MATH 247 - Numerical solutions of Differential Equations

Instructor: Roberto De Leo.     Term: Spring 2020.

Euler
Cauchy

When & Where: Tuesday 4:30-7:30pm in ASB-B, Room 213.

Office: Academic Support Building B, Room 214

Email: roberto DOT deleo AT howard DOT edu

This class aims at teaching the basic methods for the numerical solution of Ordinary and Partial Differential Equations and at the same time at introducing the languages MATLAB and python with the goal of using these languages throught the semester to implement some of the algorithms and ideas covered in class. Because of this we will be able to cover less theory than a normal Numerical Analysis class does but in the end you will be able to write effective code to study Differential Equations.

Topics covered

  • Numerical Solution of Ordinary Differential Equations.
    1. one-step and multi-step methods for non-stiff systems of ordinary differential equations,
    2. extrapolation techniques and automatic step size selection,
    3. one-step and multi-step methods for stiff systems of ordinary differential equations and differential-algebraic equations,
    4. single and multiple shooting techniques, finite difference approximations, and the Ritz-Galerkin method for boundary value problems
  • Numerical Solution of Partial Differential Equations.
    1. Initial/boundary value problems for parabolic and hyperbolic PDEs (one space and one time dimension).
    2. Explicit finite-difference schemes. Implicit finite-difference schemes. Stability.
    3. Parabolic and hyperbolic PDEs in two and three space dimensions.
    4. Boundary value problems for elliptic PDEs.

Tentative Syllabus

  • Week 1: Motivation and introduction to Scientific Computing. Floating point system.
  • Week 2: Unix/Linux, introduction to Python & MATLAB.
  • Week 3 & 4: Initial value problems for ODEs: singlestep and multistep methods, accuracy & stability, explicit vs implicit methods, geometrical integration.
  • Week 5 & 6: Boundary value problems for ODEs: shooting method, finite difference method, Galerkin method.
  • Week 7 & 8: Parabolic PDEs.
  • Week 9 & 10: Hyperbolic PDEs.
  • Week 11 & 12: Elliptic PDEs.
  • Week 13 & 14: Review and writing code for complex problems.

Languages used

  • MATLAB
  • Python
  • Any other language registered students might be interested in.

Reading list

  • A. Iserles, Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics
    • Lecture Notes
  • D. Lee, Lecture Notes on Numerical Solutions of Differential Equations.
  • R. Hoppe, Numerical Differential Equations.
  • J.M. McDonough, LECTURES on. COMPUTATIONAL NUMERICAL. ANALYSIS of. PARTIAL DIFFERENTIAL EQUATIONS
  • B. Storey, Numerical solutions to differential equations
    • Other texts
  • D. Kinkaid & W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, Brooks/Cole Publishing Company
  • L. Burden and J.D. Faires, Numerical Analysis, 7th ed. (2001), Brooks/Cole.
  • U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Diferential Equations and Differential-Algebraic Equations (1998), SIAM.
  • L.F. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with Matlab (2003), Cambridge.
  • J.D. Lambert, Numerical Methods for Ordinary Differential Systems (1991), Wiley.
  • R. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (2007), SIAM.
  • J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations (2004), SIAM.