Teaching Partial Differential Equations Teaching Partial Differential Equations Using Using MathematicaMathematica

Katarina JegdicKatarina Jegdic

Assistant ProfessorAssistant Professor

Computer and Mathematical Sciences DepartmentComputer and Mathematical Sciences Department

University of Houston – DowntownUniversity of Houston – Downtown

Wolfram Technology Conference, ChampaignWolfram Technology Conference, Champaign

October 21, 2011October 21, 2011

IntroductionIntroduction Traditional way of teaching upper level mathematics coursesTraditional way of teaching upper level mathematics courses

One way communicationOne way communication Instructor lecturingInstructor lecturing Students watching, listening, taking notes and working individually on Students watching, listening, taking notes and working individually on

assignmentsassignments

Proposal at UHD for the Quality Enhancement Plan (QEP) grantProposal at UHD for the Quality Enhancement Plan (QEP) grant Students taking a more active role in the courseStudents taking a more active role in the course Students working in small teams on solving research problemsStudents working in small teams on solving research problems Active learning and cooperative learning, learning by teaching others, Active learning and cooperative learning, learning by teaching others,

discussing, using technology, presenting results discussing, using technology, presenting results

Math 4304 (Introduction to Partial Differential Equations) at UHDMath 4304 (Introduction to Partial Differential Equations) at UHD Differential equations play a prominent role in physics, chemistry, Differential equations play a prominent role in physics, chemistry,

engineering, biology, economics, etc.engineering, biology, economics, etc. Applications in fluid dynamics, oil industry, medicine, aerospace Applications in fluid dynamics, oil industry, medicine, aerospace

engineering, traffic control, etc. (Houston oil sector, Texas Medical engineering, traffic control, etc. (Houston oil sector, Texas Medical Center, NASA)Center, NASA)

Course ImplementationCourse Implementation

Math 4304 syllabus:Math 4304 syllabus: Heat equation in one-dimensionHeat equation in one-dimension

• Derivation, boundary conditions, derivation in 2d and 3dDerivation, boundary conditions, derivation in 2d and 3d Method of separation of variablesMethod of separation of variables

• Introduction, linearity, examples with the heat equationIntroduction, linearity, examples with the heat equation Laplace equationLaplace equation

• Equation inside a rectangle, equation for a circular diskEquation inside a rectangle, equation for a circular disk Fourier seriesFourier series

• Introduction, convergence theorem, Fourier Cosine and Sine series, term-by-Introduction, convergence theorem, Fourier Cosine and Sine series, term-by-term differentiation and integrationterm differentiation and integration

Wave equationWave equation• Introduction, derivation, boundary conditionsIntroduction, derivation, boundary conditions

Sturm-Liouville eigenvalue problemsSturm-Liouville eigenvalue problems• Introduction, examples, general classification, self-adjoint operators, Introduction, examples, general classification, self-adjoint operators,

Rayleigh quotientRayleigh quotient Nonhomogenous problemsNonhomogenous problems Fourier Transform solutions of PDEsFourier Transform solutions of PDEs Laplace Transform solutions of PDEsLaplace Transform solutions of PDEs

Course Implementation - continuedCourse Implementation - continued

Real world problems motivate students and give meaning to the Real world problems motivate students and give meaning to the material studied in classmaterial studied in class

Often too complex for explicit solvingOften too complex for explicit solving Need to develop numerical methodsNeed to develop numerical methods

Projects consist of Projects consist of Derivation of differential equations which describe particular Derivation of differential equations which describe particular

physical phenomenonphysical phenomenon• The equations are simplified enough so that their basic properties The equations are simplified enough so that their basic properties

could be understood and analyzed theoreticallycould be understood and analyzed theoretically Study of numerical methods for the approximate solvingStudy of numerical methods for the approximate solving

• Students derive numerical methods based on the equation Students derive numerical methods based on the equation (finite difference, finite volume)(finite difference, finite volume)

Implementation of codes using Mathematica, Matlab or MapleImplementation of codes using Mathematica, Matlab or Maple

Course Implementation - continuedCourse Implementation - continued

ApplicationsApplications Aerodynamics Oil Flow Traffic FlowAerodynamics Oil Flow Traffic Flow

Shallow Water Equations Heat Equation Wave EquationShallow Water Equations Heat Equation Wave Equation

Time tableTime table First weekFirst week

The instructor provides a detailed study guide for each project The instructor provides a detailed study guide for each project consisting of book chapters, research articles, codes and any consisting of book chapters, research articles, codes and any additional informationadditional 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 weeksRemaining three weeks Problem driven, student-orientedProblem driven, student-oriented Students work in teamsStudents work in teams

• Understand particular example, derive equations, derive numerical Understand particular example, derive equations, derive numerical methodsmethods

• Use existing codes and/or implement their own codes in Use existing codes and/or implement their own codes in Mathematica, Matlab, MapleMathematica, Matlab, Maple

Discuss the results among themselves, with other teams and Discuss the results among themselves, with other teams and with the instructorwith the instructor

Students type their results using Microsoft Word or LatexStudents type their results using Microsoft Word or Latex

Example I: Wave EquationExample I: Wave Equation

- Information about the equation - Information about the equation

uutttt = c = c22 u uxxxx, , wherewhere 0<x<l 0<x<l and and 0<t<t0<t<t00

u(0,t)=0 u(l,t)=0 u(x,0)=u(0,t)=0 u(l,t)=0 u(x,0)=φφ(x) u(x) utt(x,0)=(x,0)=ψψ(x)(x)

- Solve the above problem using the method of separation of - Solve the above problem using the method of separation of variablesvariables u(x,t)=X(x)T(t)u(x,t)=X(x)T(t) - Solve the eigenvalue problems for - Solve the eigenvalue problems for T(t)T(t) and and X(x)X(x) with given with given boundary conditionsboundary conditions

- Plot - Plot TTnn(t)(t) and and XXnn(t)(t) for various for various nn

- - Find and plot solutions to various initial conditionsFind and plot solutions to various initial conditions

Study guideStudy guide

Example1: Example1:

Initial ConditionsInitial Conditions SolutionSolution

φφ(x)= x(1-x)(x)= x(1-x)

ψψ(x)=0(x)=0

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

0 . 2 5x

Example 2:Example 2:

Initial Conditions Solution Initial Conditions Solution

φφ(x)= sin(5(x)= sin(5ππx) + 2 sin(7x) + 2 sin(7ππx)x)

ψψ(x)=0(x)=0

0.2 0.4 0.6 0.8 1.0

2

1

1

2

3x

Example 3: (The plucked string)Example 3: (The plucked string)

Initial Conditions SolutionInitial Conditions Solution φφ((x) = 3x/2, xx) = 3x/2, x≤≤2/32/3

3(1-x), x>2/33(1-x), x>2/3

ψψ(x)=0(x)=0

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0x

Example 4: (Localized Plucking)Example 4: (Localized Plucking)

Initial Conditions SolutionInitial Conditions Solution

φφ(x)= 0, x(x)= 0, x≤≤aa

h(x-a)/(p-a), a<x<ph(x-a)/(p-a), a<x<p

h(x-b)/(p-b), p<x<bh(x-b)/(p-b), p<x<b

0, x>b0, x>b

ψψ(x)=0(x)=0

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5x

Example II: Secondary Oil Recovery Example II: Secondary Oil Recovery Study guide:Study guide:

Information about the equations Information about the equations

(s(sww))tt + f (s + f (sgg , s , sww))xx = 0 = 0

(s(sgg))tt + g (s + g (sgg, s, sww))xx = 0 = 0

x x – space, – space, tt – time – time

unknown functions: unknown functions: ssw w - saturation of water - saturation of water

ssgg - saturation of gas - saturation of gas

ssoo - saturation of oil - saturation of oil ssoo = 1 – (s = 1 – (sww + s + sgg))

Find the eigenvalues of the systemFind the eigenvalues of the system Show that the system is hyperbolicShow that the system is hyperbolic Find the umbilic points of the system Find the umbilic points of the system

Eigenvalues of the system

Umbilic pointsUmbilic points

Approximate SolutionsApproximate Solutions

Students’ ExperiencesStudents’ Experiences Team workTeam work Research oriented projectsResearch oriented projects Using technologyUsing technology Active, collaborative and cooperative learningActive, collaborative and cooperative learning Development of communication and writing skillsDevelopment of communication and writing skills

AcknowledgementsAcknowledgements QEP (Quality Enhancement Plan) Grant at UHDQEP (Quality Enhancement Plan) Grant at UHD

during the spring semester of 2011during the spring semester of 2011

http://cms.uhd.edu/qep/http://cms.uhd.edu/qep/ Dr. Linda Becerra, UHDDr. Linda Becerra, UHD Dr. Jeong-Mi Yoon, UHDDr. Jeong-Mi Yoon, UHD Dr. Volodymyr Hrynkiv, UHDDr. Volodymyr Hrynkiv, UHD