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Prerequisites

Calculus of Several Variables (18.02), Differential Equations (18.03) or Honors Differential Equations (18.034)

Course Outline

This course has three major topics:

Applied Linear Algebra

• Second Difference Matrices K, T, B, C
• Positive Definiteness: Pivots, Eigenvalues, Energy
• A^TCA Framework for Equilibrium Problems
  • Springs and Masses
  • Least Squares and Covariance Matrix
  • Graphs, Networks, Kirchhoff's Laws
  • Deformation of Trusses (and Mesh Generation)
  • Minimum Principles and Constraints
  • Finite Elements in One Dimension

Boundary Value Problems

• Ordinary Differential Equations
  • Boundary Conditions and Delta Functions
  • Dynamics: Mu'' + Ku = F(t)
  • Beam Equations and Cubic Splines
• Partial Differential Equations
  • Laplace and Poisson Equations
  • Divergence, Gradient, and Curl
  • Special Solutions from (x + iy)ⁿ and f(x + iy)
  • Potential, Stream Function, Cauchy-Riemann Equations
  • Finite Differences and Boundary Conditions
  • Finite Element Method and Weak Form

Fourier Methods and the FFT

• Fourier Series (and Orthogonal Polynomials)
  • Orthogonality and Parseval's Formula
  • Laplace Equation on a Circle
• Discrete Fourier Series
  • Fourier Matrix and the Fast Fourier Transform
  • Convolution and Filtering in Signal Processing
• Fourier Integral
  • Shannon Sampling Theorem
  • Differential Equations
  • Integral Equations (Convolution Kernel)

Assignments and Exams

18.085 has regular homeworks, three one-hour quizzes, and no final exam.

Texts

This course as taught during the Fall 2005 term on the MIT campus used the following text:

Strang, Gilbert. Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press, 1986. ISBN: 9780961408800. (Table of Contents)

Since that time, Professor Strang has published a new textbook that is being used for this course as it is currently taught on the MIT campus, as well as for Mathematical Methods for Engineers II (18.086). Information about the new book can be found at the Wellesley-Cambridge Press Web site, along with a link to Prof. Strang's new "Computational Science and Engineering" Web page developed as a resource for everyone learning and doing Computational Science and Engineering.

Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.

Grading