HomeARITHMETICApplying the Finite Difference...

Applying the Finite Difference Method to Boundary-Value Problems 2 |

Applying the Finite Difference Method to Boundary-Value Problems 2 |

Applying the Finite Difference Method to Boundary-Value Problems 2 |

Let’s examine the following two-point boundary-value problem:

\displaystyle \begin{cases} v''(x) = c(x) v(x) + d(x) \\ v(a) = \alpha \\ v(b) = \beta \end{cases}.

Notice the first-order derivative term is missing from the differential equation. This omission is made without the loss of generality, as demonstrated in “z”(t) + p(t)z'(t) + q(t)z(t) = f(t) transformed“.

Following the approach in “Applying the Finite Difference Method to Boundary-Value Problems 1“, we first divide the interval [a, b] into n sub-intervals, 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, the derivative v''(x) is replaced by its finite difference approximation from “Deriving Finite Difference Approximations of the Derivatives“:

\displaystyle v''(x_i) = \frac{v(x_{i+1})-2v(x_i) + v(x_{i-1})}{h^2} + O(h^2).

Since the solution satisfies the following relationships:

\displaystyle \begin{cases} \frac{v(x_{i+1})-2v(x_{i})+v(x_{i-1})}{h^2} \approx c(x_i)v(x_{i}) + d(x_i), \quad 1 \le i \le n-1\\ v(x_0 = a) = \alpha \\ v(x_n = b)=\beta,\end{cases}

we consider v_1..., v_{n-1} to be the approximations of the solution v(x) at the the grid points x_1, ... , x_{n-1} if they satisfy the equations:

\displaystyle \begin{cases} \displaystyle \frac{v_{i+1}-2v_i+v_{i-1}}{h^2} = c_i v_i + d_i \quad 1 \le i \le n-1\\ v_0=\alpha \\ v_n = \beta\end{cases}\quad\quad\quad(*)

where c_i = c(x_i) and d_i = d(x_i).

In matrix form, (*) is a system of n-1 linear equations in the n-1 unknowns v_1, ..., v_{n-1}:

\displaystyle \begin{pmatrix} 2+c_1 h^2& -1\\ \ddots & \ddots & \ddots \\ & -1 & 2 + c_{k}h^2 & -1 \\ & & \ddots & \ddots & \ddots\\ & & &-1 & 2 + c_{n-1}h^2&\end{pmatrix} \begin{pmatrix} v_1 \\ \vdots \\v_k \\ \vdots \\ v_{n-1} \end{pmatrix} = \begin{pmatrix} -d_1 h^2+\alpha \\ \vdots \\ -d_k h^2 \\ \vdots \\ -d_{n-1}h^2 + \beta\end{pmatrix}

Solving this linear system yields v_i for 1 \le i \le n-1.

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