Using Mathematica at the
University of Houston - Downtown

Katarina Jegdic
Associate Professor
Department of Mathematics and Statistics
University of Houston - Downtown

ABSTRACT

We give an overview of topics, courses, and projects in the undergraduate curriculum at the University of Houston - Downtown (UHD) where the students have the opportunity to use Mathematica.  The emphasis is on students’ discovery and active learning through exploration of mathematical concepts. 
Examples include: 
- analyzing graphs of functions in College Algebra (Math 1301) and Calculus I (Math 2401), 
- finding eigenvalues, eigenvectors, and inverse matrices in Linear Algebra (Math 2407), 
- understanding solutions of ordinary and partial differential equations in Differential Equations (Math 3301) and Partial Differential Equations (Math 4304) courses, 
as well as using Mathematica in high-impact experiences, such as, 
- Undergraduate Research (Math 3394), 
- Directed Study (Math 4399), 
- Senior Project (Math 4395), and 
- undergraduate research projects sponsored by UHD Scholars Academy. These high-impact venues are taught one-on-one and are on a certain advanced topic not covered in the class. Typically, the topics include real world applications and computational mathematics where students could use Mathematica, Matlab, or Maple. 

Acknowledgments:  UHD Scholars Academy
                                U.S. Nuclear Regulatory Commission Grant

Introduction

University of Houston - Downtown 
- four-year state university 
- one of four distinct institutions in the University of Houston System (UH - main campus, UH - Downtown, UH - Clear Lake, and UH - Victoria)
- founded in 1974
- the second-largest university in Houston with more than 14,000 students
- Hispanic Serving Institution and Minority Serving Institution (48% Hispanic, 22% African-American, 16% white)
- four academic colleges (Science and Technology, Business, Humanities and Social Sciences, and Public Service)
- offers 45 bachelors and 8 masters degree programs

Department of Mathematics
- majors: 
            B.S. in Mathematics
            B.S. in Statistics
            B.S. in Data Science (first and still the only undergraduate degree in Texas)
            
- M.S. in Data Analytics

Scholars Academy
- about 150 students from College of Science and Technology, STEM majors
- about 10 groups, each group mentored by a Peer Mentor and a Faculty Mentor
- responsibilities include attending and participating in various events: 
             orientations, networking activities, volunteering activities, fild trips,
             Student Research Conference, Graduate School & Internship Fair

Using Mathematica in classroom

College Algebra - Math 1301
- investigating properties of functions by graphing them (domain, range, intercepts, asymptotes, monotonicity, extrema)

Plot[x^3-x, {x,-2,2}]

Plot[Exp[x],{x,-2,2}]

Plot[Log[x],{x,-5,5}]

- investigating properties of polynomial functions depending on the degree and the sign of the leading coefficient (number of zeros, end-behavior)

Plot[x^4-2 x^3-x^2+2 x,{x,-2.5,2.5}]

Plot[x^5-5x^3+4 x,{x,-3,3}]

Plot[x^5-5x^3+4 x+10,{x,-3,3}]

Plot[-x^4-2 x^3-x^2+2 x,{x,-2.5,2.5}]

Calculus - Math 2401
- investigating graphs of functions (domain, range, asymptotes, zeroes, monotonicity, extrema, concavity, inflection points, etc.)

Plot[x/(x+1), {x,-5,5}]

- finding derivatives

D[3x^2-1,x]

- finding integrals - indefinite and definite

Integrate[3x^2,x]

Integrate[3x^2,{x,0,1}]

Plot[3x^2, {x,0,1}, Filling->Axis]

Differential Equations - Math 3301
- understanding solutions to ordinary differential equations

DSolve[y'[x]==x,y[x],x]

Plot[{x^2/2,x^2/2+1,x^2/2-1, x^2/2+2},{x,0,5},PlotLegends->"Expressions"]

Partial Differential Equations - Math 4304
- understanding solutions to partial differential equations
- transport equation u_t+ 2 u_x=0
  (analytically solve it using the method of characteristics)

pde=D[u[x,t],t]+ 2D[u[x,t],x]==0

sol=DSolve[pde,u[x,t],{x,t}]

sol1=DSolve[{pde,u[x,0]==Sin[x]},u[x,t],{x,t}]

Plot3D[u[x,t]/.sol1,{x,-5,5},{t,-5,5}]

- wave equation u_tt = u_xx 
           (analytically solve it suing separation of variables)

pde=D[u[x,t],{t,2}]==D[u[x,t],{x,2}]

sol=DSolve[pde,u[x,t],{x,t}]

ic={u[x,0]==x(1-x),Derivative[0,1][u][x,0]==1}

DSolveValue[{pde,ic},u[x,t],{x,t}]

DSolveValue[{pde,ic},u[x,t],{x,t}];
Plot3D[%,{x,-7,7},{t,0,4}]

Using Mathematica in High-Impact-Experience Projects

Undergraduate Research (Math 3394), Directed Study (Math 4399), and Senior Project (Math 4395)
- a semester long course
- one-to-one or in a small team of two-three students
- real world applications: gas dynamics, secondary oil recovery, traffic flow, shallow water equations

- students are given a detailed study guide (theory, numerical methods, etc.) that instructors goes over during the two weeks of the semester
- students develop a Mathematica code that computes approximate solutions for a variety of initial value problems

Example 1: Secondary oil recovery

Equations: 
        (Subscript[(sw), t])  + f (sg , (Subscript[sw), x]) = 0 
        (Subscript[(sg), t]) + g (sg, (Subscript[sw), x]) = 0
         
        sw = saturation of water
        sg = saturation of gas
        so = saturation of oil = 1-sw-sg
        
- Find the eigenvalues of the system
- Show that the system is hyperbolic
- Find the umbilic points of the system     
- Numerical solutions


Example 2: Leslie Matrix

Application of Linear Algebra to a linear deterministic discrete population dynamics model. 

The population of certain species is divided into a discrete number of age groups and represented by a vector: (n^t)=((Subscript[n, 1]), (Subscript[n, 2]), ..., (Subscript[n, k]))
If at time period i, the population is given by the vector (n^i), then at the period i+1, the population is given by 
           (n^(i+1))= P (n^(i ))
where P is a fixed k x k matrix. 

- Assume k=3, where the age of the individuals from the first group is greater than 0, but less than 1, the age of the individuals from the second group is at least 1, but less than 2, and the age of the individuals in the third group is at least 2, but less than 3. Then 
        (Subscript[n, 1 ])+ = (Subscript[P, 11]) * (Subscript[n, 1]) + (Subscript[P, 12]) * (Subscript[n, 2]) + (Subscript[P, 13]) * (Subscript[n, 3])
        (Subscript[n, 2 ])+ = (Subscript[P, 21]) * (Subscript[n, 1]) + (Subscript[P, 22]) * (Subscript[n, 2]) + (Subscript[P, 23]) * (Subscript[n, 3])
        (Subscript[n, 3 ])+ = (Subscript[P, 31]) * (Subscript[n, 1]) + (Subscript[P, 32]) * (Subscript[n, 2]) + (Subscript[P, 33]) * (Subscript[n, 3])
- We interpret  (Subscript[P, 11]), (Subscript[P, 12]), and (Subscript[P, 13]) as fertility rates and denote them by (Subscript[f, 1 ]), (Subscript[f, 2 ])  and (Subscript[f, 3])
- We intepret (Subscript[P, 2i]) and (Subscript[P, 3j])  as survival rates, assume (Subscript[P, 22])=(Subscript[P, 23])=(Subscript[P, 31])=(Subscript[P, 33])=0, and denote (Subscript[P, 21]) = (Subscript[P, 1]) and (Subscript[P, 32]) = (Subscript[P, 2])   
- We obtain a Leslie matrix

MatrixForm[{{Subscript[f,1], Subscript[f,2], Subscript[f,3]},{Subscript[P,1], 0, 0},{0, Subscript[P,2], 0}}]

L1:={{0,0,8},{0.25,0,0},{0,0.5,0}}

MatrixForm[L1]

Eigenvalues[L1]

Eigenvectors[L1]

MatrixForm[Power[L1,2]]

MatrixForm[Power[L1,10]]

Undergraduate Research Projects with Scholars Academy

Research supported by U. S. Nuclear Regulatory Commission Grant
Title of the Project: Preparing Minorities for Nuclear-related Jobs through Research and Certification

Example: Two-Phase liquid-gas flows in vertical pipes 


                    https://whatisnuclear.com/reactors.html


https://www.nuclear-power.net/nuclear-engineering/fluid-dynamics/two-phase-fluid-flow/mist-flow/

Equations:

(Subscript[((Subscript[α, l])), t ]) + h(Subscript[((Subscript[α, l])), x]) = 0

(Subscript[((Subscript[α, g])), t ]) + g(Subscript[((Subscript[α, g])), x]) = 0

where t = time
          x= space
          (Subscript[α, l]) = liquid volume fraction 
          (Subscript[α, g]) = gas volume fraction

- Solve the system by implementing a finite difference method