Homework 7
1. Use finite differences to solve the equation 0.1d2u+u−u3 =sin(x)
u(0) = 1, u(10) = 1
The solution should look like:
2. The following differential equation can be used to model diffusion within a catalyst d2c(x) = Φ2c2(x)
dx2 subject to the boundary conditions:
c(0)=1, d c(1)=0 dx
The equation models the reaction 2A → B. The dimensionless parameter Φ is known as the Thiele modulus and captures the relative rates of reaction versus diffusion. Use finite differences to determine the effectiveness factor (η) as a function of the Thiele modulus Φ. Recall, the effectiveness factor is defined as
dx2 subject to the boundary conditions:
The solution should look like:
η=−1 dc(0) Φ2 dx
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Homework 7
3. Suppose now that the reaction is reversible: 2A ↔ B. The corresponding equations are d2cA(x) = −Φ2c2A(x) + 1 ΦcB(x), d2cB(x) = 1Φ2c2A(x) − 1 ΦcB(x),
dx2 K dx2 2 2K subject to the boundary conditions:
cA(0)=1, d cA(1)=0, cB(0)=1, d cB(1)=0, dx dx
Determine how the effectiveness factor (using the same definition for cA(x)) varies as a function of Φ and K. You should generate the following plot:
4. Solve the partial differential equation
∂u(t,x) = ∂2u(t,x) +u(1−u)
∂t ∂x2 subject to the boundary conditions:
u(t,0) = 1, 2
u(t,60) = 0,
Homework 7
and the initial condition
This equation is known as Fisher’s equation. You should generate the following plots:
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u(0, x) = 0, 0 < x < 60.