Solving 1D Poisson's equation approximation using a linear system

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 M**T**=**b**, where M is the tridiagonal matrix below and **b** has 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 M**T**=**b** using 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 vector **b**. 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!

Re: Solving 1D Poisson's equation approximation using a linear system

This is a linear differential eigenvalue problem to an approximating algebraic system.

$\displaystyle y"{+$\lambda $y=0}$

y(0)=y(1)=0

$\displaystyle y_j+\left({$\lambda $h}^2-2\right)y_{j-1}+y_{j+1}=0$ or similar.

Requiring this to hold at the interior points j=1,...N, we have an algebraic eigenvalue problem:

$\displaystyle M y=h^2 \lambda y$

with the band matrix M and $\displaystyle y^T=\left(y_1,\text{...}.y_N\right)$

The exact solution is $\displaystyle y_j=\sin {n$\pi $x}_j$ with $\displaystyle \lambda _n=\left(4\left/h^2\right.\right){Sin}^2({n$\pi $h}/2)$

as h tends to zero these results converge to those of the target differential problem.