Implicit Method

Instead of computing the value of the current neighboring values to determine the future value, implicit method uses the future neighboring values, so the equation to approximate the values are now:

$$\frac{u^{k+1}_{i,j} - u^{k}_{i,j}}{s} = k \frac{u^{k+1}_{... ... + u^{k+1}_{i,j-1} + u^{k+1}_{i-1,j}- 4u^{k+1}_{i,j}} {h^2}$$ (2.11)

Now, instead of computing the unknown element using the known values, we now have to compute the unknown using the unknowns, or we have a system of equations to solve all the unknown elements in the equation.

At this point, we have not explored this area throughly to fully explain these concepts, but to solve such systems, the unknowns and their coefficient should be separated from the known values with which we attain the b portion of the typical linear system of equation Ax = b where A is the matrix holding the coefficients of the unknowns, b is the list of known values, and x is the list of unknowns.