MATLAB: Integrate a function after having differenciated it

handlesintegrate

I would like to differenciate a function (2 variables) and after that integrate it, but its giving me errors all the time. You can see here an extract of the code. Does anybody now how can I do this?

  u = @(x,y) sqrt(1-x).*y;
  duxx = @(x,y) diff(u(x,y),x,2);
  k = (1/2)*integral2(duxx,0,1,0,1);

Best Answer

integral2 is a numeric integration routine. It will pass the function handle two arrays of values, and must return something the same size.

With your function handle, u(x, y) will be invoked, passing in the arrays. Your u is vectorized so it will return a numeric array.

The numeric array will then be passed as the first parameter to diff(), and the numeric x will be passed as the second parameter, and 2 will be passed as the third.

https://www.mathworks.com/help/matlab/ref/diff.html shows that three parameters are possible for diff() but only when the second is a positive integer representing the difference number and the third is the dimension number. The array of x values is not a dimension number.

You have gotten confused between the numeric diff() function and the symbolic diff() function which is calculus. The calculus diff does not work on function handles, just symbolic expressions and symbolic functions.

sym x y
duxx = matlabFunction(diff(u(x,y), x, 2), 'vars', [x, y])
integral2(duxx, etc)

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MATLAB: I need to plot the following functions but I receive this error “Invalid indexing or function definition. When defining a function ensure that the arguments are symbolic variables ….”

You have defined

 x = 2:30;
 y = 40:60;

but then you have

syms x y

That is equivalent to

x = sym('x');
y = sym('y');

which is to say that it overwrites x and y, and afterwards they are no longer numeric. But afterwards you sometimes treat them as symbols and sometimes treat them as numeric. In particular,

 [X,Y] = meshgrid(x,y);

attempts to treat them as numeric.

You also have

u = @(x,y) 10*sin(200*x)+0.1*cos(10*y)+(100*x)./y;           
v = @(x,y) 10*cos(200*x)+0.1*sin(10*y)-100*(x./y);
ex =  diff(u,x);
ey = diff(v,y);

The two diff() lines attempt to take the symbolic differentiation of a MATLAB function handle. You probably cannot do that. You can differentiate a symbolic expression, and you might be able to differentiate a symbolic function, but probably not a function handle.

ex = diff(u(x,y), x);
ey = diff(v(x,y), y)

would invoke the function handles on the symbolic parameters x and y, giving a symbolic expression as a result, and that symbolic expression can be differentiated.

MATLAB: Double Integral of Several Functions where Integrand is in Symbolic Array Format

Hi Saeid,

In order to perform integration of symbolic functions, you can use the int function from Symbolic Math Toolbox. This function can directly operate on a matrix of symbolic functions too. If you need to do a double integration (say integrate w.r.t x and y), you need to use the int function twice, first integrate your function w.r.t x and then integrate the result hence obtained w.r.t y (or vice-versa). For example, to perform the double integral of f(x,y) =xy over the region (x = 0 to x = 1 and y = 0 to y =1), we can use:

    syms x y
    f = x*y;
    value = int(int(f,x,0,1),y,0,1)
    value =
          1/4

Here the inner integral is w.r.t to x and the outer integral is w.r.t y.

The same applies for a matrix of symbolic functions also, for example

    syms x y
    A = [x + y, x^2 + y^2; x - y, x^2 - y^2];

    value = int(int(A,x,0,1),y,0,1)
    value = 
           [ 1, 2/3]
           [ 0,   0]

Best Answer

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MATLAB: I need to plot the following functions but I receive this error “Invalid indexing or function definition. When defining a function ensure that the arguments are symbolic variables ….”

MATLAB: Double Integral of Several Functions where Integrand is in Symbolic Array Format

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