$6\mathrm{.}18$ In this problem, we will look at the convergence of the solution to a boundary

value problem as a function of the grid spacing. Consider the reaction$-$diffusion equation

$\frac{d^{2}c}{d{x}^2}=k(x)c$

where the spatially dependent reaction term is

$k(x)=10[1+sin(x)]$

The boundary conditions for the problem are $c(-1)=1$ and $c(1)=0\mathrm{.}5.$

You should write a MATLAB code that solves this problem using centered finite

differences for grid spacings $x=\frac{1}{2},\frac{1}{4},$dots,$\frac{1}{1024}.$ For the smallest value of

$x=x_{\text{m}\text{i}\text{n}},$ make a plot of the concentration versus position. We will estimate the

error in the solution by comparing the value of the solution at $x=0$ for the different

values of $x.$ Define the error of the solution as

$lon(x)=\frac{c(0,x)}{c(0,x_{\text{m}\text{i}\text{n}})}-1$

In other words, lets assume that the finest grid spacing corresponds to a "perfect" solu$-$

tion and assess the fractional error at the other values of $x.$ Make a semilog$-$x plot of $lon$

versus $x.$

Please write code in MATLAB to solve this problem and do not use any automatic matlab functions