Suppose that,

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,

.

Thus,

.

With,

.

And,

.

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.