a closed book quiz consisting of some past homework problems, definitions, and theorems.

Grading:

- Quizzes: 35 %
- Homework assignments: 25 %

- Midterm: 10 %

- Final: 30 %
Textbook: Advanced Engineering Mathematics by Peter V. O'Neil, 6th Edition, ISBN: 0534552080

Course programCourse program (PDF)

General comments:The major goal of this course is to cover practical mathematical methods important to engineering applications. It is expected that students already have a mathematical background, including undergraduate courses in calculus, linear algebra and differential equations. However, the course will start with a review of integration methods and first order differential equations; and will also review linear algebra, so students with some lack in their background can succeed in this course. In general, this course will be focused on more advanced topics, including systems of linear differential equations, qualitative methods, the Fourier transform, and partial differential equations. An important goal of this course is to learn how to use analytical methods in concert with computer calculations. Computational parts of the course include the use of the Mathematica package: all students must purchase the student edition of this program (the semester edition is less expensive) at

http://www.wolfram.com/products/student/mathforstudents/licenses.html

Course program:Aug 30. Lecture 1.- Introduction. Rewiev of Integration.

1. Different approaches in mathematical modeling.

2. Individual-based models and differential equations.

3. Implicit and explicit assumptions in modeling.

4. First Order Differential Equations.

5. The Initial Value Problem.

6. Separable Equations.

7. Review of Integration Methods.

Homework 1.pdf

Sep 6. Lecture 2.- Review of First Order Equations.

8) Linear Differential Equations.

9) Exact Differential Equations.

10) Integrating Factors.

11) Separable, Exact and Linear equations from a general point of view.

12) Homogeneous, Bernoulli and Riccati Equations.

13) Applications to Mechanics, and Electrical Circuits.

14) Existence and Uniqueness for Solutions of Initial Value Problems.

Homework 2.pdf Quiz 1.pdf

Sep 13. Lecture 3.- Introduction to Mathematica. Second Order Differential Equations.

15) Symbolic and numerical calculations in Mathematica.

16) Differential equations in Mathematica.

17) Linear algebra in Mathematica.

18) Programming in Mathematica.

19) Linear Second Order Differential Equations.

20) Homogeneous and Nonhomogeneous Equations.

21) The Constant Coefficient Homogeneous Linear Equation.

Homework 3.pdf Mathematica tutorial 1.nb Quiz 2.pdf

Sep 20. Lecture 4.- Nonhomogeneous Linear Second Order Differential Equations.

22) The Method of Variation of Parameters.

23) The Method of Undetermined Coefficients.

24) The Principle of Superposition.

25) Application of Second Order Differential Equations.

26) Unforced and Forced Motions.

27) Resonance.

Homework 4.pdf Mathematica tutorial 2.nb Quiz 3.pdf

Sep 27. Lecture 5.- The Laplace Transform.

28) Definition and Basic Properties.

29) Solution of Initial Value Problems Using the Laplace Transform.

30) Shifting Theorems and the Heaviside Function .

31) Convolution.

32) Laplace Transform Solution of Systems.

33) Differential Equations With Polynomial Coefficients.

Homework 5.pdf Mathematica tutorial 3.nb Quiz 4.pdf

Oct 4. Lecture 6.- Series Solutions.

34) Convergence and algebra of Power Series.

35) Taylor and Maclaurin Expansions.

36) Power Series Solutions of Initial Value Problems.

37) Frobenius method.

38) Logarithm Factors.

Homework 6.pdf Mathematica tutorial 4.nb Quiz 5.pdf

Oct 11. Lecture 7.- Review of linear algebra.

39) Vector spaces, linear dependence.

40) Basis and dimensions.

41) Matrices, algebraic operations, inverses and transposes.

42) Determinants.

43) Eigenvalues and Eigenvectors.

Homework 7.pdf Mathematica tutorial 5.nb Quiz 6.pdf

Oct 18. Lecture 8.- Systems of Linear Differential Equations.

44) Theory of Systems of Linear First Order Differential Equations.

45) Homogeneous and Nonhomogeneous Systems.

46) Solution of Linear Homogeneous Systems with Constant Coefficients.

47) Solution of Nonhomogeneous Systems.

48) Variation of Parameters.

Homework 8.pdf Mathematica tutorial 6.nb Quiz 7.pdf

Oct 25.- Midterm Exam.

Midterm Exam.pdf

Nov 1. Lecture 9.- Qualitative Methods and Systems of Linear Differential Equations.

49) Existence and Uniqueness of Solutions.

50) Phase Portraits of Linear Systems.

51) Critical Points and Stability.

Homework 9.pdf Mathematica tutorial 7.nb Quiz 8.pdf

Nov 8. Lecture 10.- Systems of Nonlinear Differential Equations.

52) Almost Linear Systems.

53) Lyapunov's Stability Criteria.

54) Limit Cycles and Periodic Solutions.

Homework 10.pdf Mathematica tutorial 8.nb Quiz 9.pdf

Nov 15. Lecture 11.- Fourier Series.

55) The Fourier Series of a Function.

56) Convergence of Fourier Series.

57) Fourier Cosine and Sine Series.

58) The Fourier Integral.

59) The Fourier Transform.

Homework 11.pdf Quiz 10.pdf

Nov 22. Thanksgiving Recess

Nov 29. Lecture 12.- Partial Differential Equations 1.

71) The Wave Equation and Initial and Boundary Conditions.

72) Fourier Series Solutions of the Wave Equation.

73) Wave Motion Along an Infinite String.

74) Characteristics and d'Alembert's Solution.

75) The Heat Equation.

76) Fourier Series Solutions of the Heat Equation.

Homework 12.pdf Quiz 11.pdf

Dec 6. Lecture 13.- Partial Differential Equations 2.

77) Fourier Series Solutions of the Heat Equation.

78) Heat Conduction in Infinite Media.

79) The Potential Equation.

80) Harmonic Functions and the Dirichlet Problem.

81) Dirichlet Problem for a Rectangle.

82) A Neumann Problem for a Rectangle.

Homework 13.pdf Quiz 12.pdf

Dec 13. Final exam.

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