Problem: Find the temperature distribution T(x) along a one-dimensional rod heated by a heat source f(x). The rod has unit length, with boundary conditions T(0) = 0 and T(1) = 0. The heat source is represented by f(x) = max(0, 1-(x-xc)^2/d^2), where d is the half-width of the heated region and xc is the center of the heated region.

The numerical approach being taken is to discretize the rod into n interior points x[i] = i*h for i = 1,2,...,n, where h = 1/(n+1) is the distance between points. Poission's equation is approximated as T[i-1] - 2T[i] + T[i+1] = -A*h^2*f[i], where T[i] ~= T(x[i]) and f[i] = f(x[i]). This equation is the ith equation of n equations (one for each value of i) that can be represented as the linear system MT=b, where M is the tridiagonal matrix below andbhas components b[i] = A*h^2*f[i]. It is implied that T[0] = 0 and T[n+1] = 0, which corresponds to the boundary conditions T(0)=0 and T(1) = 0.

M=

2 -1 -1 2 -1 -1 2 -1 ... -1 2 -1 -1 2

We are asked to write a function (in MATLAB) to solve the system MT=busing the MATLAB backslash operator (T= M\b).

The problem I'm having is that I do not see how it's possible to numerically solve this system without having values for the vectorb. Although we are provided with an equation for finding the values of the elements b[i] of this vector, I don't see how that can be done since the values of the constants A, xc, and d have not been specified. It's certainly possible to solve this system analytically if you allow the unspecified constants to remain that way, but I can't see how I can solve this numerically.

I've attempted to try to express the unknown constants xc and d in terms of x or h, but this approach did not work. I've also attempted to examine what the upper and lower bounds for f(x) and other constants might be, but I do not see how the results I obtained help me solve the problem. Even so, A is only stated to be a "proportionality constant" so I do not see how I can get around being told what A is, since it is not equated to any other variables of the problem.

Anyone have any ideas or thoughts about what it is I'm not seeing and what I need to do to get on the right path?

Thank you!