Double Integral for Discrete Function

Dear All, I am solving a PDE in which I have to find the double integral of a discreet function. PDE has the following form:

where F(x,y) is a matrix of size (M,N), obtained by solving another PDE.

 [F,x,y]=some_funtion_to_solve_PDE(some inputs);

I'll use some initial guess for G(x,y) within the integral which also gives me a matrix of the same size as F(x,y).

Here the integrand is a product of two discrete functions and the integral limits are numbers. Matlab built-in routine quad2d() requires the integrand to be a continuous function of x & y. Similarly, the other routines like integral2() also require continuous function expressions. I need help and suggestions on solving this double integral with discrete function, which will enable me to solve the PDE. Many Thanks.

Answer 1

[Torsten]

Take a look at the example "Multiple Numerical Integrations" under

http://de.mathworks.com/help/matlab/ref/trapz.html

Best wishes

Torsten.

[Raees]

@Torsten:

Thanks for your answer. I've used the following command:

x=0:0.1:2; %limits of integration for x
y=0:0.1:2; %limits of integration for y
I=trapz(y,trapz(x,F,2));

But it gives the following error,

LENGTH(X) must equal the length of Y in dim 2.

I cannot understand what I am doing wrong to evaluate this integral. If possible please help me to correct this code. Thanks.

[Torsten]

You are aware that in your case, F must be a 21x21 matrix ?

Best wishes

Torsten.

[Raees]

@Torsten Thanks for reply.

I just used it as an example. F can be 41x41 matrix or 20x20 matrix. As you see in the original question, F comes from the solution of another PDE which i've solved earlier.

Thanks.

[Torsten]

And does it work now with F being a 21x21 matrix ?

x=0:0.1:2;
y=0:0.1:2;
[X,Y] = meshgrid(x,y);
F = X.^2 + Y.^2;
I = trapz(y,trapz(x,F,2))

Note that F is 21x21.

Best wishes

Torsten.

[Raees]

What i understood from your above code, you wrote F to be a continous function. I cannot understand the reason of defining F = X.^2 + Y.^2?

In my problem, F is not a continuous function of x and y, in fact its a discrete data. F is a matrix [f_{i,j}] and each f_{i,j} is the value of F at some point (x,y). Hope I have conveyed you clearly my problem.

Thanks.

[Torsten]

I just chose an analytical function form to create the matrix F for the sake of convenience.

F can be any matrix you like, as long as F has dimension

(length(y)) x (length(x))

Best wishes

Torsten.

[Raees]

Thanks, Torsten. This time, it worked. So nice of you.