A water heater draws water from a well at temperature T = 5 °C and uses thermal flux Q from the sun to heat up water to temperature Tin = 60 °C. A schematic is provided below. For this system, the steady-state heat equations reduce to \frac{d^{2} T}{d x^{2}}=-\dot{Q} where Q =50 sin(2pie x), along with the boundary conditions T(x = 0) = Tw = 5

T(x = 1) =T₁n = 60

(a) Start by discretizing Eq. (1) using the second order central difference method \frac{d^{2} T}{\partial x^{2}} \approx \frac{T_{i+1}-2 T_{i}+T_{i-1}}{\Delta x^{2}} and construct the coefficient matrix and solution vectors. Show these matrices for a domain discretized with M = 8 equidistant elements.

(b) Solve Eq.(1) using a tri diagonal solver coded in Python or MATLAB. A tri-diagonal solver is a simplified Gaussian elemination solver that makes use of the banded nature of the matrix to reduce the amount of storage and computation required. We give you the following pseudo-code to get started:

(I) Store four ID vectors

(II)Elimination

for i = 1 to N do

b(i) = b(i)- c(i-1) *a(i)/b(i-1)

d(i)=d(i)-d(i-1)*a(i)/b(i-1)

end for

III) Back substitution.

d(N) = d(N)/b(N)

for i N-1 to 0 do

d(i) = ((d(i) - c(i)*d(i+1))/b(i))

end for

(c) Plot the numerical solution (i.e. T(x) vs x) for M = 4, 8, 16, 32, and 64 alongside the exact analytical solution to Eq. (1). How does the numerical result change with increasing mesh elements M?

(d) Conduct a performance study on your tri-diagonal solver by solving Eq.(1) with M = 10, 100, 1000, 5000, 10000 elements and plotting mesh size versus time. Briefly discuss your results.