Partial Differential Equations/Heat equation
In this chapter, we will consider the heat equation:
Lemma 4.1: Multi-dimensional Gaussian integral
But with one-dimensional transformation of variables, with the diffeomorphism , we obtain from lemma 3.1, that:
Multiplying this with , we obtain that
. This finishes the proof.
A Green's kernel for the Heat Equation is given by
Proof: First we show that is locally integrable: Let a compact set, and let's choose such that . Then:
Now transformation of variables in the inner integral, applying the diffeomorphism , and lemma 4.1 give us:
,which shows us that is indeed locally integrable.
Theorem 1.3 gives us, that is a distribution, and lemma 2.4 tells us that it depends continuously on .
We will now calculate some derivatives of , because we will need them later:
Let now , .
Let's firstly choose , , such that:
We furthermore define the following integrals and calculate a little:
, the dominated convergence theorem tells us that
Furthermore, the following two calculations help us: First, one-dimensional integration by parts of with respect to :
And also twice multi-dimensional integration by parts of with respect to :
From the derivative calculations from above, we see that
. Thus, we obtain, by combining the calculations for and :
With transformation of variables with the diffeomorphism we also obtain:
But with the dominated convergence theorem, continuity of , lemma 4.1 and this last calculation, we obtain the last thing, which finishes the proof:
Let's suppose that . Then
solves the heat equation in the classical sense.
Proof: Since all the derivatives of are integrable and is bounded, we know due to the Leibniz integral rule that we may interchange integration and differentiation, which is why we obtain that is continuous. Due to theorem 2.5, the claim follows.
Initial Value Problem
The initial value problem for a heat equation is given by
If such that meets the requirements for the solvability of the heat equation stated above, and , then
solves the initial value problem, where is the extension of to where , and . This is a very nice solution formula.
Proof: By the definition of , we have that on . Furthermore, is a well-defined solution for the heat equation . Therefore, since is well-defined as is integrable for every (indeed the integral is always equal to as shown in the calculation for the Green's kernel), if we show that , then we know that the first equation of the problem is solved, and if we show that for every , we have shown that the second equation is solved such that .
But since all the derivatives of are integrable, we may use differentiation under the integral sign to obtain:
This shows the first equation.
For the second equation, we first note that
. Furthermore, due to the continuity of , we may choose for arbitrary and any a such that .
Due to the triangle inequality, we may further estimate:
, where in the last line we use the fact we have first noted. But due to transformation of variables, applying the diffeomorphism , we obtain
which is why
Since was arbitrary, this finishes the proof.
- In this formula, is the convolution with respect to .