Calculate the acceleration of an object from a vector of position data.

Create a vector of position data.

p = [1 3 6 10 16 18 29];

To find the acceleration of the object, use del2 to calculate the second numerical derivative of p. Use the default spacing h = 1 between data points.

L = 4*del2(p)

L = 1×7

     1     1     1     2    -4     9    22

Each value of L is an approximation of the instantaneous acceleration at that point.

Calculate the discrete 1-D Laplacian of a cosine vector.

Define the domain of the function.

x = linspace(-2*pi,2*pi);

This produces 100 evenly spaced points in the range $$-2\pi \le x\le 2\pi$$.

Create a vector of cosine values in this domain.

U = cos(x);

Calculate the Laplacian of U using del2. Use the domain vector x to define the 1-D coordinate of each point in U.

L = 4*del2(U,x);

Analytically, the Laplacian of this function is equal to $$\Delta U=-\cos (x)$$.

Plot the results.

plot(x,U,x,L)
legend('U(x)','L(x)','Location','Best')

The graph of U and L agrees with the analytic result for the Laplacian.

Calculate and plot the discrete Laplacian of a multivariate function.

Define the x and y domain of the function.

[x,y] = meshgrid(-5:0.25:5,-5:0.25:5);

Define the function $$U(x,y)=\frac{1}{3}(x^4+y^4)$$ over this domain.

U = 1/3.*(x.^4+y.^4);

Calculate the Laplacian of this function using del2. The spacing between the points in U is equal in all directions, so you can specify a single spacing input, h.

h = 0.25;
L = 4*del2(U,h);

Analytically, the Laplacian of this function is equal to $$\Delta U(x,y)=4x^2+4y^2$$.

Plot the discrete Laplacian, L.

figure
surf(x,y,L)
grid on
title('Plot of $\Delta U(x,y) = 4x^2+4y^2$','Interpreter','latex')
xlabel('x')
ylabel('y')
zlabel('z')
view(35,14)

The graph of L agrees with the analytic result for the Laplacian.

Calculate the discrete Laplacian of a natural logarithm function.

Define the x and y domain of the function on a grid of real numbers.

[x,y] = meshgrid(-5:5,-5:0.5:5);

Define the function $$U(x,y)=\frac{1}{2}\log \left(x^{2}y\right)$$ over this domain.

U = 0.5*log(x.^2.*y);

The logarithm is complex-valued when the argument y is negative.

Use del2 to calculate the discrete Laplacian of this function. Specify the spacing between grid points in each direction.

hx = 1; 
hy = 0.5;
L = 4*del2(U,hx,hy);

Analytically, the Laplacian is equal to $$\Delta U(x,y)=-(1/x^2+1/2y^2)$$. This function is not defined on the lines $$x=0$$ or $$y=0$$.

Plot the real parts of U and L on the same graph.

figure
surf(x,y,real(L))
hold on
surf(x,y,real(U))
grid on
title('Plot of U(x,y) and $\Delta$ U(x,y)','Interpreter','latex')
xlabel('x')
ylabel('y')
zlabel('z')
view(41,58)

The top surface is U and the bottom surface is L.