With boundary value problem .
And initial value problem .
Write the function as such that satisfies the boundary value problem. For example, . And is the modified function that is needed to solve the equation.
Substitute that into the equation,
Therefore, we have transformed an equation with a non-homogenous boundary value problem into an equation with a homogenous boundary value problem but it has an 'uglier' form. Meaning we have to solve the inhomogenous heat equation.
Now there is a method for solving (with homogenous boundary),
For a given differenciable function .
But that is a totally different question.