Chapter 30, Problem 15P

Solve the following PDE:

$\frac{{\partial }^{2}u}{\partial x^2}+b\frac{\partial u}{\partial x}=\frac{\partial u}{\partial t}$

Boundary conditions	$u\left(0,t\right)=0$	$u\left(1,t\right)=0$
Initial conditions	$u\left(x,0\right)=0$	$0\le x\le 1$

Use second-order accurate finite-difference analogues for the derivatives with a Crank-

Nicolson formulation to integrate in time. Write a computer program for the solution.

Increase the value of $\Delta t$ by 10% for each time step to more quickly obtain the steady-

state solution, and select values of $\Delta t$ and $\Delta t$ for good accuracy. Plot u versus x for various values of t. Solve for values of $b=54,2,0,-2,-4$.