To discretize the given second-order ODE using second-order accurate finite difference schemes, let's break the problem into the specified tasks:

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1. Problem Setup

The second-order ODE is:

$\frac{d^2y}{dx^2} + ay + by^4 = 0 \quad \text{for } 0 \leq x \leq 1,$

with boundary conditions: $\frac{dy}{dx}\bigg|_{x=0} = 0, \quad y(1) = 1.$

We will approximate derivatives using finite difference methods over a grid $x_0, x_1, \ldots, x_N$, where $x_0 = 0$, $x_N = 1$, and $h$ is the grid spacing ($h = \frac{1}{N}$).

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2. Discretization of the Second-Order ODE

a) Second-Order Accurate Forward Difference

For the second derivative $\frac{d^2y}{dx^2}$, the second-order forward difference formula is:

$\frac{d^2y}{dx^2} \bigg|_{x_i} \approx \frac{-y_{i+2} + 4y_{i+1} - 3y_i}{2h^2}.$

Substitute this into the ODE:

$\frac{-y_{i+2} + 4y_{i+1} - 3y_i}{2h^2} + ay_i + b(y_i)^4 = 0, \quad \text{for } i = 0, 1, \ldots, N-2.$

b) Second-Order Accurate Backward Difference

For the second derivative $\frac{d^2y}{dx^2}$, the second-order backward difference formula is:

$\frac{d^2y}{dx^2} \bigg|_{x_i} \approx \frac{3y_i - 4y_{i-1} + y_{i-2}}{2h^2}.$

Substitute this into the ODE:

$\frac{3y_i - 4y_{i-1} + y_{i-2}}{2h^2} + ay_i + b(y_i)^4 = 0, \quad \text{for } i = 2, \ldots, N.$

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3. Discretizing the Boundary Condition

c) At $x = 0$, use the second-order accurate forward difference for $\frac{dy}{dx}$:

The first derivative $\frac{dy}{dx} \big|_{x=0}$ can be approximated using the second-order forward difference formula:

$\frac{dy}{dx} \bigg|_{x_0} \approx \frac{-3y_0 + 4y_1 - y_2}{2h}.$

The boundary condition $\frac{dy}{dx} = 0$ at $x = 0$ becomes:

$\frac{-3y_0 + 4y_1 - y_2}{2h} = 0.$

This can be rearranged as:

$-3y_0 + 4y_1 - y_2 = 0.$

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Summary

• Forward difference (ODE): $\frac{-y_{i+2} + 4y_{i+1} - 3y_i}{2h^2} + ay_i + b(y_i)^4 = 0, \quad i = 0, 1, \ldots, N-2.$

• Backward difference (ODE): $\frac{3y_i - 4y_{i-1} + y_{i-2}}{2h^2} + ay_i + b(y_i)^4 = 0, \quad i = 2, \ldots, N.$

• Boundary condition at $x = 0$: $-3y_0 + 4y_1 - y_2 = 0.$

Would you like numerical examples or further explanation on implementing these discretizations?

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Expanding Questions:

1. How can we solve the resulting algebraic equations using matrix methods?
2. How would the discretization change for non-uniform grid spacing?
3. Can you compare the forward and backward difference approximations for accuracy?
4. How would you incorporate higher-order terms for better accuracy in these schemes?
5. How does the stability of the numerical solution depend on the choice of $h$?

Tip: Ensure the grid spacing $h$ is sufficiently small to achieve high accuracy in numerical solutions, especially for nonlinear terms like $by^4$.