The axial dispersion model in an isothermal tubular reactor can be described by:

$\frac{1}{Pe}\frac{d^{2}C}{d{x}^2}-\frac{dC}{dx}-F(C)=0$

$x=0,\frac{1}{Pe}\frac{dC}{dx}=C-1$

$x=1,\frac{dC}{dx}=0$

Where $x$ is the dimensionless distance through the reactor, $C$ is the reactant dimensionless

concentration, $Pe$ is the Peclet number and $F(C)$ is the dimensionless reaction rate. Using $Pe=$

$10,$ perform the following:

a$)$ For the linear case of first order reaction with $F(C)=4C,$ and using the derivative matrix

approach to represent finite difference derivatives, write the general form of matrix $A$

and vector $b$ that represent the discretization of this problem.

b$)$ For the nonlinear case of second order reaction with $F(C)=4C^2,$ and using the derivative

matrix approach to represent finite difference derivatives, write the general form of the

residual equations and Jacobian matrix that represent the discretization of this problem.

c$)$ Modify the sample code provided to you in class to solve this problem for part b$)$ using $N$

$=10.$ Note that in order to get full credit for this part, your code should run with flag $=0.$