6.21.2 Laplacian: laplacian

Recall, the Laplacian of a function F of n variables x₁,…,x_n is

Also, the n× n discrete Laplacian matrix (also called the second difference matrix) is the n × n tridiagonal matrix with 2s on the main diagonal, −1s just above and below the main diagonal;

If L is the n× n discrete Laplacian matrix and Y is an n × 1 column vector whose kth coordinate is y_i = y(a + kΔ x) for a twice differential function y, then the kth coordinate of L Y will be −y(a + (k−1)Δ x) + 2 y(a + kΔ x) − y (a + (k−1)Δ x) (implicitly assuming that y(a) = y(a + (N+1)Δ x) = 0), which approximates y″(a + kΔ x). So L Y is approximately −Δ x² Y″, where Y″ is the n × 1 column vector whose kth coordinate is y″(a + kδ x).

The laplacian command can compute the Laplacian operator or the discrete Laplacian matrix.

To compute the Laplacian operator:

• laplacian takes two arguments:
  • expr, an expression involving several variables.
  • vars, a list of the variable names.
• laplacian(expr,vars) returns the Laplacian of the expression.

Example
Find the Laplacian of F(x,y,z)=2x²y−xz³.
Input:

Output:

To compute the discrete Laplacian matrix:

• laplacian takes one argument:
  n, an integer or floating-point integer.
• laplacian(n) returns the n× n discrete Laplacian matrix.

Examples.

• Input:
  laplacian(3)
  Output:

  ⎡ ⎢ ⎢ ⎣

  2 −1 0 −1 2 −1 0 −1 2

  ⎤ ⎥ ⎥ ⎦

• Input:
  laplacian(2.0)
  Output:

  ⎡ ⎢ ⎣

  2.0 −1.0 −1.0 2.0

  ⎤ ⎥ ⎦