Mathematical Modeling Assignment-

Q1. For the p.d.e. U_xy - 2U_yy + 3U_y = 0.

a. Show that U = f(x) + e^-3xg(2x+y), where f and g are arbitrary smooth functions, is a general solution of  Do not solve!

b. How many x-conditions and y-conditions would we need to possibly solve the problem for a unique solution?

Q2. Consider the following pde:

α₁U_xx + α₂U_xy + α₃U_yy + α₄U_x + α₅U_y + α₆U + α₇ = 0

Complete the following sentences in terms of α_i: The linearity classification of p.d.e. is:

(a) semilinear if . . .

(b) quasilinear if . . .

(c) nonlinear if . . .

(d) homogeneous if . . .

Q3. Given that the equation αU_x + βU_y = χ  is quasilinear and its general solution is

U = x² + e^-xF(y² - x²),

where F is arbitrary:

(a) Could β = β(x, y, U)? Why/why not?

(b) Write down two independent families of characteristics for the pde.

(c) For the initial condition U = e^1-x + x² on the initial curve y = -x, determine the number of solutions for the IVP. Your method of determination is important.

Q4. Solve:

(a) U_yy - 9U = 0.               

(b) U_xx + U_xy - 12U_yy = 0.

(c) U_xy - 1/x(U_y) = x².

(d) 2U_xy + 5U_yy = 0.

Q5. Solve: 3U_x - 4U_y = 0; U = 3x - 1 on y = (x/2) + 1.

Q6. Solve: xU_x + (y+1)U_y + 1 - U = 0; x > 0 with U(x, x) = x². 

Q7. Revisit the Framework for Mathematical Modeling as discussed in your class. Re-organize that framework into THREE distinct headings/classes, listing the several steps in each class. Name each of your classes, keeping in mind that in such a case, a good (functional) name is suggestive . . ..[This demands more thought and less action. Your responses could be seen in terms of: not appropriate, acceptable, or excellent.]

Q8. The following "issue/problem" is vaguely stated:

• The U.S. Food & Drug Administration is interested in knowing if a new drug, Modex, is effective in the control of diabetes in the U.S.

(a) As is, we cannot utilize mathematical modeling.  Name 2 words in the stated problem that cause this vagueness.

(b) From a modeling standpoint, pose 3 strategic questions that you would ask (for clarification) in order to determine how to help the government to decide how to proceed in their quest.

(c) Assume that you have "satisfactory" answers to your questions in (b). State those answers.

(d) On the basis of those answers, reframe the "issue/problem" in a manner suitable to model the problem mathematically. In essence, "identify the quantity of interest unambiguously". [Be sure to avoid exclusive dependence on the Scientific Method.]

(e) What would you name as potential benefits, if any, of using Mathematical Modeling here as opposed to the Scientific Method? Explain briefly.

[Note: This question is seeking to gauge your understanding/perception of "how to get started" in the process of mathematical modeling. It does not call for massive literature search even though a cursory knowledge of the central issue may be helpful]