vSize =12;
hSize = 10;
V = 10;
V_index = 75;

U = calculatePotentials(vSize,hSize,V,V_index);
plotPotentials(U,vSize,hSize,V_index);
----------------------------------------------------------
function U = calculatePotentials(hSize, vSize,U,uIndex)
pCount = hSize*vSize;
if uIndex>pCount
   error('Index of point with potential(%d) is out of range(%d)',uIndex,pCount);
end

if uIndex <=vSize || uIndex> vSize*(hSize-1) || mod(uIndex,vSize) == 0 || mod(uIndex,vSize) == 1
    error('Point must be inside area');
end

I = eye(vSize);
I1 = diag([0 -ones(1,vSize-2) 0]);
T = diag([1 4*ones(1,vSize-2) 1]) + diag([0 -ones(1,vSize-2)],1)+diag([-ones(1,vSize-2) 0],-1);
C = repmat({T},hSize-2,1);
H = kron(diag([1 zeros(1,hSize-2) 1]),I)+kron(diag([0 ones(1,hSize-2) 0]),T)+kron(diag([0 ones(1,hSize-2)],1),I1)+kron(diag([ones(1,hSize-2) 0],-1),I1);

H(uIndex,:) = 0;
H(uIndex,uIndex) = 1;
B = zeros(pCount,1);
B(uIndex) = U;

U = gmres(H,B);
end
-----------------------------------------------------------------------------------
function plotPotentials(U,hSize, vSize,V0_index)
close all;

u_2d= reshape(U,vSize,hSize);
contour(u_2d,50);
colorbar;
grid on;
hold on;
plot(floor(V0_index/vSize)+1,mod(V0_index,vSize),'k.','MarkerSize',10);
title('Rozklad pola elektrycznego U($x_{i}$,$y_{i}$)','interpreter','latex')
xlabel('i','interpreter','latex');
ylabel('j','interpreter','latex');
end