How to write FDM matlab code for nonlocal source of the form u(1-integral u) in two dimension ?

How to write FDM matlab code for nonlocal source of the form ?

Answer 1

[Torsten]

Hint:

A possible approximation for the integral over a rectangle [x(i),x(i)+dx] x [y(j),y(j)+dy] is

dx*dy*(u(i,j)+u(i,j+1)+u(i+1,j)+u(i+1,j+1))/4

[GNANASEKARAN S]

Thank you. If I am right, u(i,j) is discreate data so we an use $\sum_{i,j=2}^{m,n} u(i,j)*dx*dy$ Isn't it?

[Torsten]

Yes, you must sum over all rectangles the domain is made of.

And of course, you will have to write out the integral correctly. If e.g. the domain of integration Omega is a circle in polar coordinates, you will have to write

integral over Omega u(omega) dOmega = integral_{r=0}^{r=R} integral_{theta=0}^{theta=2*pi} r*u(r,theta) dtheta dr

thus there is an additional factor r from the coordinate transformation in the integral that has to be included in the discrete form, too.

[GNANASEKARAN S]

Thank you.

[GNANASEKARAN S]

Dear Torsten,

how to run the matlab program in the excutable (eg. heat.exe) format ? in HPC cluster

[Torsten]

I don't know. You should open a new question for that.