Question 1:

First order hyperbolic partial differential equation: Consider the following partial differential equation:

3∂u/∂x + 2∂u/∂t = sin(x)

such that for 0 ≤ x ≤ 1.5, u(x, 0) = x².

1. Solve the equation using the method of characteristics (check your answer by subbing back in).

2. Sketch the region in the x, t plane where the solution to this equation is unique. 3.We wish to evaluate u(1, 0.5). Consider the following three numerical approaches:
• FTFS explicit forward in time, forward in space finite difference method with h = 0.2, k = 0.1.
• FTBS explicit forward in time, backward in space finite difference method with h = 0.2, k = 0.1.
• Lax-Wendroff explicit finite difference method with h = 0.1, k = 0.1.
State which of these three methods are stable for the given equation. (Give reasons for your answer by checking stability for all cases).

4. Use an appropriate method (from these three) to estimate u(1, 0.5).

Hint: you can use the Maple file chp6_hyp1_methods.mw to help with the numerical method.

Question 2:

Elliptic partial differential equation:

Consider the following partial differential equation defined for the region 0 ≤ x ≤ 3, 0 ≤ y ≤ 3:

u_xx+ u_yy = 4, u(x, 0) = 3(x+1)², u(x, 3) = 3(x+1)²-15, u(0, y) = (3+y)(1-y), u(3, y) = (8+y)(6-y)

1. By trying separation of variables, u(x, y) = F (x) + tt(y), solve this equation analytically. [Check your answer by subbing back into the original equation.]

2. Solve this equation numerically by converting the differential equation into centred finite differences and using step sizes in the x and y direction of 1 via the following methods.

(a)(Linear system) Write down the 4 linear equations in the 4 unknown interior points.

Then solve this linear system to find the values at the unknown points.
Hint: you can use a computer to solve the equations for you if you want. In Maple, try code something like:
A := <<-4|1|1|0>,<1|-4|0|1>,<1|0|-4|1>,<0|1|1|-4>>:

b := <-10/3, 0, -35/3, -5>:

with(LinearAlgebra): v := LinearSolve(A,b); evalf(v);

(b) (Gauss-Seidel) Perform the Gauss-Seidel method twice to estimate the values. Use initial guesses of 0 for values, and a relaxation parameter of 1. Do these calculations MANUALLY. (Make sure you update values as you go.) You may check your answers using a computer.

3. Compare the effectiveness of the above two numerical methods as the step size is decreased. You should consider accuracy, speed and storage space (for example the matrix of the linear system must be stored for the first case).

Question 3

Data modelling:

The Arrhenius equation describes an empirical relationship between the rate of a chemical reaction (k) and the absolute temperature (T ) for some chemical reactions. The equation is:

k = eA-1000E/T ,

where A and E are constants (commonly the second constant is written as 1000E = Ea/R (or E = Ek/R) where Ea is activation energy (in units of J/mol), (Ek is activation energy in kJ/mol) and R is the universal gas constant R = 8.314J/mol.K; P = eA is called the pre-multiplication factor). You may have seen this equation in first year chemistry. [The slight modification of the parameters from the common way this equation is presented is designed to increase numerical stability and make the calculations slightly neater.]

The following data have been gathered from a series of experiments of a certain chemical reaction:

T (Kelvin) 400 510 700
k (mol/L.sec) 0.011 0.789 145

1. Obtain approximate values for A and E by fitting the model to the outermost points. 2.Without transforming the data, use the Gauss-Newton method to obtain a least-squares regression fit for the model. Use the values from the previous part as your starting values.

• In order to understand the process (and get practice for the exam) perform 1 iteration by hand.

• Then get a computer program to complete the fit. How many iterations did it take to converge? Hint: you can use the Maple file chp8_data_modelling.mw.

• This particular model is usually fitted using a transformation (let y = ln(k) and x = 1/T ) to get a linear model y dependent on x. Use linear least squares regression to fit the linear model to the data. Compare the results with Gauss-Newton. Also compare efficiency of the method.

Hint: linear least squares regression can be done use the regression mode of a scientific calculator.

• If you try the Gauss-Newton method directly to the standard way of express- ing this equation: k = Pe-Ea/(8.314T ) (which appears to be equivalent to the equation above), then the method performs badly. Identify why the method does not do well in this case.

Note: you should check your answers - for example plot the points against the fitted equation to see if the fit is good.