Dear Colleagues;
I want to solve an equation like \nabla \Psi=0 with a Dirichlet boundary condition on the dashed-blue line border, where ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Psi(y)=\int_0^y f(y) dy

Indeed I want to integrate over a boundary to return value at each node, while the existing functions return the integral on whole boundary. In other words I want F(y)=\int_0^y f(y) dy , while the existing function gives F(L)=\int_0^L f(y) dy.

Thanks so much for your answer; Your understanding is correct. But: \psi(y)=\int_0^y f(y)dy . where f(y) is indeed a data nodes [There is not a explicit relation to integrate and f(y) comes out of iteration on data]

What I think I need to do is:

Define new variable like \alpha where \alpha=\int_0^y f(y)dy.

In my weak form add the Dirichlet BC as …+ on(3,\psi=\alpha)+…

As we can see I need to calculate the a cumulative integral for each node,

The only integration in FreeFEM is int1D which returns the integral one whole boundary i.e. \int_0^L f(y)dy, and returns one number as result.
I am wondering how can I calculate a cumulative integral (i.e. \int_0^y for each point, y \in [0, L], which returns results for all nodes)

Thanks so much for your response;
As I said an equation like\nabla \psi =0, not exactly this one;

The question is how to make a Dirichlet BC as: \psi(y)=\int_0^y f(y) dy~~~~~~~~~~~\text{on}~x=0

I think I need to define a variable \alpha where \alpha=\int_0^y f(y)dy and in my weak form add
…+on(3,\psi=\alpha)…

However I don’t know if there is any function defined in FreeFEM to calculate a cumulative integral, i.e., F(y)=\int_0^yf(y)dy~~~y \in [0,L]? The only function that I found is int1D which returns F(L)=\int_0^Lf(y)dy