APPM 2360 Lab 2
Linear Systems A Diffusive System

This lab demonstrates the use of linear systems to model physical processes. Many linear and nonlinear systems of first-order differential equations are used to analyze modern physical problems such as the spread of viruses (hepatitis c virus (HCV) and human immunodeficiency virus (HIV)), fluid flow through porous media, diffusive systems, and countless other physical situations.

In this lab, we will use techniques from linear algebra to solve a system of differential equations modeling a diffusive system. The steps taken to determine the solution to the system will incorporate all of your knowledge of linear algebra. Individually each step is a relatively straightforward problem from linear algebra, but combined, these steps form a very powerful method for solving linear systems of differential equations. This method will be further addressed during the course of the semester.

This lab is due, in class, on November 2nd.

Follow these guidelines:

• Three (3) students is the maximum group size.
• Use a cover page.
• List the name, recitation number, TA name, and instructor for each student in the group on the cover page.
• There are some lengthy calculations in this lab that will need to be put in an appendix. Do not include these in the body - reference them and summarize as you present your results in the body, but do not show every step in the body of the report. Simillarly, there are questions that you can use a computer to help you answer. If you do,  please attach the code, and not the output, in the appendix. Please be aware of how much space this takes up and consider ways to reduce the amount of paper used. Try 2 pages to a side or a small font.
• Submit the report in class, do not put reports in mailboxes unless specifically told to do so by a TA or a professor.
• Three decimal point accuracy will suffice for this lab.
• Refer to the links here for more information.
  

Free advice:

• Start early.
• Work in a group, it can help.

The Diffusive System

Consider three containers separated by a semi-permeable membrane as demonstrated below:

Assume that the containers have the same volume, V, and that they each contain some chemical in a solution with water. The boundaries between these containers are semi-permeable, that is, the chemical is allowed to pass through the membrane by the process of diffusion, but no water is allowed to pass between the containers. Further assume that the chemical is not saturated and remains in solution. We will investigate the diffusion of a chemical in this system using a system of first order differential equations.

Derivation of the Model

Using conservation laws, we can derive a system of differential equations to model the flow of the chemical through the membranes. The change in concentration of the chemical in one of the containers is proportional to the difference between the concentration of the adjacent containers. Let y₁(t), y₂(t), and y₃(t) be the concentrations of the chemical in the containers 1, 2, 3, respectively. Using the conservation law and concentration defined by mass divided by volume of solution, we have the following system of differential equations:

    y₁' = k(-y₁ + y₂)

    y₂' = k(y₁ - 2y₂ + y₃)

    y₃' = k(y₂ - y₃)

where the constant of proportionality, k, depends upon the type of membrane.

Questions and Issues to Address

1. Write the system in matrix form (see page 115 of section 3.1 in Farlow et al.). Call the coefficient matrix A, let x = [y₁(t), y₂(t), y₃(t)]^T. Classify the system using the terms on page 125 of section 3.1 in Farlow et al.
2. Find all equilibrium solutions for this system, that is, solve Ax = 0, where A is the coefficient matrix and x is as in problem 1. Explain the physical meaning of the equilibrium solution. Is this system consistent or inconsistent? (For help see page 137 of Section 3.2 in Farlow et al.) Be sure to explain your answer.
3. Calculate the determinant of A. You should do this by hand and show your work in the appendix, but you may refer to a software program to check your work. What does this say about the matrix A?
4. For the remainder of this lab, set k = 3. Solve det(A - λI) = 0. To do this, first calculate the determinant. This will yield a cubic polynomial. Set the polynomial equal to zero and solve for the roots. These roots are called eigenvalues and are important for determining the analytical solution to the system.
5. The next step in determining the analytical solution to this system is to find the corresponding eigenvectors. Eigenvectors are linearly independent vectors that make up one component of the analytical solution. In this case we must find three eigenvectors. To do this, we use Gauss-Jordan elimination to find the reduced row echelon form for three problems, i.e., we solve Bv = 0 for three different coefficient matrices corresponding to the three eigenvalues determined in the previous problem. The system Bv = 0 can also be solved using Gaussian elimination with back substitution. The first system is Bv₁ = 0, where B = (A + 3I).
   • Determine the column vector v₁.
   • Be sure to clearly show the steps in Gauss-Jordan elimination or in Gaussian elimination with back substitution in the appendix.
   • Be sure to state which method you are using before solving, and include all calculations in the appendix - no credit will be given without the work.
   • Classify this system as consistent or inconsistent?
   • Choose one solution, that is, fix the arbitrary parameter in eigenvector, v₁.
6. Next find v₂ by solving Bv₂ = 0, this time where B = (A + 9I). Follow the same guidelines as in the previous problem.
7. One last time, determine v₃ by solving Bv₃ = 0, where B = A. Again, follow the same guidelines as before.
8. Determine if the three column vectors, v₁, v₂, and v₃, are linearly independent or not. Be sure to discuss your methodology for testing for linear independence and include any necessary calculations.
9. The general solution to this system of differential equations is given by the following:
   Y(t) = C₁v₁e^-3t + C₂v₂e^-9t + C₃v₃
   where Y(t) = [y₁(t), y₂(t), y₃(t)]^T. Use this to determine the general solution to the system by substituting the eigenvectors, that is v_{1, 2, 3}, you found in the previous problems. Note that, since there are infinitely many choices for each of the three eigenvectors, your results may not agree exactly with other students' results. Your choice depends on the scalar factor you chose when determining the column vectors v_{1, 2, 3}.
10. Use the initial condition Y(0) = [4, 3, 1]^T and the equation in question 9 to determine the arbitrary constants C_{1, 2, 3}. What happens to Y(t) as t → ∞? What does this mean physically? What is happening to the concentration of the chemical in the containers?
11. Define M(t) = y₁(t) + y₂(t) + y₃(t), and use the following
        y₁(t) = (1/6)e^-9t(-1 + 9e^6t + 16e^9t)
        y₂(t) = (1/3)e^-9t(1 + 8e^9t)
        y₃(t) = (1/6)e^-9t(-1 - 9e^6t + 16 e^9t)

    Compute M(t) and M'(t).

12. Produce a graph of each of the functions y₁(t), y₂(t), and y₃(t) from question 11. Discuss the behavior for each function as t → ∞. Be sure to
    • Label the graph and axes.
    • Choose an appropriate domain to adequately explore the function (make sure you don't miss anything!).
13. What are M(t) and M'(t)? Give a physical interpretation for these quantities. Based on what you found in problem 12, are the results of your computation expected? Explain. [Note that the question about the multiplicative factor has been removed.]
    
14. In this analysis, we chose a specific value for the parameter k in order to carry out calculations. What is the physical meaning of k? What effect will changing the value of k have on the analysis we performed? Explain your reasoning.
15. In our discussion of this diffusive system, we made several assumptions that simplified the model. For instance, we assumed that the semi-permeable membrane was the same between the containers and that only the chemical could pass through the membrane. Address any issues that you feel may improve the model. Be sure to discuss your assertions and provide justification for your claims.

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    Written by Matt Nabity 10/2002, Editted by Juliet Hougland 10/2011

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