Teaching Partial Differential Equations Using Mathematica Katarina Jegdic Assistant Professor Computer and Mathematical Sciences Department University of Houston – Downtown Wolfram Technology Conference, Champaign October 21, 2011

Introduction  Traditional way of teaching upper level mathematics courses One way communication One way communication Instructor lecturing Instructor lecturing Students watching, listening, taking notes and working individually on assignments Students watching, listening, taking notes and working individually on assignments  Proposal at UHD for the Quality Enhancement Plan (QEP) grant Students taking a more active role in the course Students taking a more active role in the course Students working in small teams on solving research problems Students working in small teams on solving research problems Active learning and cooperative learning, learning by teaching others, discussing, using technology, presenting results Active learning and cooperative learning, learning by teaching others, discussing, using technology, presenting results  Math 4304 (Introduction to Partial Differential Equations) at UHD Differential equations play a prominent role in physics, chemistry, engineering, biology, economics, etc. Differential equations play a prominent role in physics, chemistry, engineering, biology, economics, etc. Applications in fluid dynamics, oil industry, medicine, aerospace engineering, traffic control, etc. (Houston oil sector, Texas Medical Center, NASA) Applications in fluid dynamics, oil industry, medicine, aerospace engineering, traffic control, etc. (Houston oil sector, Texas Medical Center, NASA)

Course Implementation  Math 4304 syllabus: Heat equation in one-dimension Heat equation in one-dimension Derivation, boundary conditions, derivation in 2d and 3dDerivation, boundary conditions, derivation in 2d and 3d Method of separation of variables Method of separation of variables Introduction, linearity, examples with the heat equationIntroduction, linearity, examples with the heat equation Laplace equation Laplace equation Equation inside a rectangle, equation for a circular diskEquation inside a rectangle, equation for a circular disk Fourier series Fourier series Introduction, convergence theorem, Fourier Cosine and Sine series, term- by-term differentiation and integrationIntroduction, convergence theorem, Fourier Cosine and Sine series, term- by-term differentiation and integration Wave equation Wave equation Introduction, derivation, boundary conditionsIntroduction, derivation, boundary conditions Sturm-Liouville eigenvalue problems Sturm-Liouville eigenvalue problems Introduction, examples, general classification, self-adjoint operators, Rayleigh quotientIntroduction, examples, general classification, self-adjoint operators, Rayleigh quotient Nonhomogenous problems Nonhomogenous problems Fourier Transform solutions of PDEs Fourier Transform solutions of PDEs Laplace Transform solutions of PDEs Laplace Transform solutions of PDEs

Course Implementation - continued  Real world problems motivate students and give meaning to the material studied in class Often too complex for explicit solving Often too complex for explicit solving Need to develop numerical methods Need to develop numerical methods  Projects consist of Derivation of differential equations which describe particular physical phenomenon Derivation of differential equations which describe particular physical phenomenon The equations are simplified enough so that their basic properties could be understood and analyzed theoreticallyThe equations are simplified enough so that their basic properties could be understood and analyzed theoretically Study of numerical methods for the approximate solving Study of numerical methods for the approximate solving Students derive numerical methods based on the equationStudents derive numerical methods based on the equation (finite difference, finite volume) (finite difference, finite volume) Implementation of codes using Mathematica, Matlab or Maple Implementation of codes using Mathematica, Matlab or Maple

Course Implementation - continued  Applications Aerodynamics Oil Flow Traffic Flow Aerodynamics Oil Flow Traffic Flow Shallow Water Equations Heat Equation Wave Equation

Time table  First week The instructor provides a detailed study guide for each project consisting of book chapters, research articles, codes and any additional information The instructor provides a detailed study guide for each project consisting of book chapters, research articles, codes and any additional information Projects consist of: Projects consist of: Derivation of the equationsDerivation of the equations Development of numerical methods for approximate solvingDevelopment of numerical methods for approximate solving Codes in Mathematica/Matlab/MapleCodes in Mathematica/Matlab/Maple  Remaining three weeks Problem driven, student-oriented Problem driven, student-oriented Students work in teams Students work in teams Understand particular example, derive equations, derive numerical methodsUnderstand particular example, derive equations, derive numerical methods Use existing codes and/or implement their own codes in Mathematica, Matlab, MapleUse existing codes and/or implement their own codes in Mathematica, Matlab, Maple Discuss the results among themselves, with other teams and with the instructor Discuss the results among themselves, with other teams and with the instructor Students type their results using Microsoft Word or Latex Students type their results using Microsoft Word or Latex

Example I: Wave Equation - Information about the equation - Information about the equation u tt = c 2 u xx, where 0<x<l and 0<t<t 0 u tt = c 2 u xx, where 0<x<l and 0<t<t 0 u(0,t)=0 u(l,t)=0 u(x,0)=φ(x) u t (x,0)=ψ(x) u(0,t)=0 u(l,t)=0 u(x,0)=φ(x) u t (x,0)=ψ(x) - Solve the above problem using the method of separation of - Solve the above problem using the method of separation of variables variables u(x,t)=X(x)T(t) - Solve the eigenvalue problems for T(t) and X(x) with given boundary conditions boundary conditions - Plot T n (t) and X n (t) for various n - Plot T n (t) and X n (t) for various n - Find and plot solutions to various initial conditions - Find and plot solutions to various initial conditions  Study guide

Example1: Initial Conditions Solution Initial Conditions Solution φ(x)= x(1-x) φ(x)= x(1-x) ψ(x)=0 ψ(x)=0

Example 2: Initial Conditions Solution Initial Conditions Solution φ(x)= sin(5πx) + 2 sin(7πx) φ(x)= sin(5πx) + 2 sin(7πx) ψ(x)=0 ψ(x)=0

 Example 3: (The plucked string) Initial Conditions Solution Initial Conditions Solution φ(x) = 3x/2, x≤2/3 φ(x) = 3x/2, x≤2/3 3(1-x), x>2/3 3(1-x), x>2/3 ψ(x)=0 ψ(x)=0

 Example 4: (Localized Plucking) Initial Conditions Solution Initial Conditions Solution φ(x)= 0, x≤a φ(x)= 0, x≤a h(x-a)/(p-a), a<x<p h(x-a)/(p-a), a<x<p h(x-b)/(p-b), p<x<b h(x-b)/(p-b), p<x<b 0, x>b 0, x>b ψ(x)=0 ψ(x)=0

Example II: Secondary Oil Recovery  Study guide: Information about the equations Information about the equations (s w ) t + f (s g, s w ) x = 0 (s w ) t + f (s g, s w ) x = 0 (s g ) t + g (s g, s w ) x = 0 (s g ) t + g (s g, s w ) x = 0 x – space, t – time unknown functions: s w - saturation of water s g - saturation of gas s g - saturation of gas s o - saturation of oil s o = 1 – (s w + s g ) s o - saturation of oil s o = 1 – (s w + s g ) Find the eigenvalues of the system Find the eigenvalues of the system Show that the system is hyperbolic Show that the system is hyperbolic Find the umbilic points of the system Find the umbilic points of the system

Eigenvalues of the system

 Umbilic points

Approximate Solutions

Students’ Experiences  Team work  Research oriented projects  Using technology  Active, collaborative and cooperative learning  Development of communication and writing skills Acknowledgements  QEP (Quality Enhancement Plan) Grant at UHD during the spring semester of 2011 during the spring semester of  Dr. Linda Becerra, UHD  Dr. Jeong-Mi Yoon, UHD  Dr. Volodymyr Hrynkiv, UHD