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I am trying to plot functions in MATLAB. In the first plot I write a function and use the gradient command to plot the function. In second plot I have taken the derivative of a function and then plotted it. But the two plots are not identical. Where is the discrepancy?

MATLAB Code:

X=-1:.05:1;
T=-1:.05:1;
m=1+1*1i;
[x,t]=meshgrid(X,T);
r1=m.*sin(x+4.*t);
dr1=gradient(r1);
dt=gradient(t);
dr1dt=dr1./min(diff(T));
td=t(2:end);

surf(x,t,abs(dr1dt));

I have plotted it.

Then I take the derivative of r1 w.r.t t and then plot the function. i.e.,

X=-1:.05:1;
T=-1:.05:1;
m=1+1*1i;
[x,t]=meshgrid(X,T);
r1=4.*m.*cos(x+4.*t);

surf(x,t,abs(r1)');

I have found that there is difference of amplitude in these plots. Why are the two plots not identical?

The function gradient, with one output argument, computes the partial derivative in the x direction. Thus, what you compute in the first bit of code is d/dx r1, not d/dt r1. With two output arguments it computes both the x and the y derivatives.

In the second bit of code, the computed derivative is transposed, leading to swapping the x and t axes.

The following bit of code fixes both these errors:

X=-1:.05:1;
T=-1:.05:1;
m=1+1*1i;
[x,t]=meshgrid(X,T);
r1=m.*sin(x+4.*t);

[dr1dx,dr1dt]=gradient(r1);
dr1dx=dr1dx/mean(diff(X));
dr1dt=dr1dt/mean(diff(T));

dr1dt_true=4.*m.*cos(x+4.*t);

figure
subplot(1,2,1)
surf(x,t,abs(dr1dt));
xlabel('x')
ylabel('t')
title('discrete approximation')
subplot(1,2,2)
surf(x,t,abs(dr1dt_true));
xlabel('x')
ylabel('t')
title('analytical')