Applying the Finite Difference Method to Boundary-Value Problems 1 |

The finite difference method is a widely used technique for solving boundary-value problems. It approximates the derivatives in the differential equations using finite differences, thereby converting the differential equations into a system of algebraic equations. By solving these algebraic equations, one determines the values of the dependent variable at specific discrete points of the independent variable.

To demonstrate this approach, let’s solve the following second-order linear two-point boundary-value problem:

$\displaystyle \begin{cases} p(x) y''(x) + q(x) y'(x)+r(x) y(x) = s(x) \\ \alpha_0y(a) + \beta_0 y'(a) = \gamma_0 \\ \alpha_1 y(b) + \beta_1 y'(b) = \gamma_1 \end{cases}$

We begin by dividing the interval $[a, b]$ into $n$ subintervals, each of length $h=\frac{b-a}{n}$ , by the grid points

$x_0 = a, x_1=a+h, ..., x_i = x_0+i h, ..., x_n=b.$

Next, we approximate the unknown function $y(x)$ at these mesh points. By replacing the derivatives $y''(x)$ and $y'(x)$ with their finite difference approximations from “Deriving Finite Difference Approximations of the Derivatives“:

$y'(x_i) = \frac{y(x_{i+1})-y(x_{i})}{h} + O(h),$

$y'(x_i) = \frac{y(x_{i+1})-y(x_{i-1})}{2h} + O(h^2),$

$y''(x_i) = \frac{y(x_{i+1})-2y(x_i) + y(x_{i-1})}{h^2} + O(h^2),$

we obtain the following relationships that the solution must satisfy:

$\displaystyle \begin{cases} p(x_i)\frac{y(x_{i+1})-2y(x_{i})+y(x_{i-1})}{h^2} + q(x_i)\frac{y(x_{i+1})-y(x_{i-1})}{2h}+r(x_i)y(x_i) \approx s(x_i), \quad 1 \le i \le n-1\\ \alpha_0 y_0 + \beta_0 \frac{y_1-y_0}{h} \approx \gamma_0 \\ \alpha_1 y_n + \beta_1\frac{y_{n}-y_{n-1}}{h} \approx \gamma_1.\end{cases}$

Suppose that we can find numbers $y_0, y_1, ... , y_{n-1}, y_n$ that satisfy the equations:

$\begin{cases} \displaystyle p_i\frac{y_{i+1}-2y_i+y_{i-1}}{h^2} + q_i\frac{y_{i+1}-y_{i-1}}{2h}+r_i y_i = s_i, \quad 1 \le i \le n-1 \\ \alpha_0 y_0 + \beta_0 \frac{y_1-y_0}{h} = \gamma_0 \\ \alpha_1 y_n + \beta_1\frac{y_{n}-y_{n-1}}{h} = \gamma_1 \end{cases}\quad\quad\quad(*)$

where $p_i = p(x_i), q_i = q(x_i), r_i = r(x_i)$ and $s_i = s(x_i)$ .

Then we can consider $y_0, y_1..., y_{n-1}, y_{n}$ to be the approximations of the solution $y(x)$ at the the grid points $x_0, x_1, ... , x_{n-1}, x_n$ .

Furthermore, (*) can be rewritten as:

$\displaystyle \begin{cases} \left(\alpha_0 -\frac{\beta_0}{h}\right)y_0 + \frac{\beta_0}{h}y_1 = \gamma_0 \\p_i\frac{y_{i+1}-2y_i+y_{i-1}}{h^2} + q_i\frac{y_{i+1}-y_{i-1}}{2h}+r_i y_i = s_i, \quad 1 \le i \le n-1 \\ -\frac{\beta_1}{h}y_{n-1} + (\alpha_1 + \frac{\beta_1}{h})y_n = \gamma_1\end{cases}$

This forms a system of $n+1$ linear equations in the $n+1$ unknowns $y_0, y_1, ..., y_{n-1}, y_{n}$ . In matrix form it is:

$\displaystyle \begin{pmatrix} \alpha_0-\frac{\beta_0}{h} & \frac{\beta_0}{h} & & & & \\ \ddots & \ddots & \ddots & \\ & \frac{p_{k}}{h^2}-\frac{q_{k}}{2h}& \frac{-2p_{k}}{h^2}+r_{k}& \frac{p_{k}}{h^2}+\frac{q_{k}}{2h} \\ & \ddots & \ddots & \ddots \\& & -\frac{\beta_1}{h} & \alpha_1+\frac{\beta_1}{h}\end{pmatrix} \begin{pmatrix} y_0 \\ \vdots \\ y_k \\ \vdots \\y_{n} \end{pmatrix}$

$= \begin{pmatrix} \gamma_0 \\ \vdots \\ s_k \\ \vdots \\ \gamma_1\end{pmatrix}$

Solving this system of linear equations yields $y_i$ for $0 \le i \le n$ .

Exercise-1 Show that discretizing the boundary-value problem

$\displaystyle \begin{cases} p(x) y''(x) + q(x) y'(x)+r(x) y(x) = s(x) \\ y(a) = \alpha \\ y(b) = \beta\end{cases}$

leads to the following system of linear equations:

$\displaystyle \begin{pmatrix} 2p_1-r_1 h^2& -p_1-\frac{q_1}{2}h\\ \ddots & \ddots & \ddots \\ & -p_{k}+\frac{q_{k}}{2}h & 2p_{k}-r_{k}h^2 & -p_k-\frac{q_k}{2}h\\ & \ddots & \ddots & \ddots\\ & & -p_{n-1}+\frac{q_{n-1}}{2}h & 2p_{n-1}-r_{n-1}h^2&\end{pmatrix} \begin{pmatrix} y_1 \\ \vdots \\y_k \\ \vdots \\ y_{n-1} \end{pmatrix}$

$\displaystyle = \begin{pmatrix} -s_1 h^2+(p_1-\frac{q_1}{2}h)\alpha \\ \vdots \\ -s_k h^2 \\ \vdots \\ -s_{n-1}h^2 + (p_{n-1}+\frac{q_{n-1}}{2}h)\beta\end{pmatrix}$

Exercise-2 Show that a more accurate approximation of boundary conditions is given by

$\displaystyle \alpha_0 y_0 + \beta_0\frac{-3y_0+4y_1-y_2}{2h}=\gamma_0\quad\quad\quad(1)$

$\displaystyle \alpha_1 y_1 + \beta_1\frac{y_{n-2}-4y_{n-1}+3y_n}{2h} = \gamma_1\quad\quad\quad(2)$

Exercise-3 Deriving the system of linear equations by discretizing boundary conditions using (1) and (2).

Exercise-4 Show that $|y_i - y(x_i)| \le O(h^2).$

Applying the Finite Difference Method to Boundary-Value Problems 1 |

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