This is one of those problems that can be solved analytically so numerical issues associated with one dimension being much larger than another need not be a problem. If you need an exact solution, look for solutions to the heat equation (the simpler versions that satisfy Laplace's equation) in the literature. You will find a series expansion for an exact answer. The zero Neumann boundary conditions are perfectly insulated boundaries in the context of the heat equation. The boundary with a non-zero Neumann condtion is a source of heat flow. If a good approximation is good enough for your application, you might be able to benefit  from one dimension being much larger than the other by treating the problem as quasi-one-dimensional.

I would suggest to non dimensionalize the problem. As it seems, your x:y ratio is 1:100000, then it follows. Only thing is that in uniform spacing  your grids in y direction would me much more compared to that in x direction.

Thanks for the answers

I have three questions/problem :

1) have a problem with the analytical solution because the heat source is a discrete function, meaning: F(x=100 nm,y)=j(y1,y2,...yn).

2)I do not familiar with the term\concept of quasi-one-dimensional. I'll try to find some material on this subject.

3)can I solve it in one dimension if the heat source is in the x direction?

thanks

As hinted by LG. Edmonds, this problem is a candidate for  perturbation analysis: If you introduce two length scales, then the Laplace equation becomes \epsilon\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}=0, where \epsilon is the ratio of length scales and

The problem is quasi-one-dimensional if the two-dimensional Laplace equation can be adequately approximated by the one-dimensional equation even though boundary conditions imply that the solution depends on both coordinates. But if your not sure if this approximation is good enough for your application, you can use an exact solution for the steady-state heat equation. For a rectangular geometry, this solution will be a sum of series expansions containing trigonometric functions and hyperbolic functions. One series produces the required the boundary conditions on one edge while producing homogeneous conditions on all other edges, and another series produces the required boundary conditions on another edge, and so on. Laplace transforms can be used to solve for the coefficients in each series expansion, even if the heat source is a discrete function. You can get the details by consulting any textbook that discusses the steady-state heat equation.

Sorry, I said Laplace transforms above when I should have said Fourier transforms.

So if I understand correctly than I can Solve this equation analytically by separating the problem to four rectangles as I did in the attached jpg and in the end sum over all solutions- U(x,y)=u1+u2+u3+u4?

and calculating numerically the fourier transform (series) for the neumann boundary?

That is almost correct. When the boundary conditions are

Ux(0,y) = 0

Ux(a,y) = f(y)

U(x,0) = 0

U(x,b) = v

the solutions to be added consists of the one produced by

Ux(0,y) = 0

Ux(a,y) =0

U(x,0) = 0

U(x,b) = v

 plus the one produced by

Ux(0,y) = 0

Ux(a,y) = f(y)

U(x,0) = 0

U(x,b) = 0

 Your application should be okay with this, because two edges are insulated, but if you might apply similar ideas to future problems you might want to be prepared for a possible difficulty that can occur when boundary conditions are discontinuous. For example, suppose the temperature at one location is forced to be some positive number while the temperature at an adjacent location (infinitely close to the first location) is forced to be zero. The heat flow flux at this location will be infinite and a series expansion will be difficult to deal with. It is okay for the heat flow (the Neumann condition) to be discontinuous but not okay for the temperature (the Dirichlet condition)  to be discontinuous. There are tricks for dealing with this but you don't have to know about them now as long as you know that some tricks are needed when dealing with some of the more disagreeable problems.  

thank you L.D. Edmonds for your answers.

I think I got to the point where I do need some tricks...

my boundary conditions are slightly different from what you wrote (also from the picture I sent)

The boundary conditions are

(1) Ux(0,y) = 0

(2) Ux(a,y) = f(y)

(3) U_y(x,0) = 0

(4) U(x,b) = v

the problem is that at the point (a,b) I have contradictions with the boundary conditions.

On the one hand, from (2) i get Ux(a,b) = f(b)>0 , but on the other hand , by saying U(x,b) = v ((4) boundary) it implies that Ux at this side should be zero, Ux(a,x)=0 and specifically Ux(a,b)=0. when I solve this analiticaly and plot it. it shows that there is a discontinuity at this point from zero to a certain value, namlay, between Ux(z,b) and Ux(z,b-\epsilon)

# Is there a problem with the fact that there is one point that contradicts?

# This problem suggest that there is a current source from one side, then it makes a turn and go out from the upper side. meaning that at the corner (a,b) the current immediately go out.

    Usually when a series that I am dealing with has bad convergence properties, because of discontinuities or singularities in some function, I look for some simple elementary function that has these discontinuities or singularities and then use an infinite series to represent the difference between the original function and the simple elementary function. The series then represents a more well-behaved function so the series has better convergence properties.

   For your example you need some simple elementary functions that are discontinuous at the boundaries. Some examples are given in the textbook [J. D. Jackson, Classical Electrodynamics, Second Edition, 1975, John Wiley & Sons, Inc.]. One example of a discontinuous function, with a known analytical solution, is on page 71 in that book, in a section called "A Two-Dimensional Potential Problem, Summation of a Fourier Series". Another example in the same book starts on page 75 in a section called "Fields and Charge Densities in Two-Dimensional Corners and Along Edges". The idea that will help you is to use one of these solutions, or some easy-to-derive variation of those solutions, as part of the answer. Then follow the first paragraph above. 

Thank u very much I will look into it tomorrow