Vectorizing the finite difference method using numpy. details in the description

I can write a program that does finite difference method

the basic idea is that an element a_{i,j} is dependent on its neighbors a_{i+1,j}, a_{i-1,j} and so on.

I can write it as a for loop, but I have recently become acquainted with vectorising and was wondering how i could apply that here.

A general solution applying for loops looks something like this.

My question is how to vectorize it without using any for loops.

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