(Solved): Problem 2,(40 points) Avial dispersion in a chemical tubular reactor is modeled using the following ...

Problem 2,(40 points) Avial dispersion in a chemical tubular reactor is modeled using the following equation: \[ c \frac{d^{2} \psi}{d \eta^{2}}-\frac{d \psi}{d \eta}-k \psi^{m}=0 \quad ; 0 \leq x \leq L \] Subject to boundary conditions: \( \psi(\eta=0)=0.8 \quad \) and \( \quad \psi(\eta=0.3)=0.4 \) Consider \( L=0.3, k=3 \), and \( c=0.1 \), where \( \eta \) is a dimensionless tubular axis, and \( \psi \) is a dimensionless concentration. Using the finite difference method, answer the following questions: (Part A) Consider the value \( m=1.0 \) : a) Transform the BVP to a system of linear equations in the form of \( A x=b \). Discretize the above equation into 3 equal sized intervals. (10 points) b) Use U decomposition method to solve the nystem of linear equarions. (si point) (Part B) Consider the value \( m=1.5 \) : c) Transform the BVP to a system of nonlinear equations. (5 points) d) To be able to solve the system of equations derived in part (c) using the Newton's method, arrange the equations in the form of: (10 points) \[ J\left(X_{i}\right)\left(X_{i+1}-X_{i}\right)=-F\left(X_{i}\right) \text { (Do not solve the system). } \] e) Explain briefly how to solve the system above. You may describe the process as a list of steps. (5 points)