Partial Differential Equations/The heat equation

Partial Differential Equations
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This chapter is about the heat equation, which looks like this:

for some . Using distribution theory, we will prove an explicit solution formula (if is often enough differentiable), and we even prove a solution formula for the initial value problem.

Green's kernel and solutionEdit

Lemma 6.1:

 

Proof:

 

Taking the square root on both sides finishes the proof. 

Lemma 6.2:

 

Proof:

 

By lemma 6.1,

 .

If we apply to this integration by substitution (theorem 5.5) with the diffeomorphism  , we obtain

 

and multiplying with  

 

Therefore, calculating the innermost integrals first and then pulling out the resulting constants,

  

Theorem 6.3:

The function

 

is a Green's kernel for the heat equation.

Proof:

1.

We show that   is locally integrable.

Let   a compact set, and let   such that  . We first show that the integral

 

exists:

 

By transformation of variables in the inner integral using the diffeomorphism  , and lemma 6.2, we obtain:

 

Therefore the integral

 

exists. But since

 

, where   is the characteristic function of  , the integral

 

exists. Since   was an arbitrary compact set, we thus have local integrability.

2.

We calculate   and   (see exercise 1).

 
 

3.

We show that

 

Let   be arbitrary.

In this last step of the proof, we will only manipulate the term  .

 

If we choose   and   such that

 

, we have even

 

Using the dominated convergence theorem (theorem 5.1), we can rewrite the term again:

 

, where   is the characteristic function of  .

We split the limit term in half to manipulate each summand separately:

 

The last integrals are taken over   for  . In this area and its boundary,   is differentiable. Therefore, we are allowed to integrate by parts.

 

In the last two manipulations, we used integration by parts where   and   exchanged the role of the function in theorem 5.4, and   and   exchanged the role of the vector field. In the latter manipulation, we did not apply theorem 5.4 directly, but instead with subtracted boundary term on both sides.

Let's also integrate the other integral by parts.

 

Now we add the two terms back together and see that

 

The derivative calculations from above show that  , which is why the last two integrals cancel and therefore

 

Using that   and with multi-dimensional integration by substitution with the diffeomorphism   we obtain:

     

Since   is continuous (even smooth), we have

 

Therefore

  

Theorem 6.4: If   is bounded, once continuously differentiable in the  -variable and twice continuously differentiable in the  -variable, then

 

solves the heat equation

 

Proof:

1.

We show that   is sufficiently often differentiable such that the equations are satisfied.

2.

We invoke theorem 5.?, which states exactly that a convolution with a Green's kernel is a solution, provided that the convolution is sufficiently often differentiable (which we showed in part 1 of the proof). 

Initial Value ProblemEdit

Definition 6.5: Let   and   be two functions. The spatial convolution of   and   is given by:

 

Theorem and definition 6.6: Let   be bounded, once continuously differentiable in the  -variable and twice continuously differentiable in the  -variable, and let   be continuous and bounded. If we define

 

, then the function

 

is a continuous solution pf the initial value problem for the heat equation, that is

 

Note that if we do not require the solution to be continuous, we may just take any solution and just set it to   at  .

Proof:

1.

We show

 

From theorem 7.4, we already know that   solves

 

Therefore, we have for  ,

 

which is why   would follow if

 

This we shall now check.

By definition of the spatial convolution, we have

 

and

 

By applying Leibniz' integral rule (see exercise 2) we find that

 

for all  .

2.

We show that   is continuous.

It is clear that   is continuous on  , since all the first-order partial derivatives exist and are continuous (see exercise 2). It remains to be shown that   is continuous on  .

To do so, we first note that for all  

 

Furthermore, due to the continuity of  , we may choose for arbitrary   and any   a   such that

 .

From these last two observations, we may conclude:

 

But due to integration by substitution using the diffeomorphism  , we obtain

 

which is why

 

Since   was arbitrary, continuity is proven. 

ExercisesEdit

SourcesEdit

Partial Differential Equations
 ← Fundamental solutions, Green's functions and Green's kernels The heat equation Poisson's equation →