1. Introduction
Modeling of physical systems in transport theory is usually based on mass balance. Mathematical description frequently leads to an appropriate partial differential equation (PDE). If this process takes place in a bounded domain, then the governing PDE must be accompanied by suitable boundary conditions (BCs) describing the behavior of the unknown quantity outside the area of consideration. The BCs are of fundamental importance since they determine the explicit form of a solution (David Hilbert outlined 23 famous mathematical problems in 1900. One of them is “The general problem of boundary values” in relation to PDEs in bounded domains. BCs connect solutions with the exterior domain with some expectations/restrictions.). The classical heat conduction theory is based on the Fourier law and it leads to the parabolic heat conduction equation
where
is the specific heat capacity,
stands for the mass density of the material,
k is the thermal conductivity and
u denotes the temperature. The initial state is described by the initial datum
. The heat equation is usually accompanied by one of the following three classical BCs
- Dirichlet
when temperature is prescribed on the surrounding surface;
- Neumann
when the normal component of the flux is given on the boundary;
- Robin/Newton
when a linear combination of the temperature and the normal component of the flux is known at the boundary.
Besides these standard types of BCs also the following two evolution BCs are known, cf. [
1]
- Carslaw
;
- Jaeger
.
All these BCs mentioned above are local, i.e., the relation between the temperature and the flux is taken at the same time and place. Researchers have already studied (under appropriate assumptions on the data functions) the well-posedness of problems involving those local boundary conditions.
On the other hand, there exist models with so-called nonlocal BCs. A. A. Samarskii and A.V. Bitsadze [
2] are originators of problems with such BCs. Investigation of problems with various types of nonlocal boundary conditions is a hot topic presently, i.a. because multi-point boundary-value problems (BVPs) for ODEs have many applications in modeling and analyzing problems arising from electric power networks, electric railway systems, telecommunication lines and also in chemistry and analyzing kinetic reaction problems. They have been intensively studied e.g., in [
3,
4,
5,
6,
7,
8,
9]. However, there are only a few papers devoted to time-dependent problems along with multi-point BCs, e.g., [
10,
11,
12]. The article [
11] deals with 3-point BCs subject to nonlinear parabolic Cauchy problem in
where
and
are given. The convergence of the solution towards the equilibrium solution was addressed in [
12].
Alikhanov [
13] studied a linear parabolic problem along with a 3-point BC. He showed the uniqueness and the continuous dependence of a solution on the initial data. A numerical finite-difference scheme was suggested and its convergence—assuming the existence of a very smooth exact solution
—was shown. The existence of a solution was not addressed. Using the method of energy inequalities, a priori estimates for the corresponding differential and finite-difference problems were obtained in a weighted
norms. This proof technique has been generalized to multi-point BCs for linear parabolic problems in [
14]. However, also here the assumption of existence of a very regular solution is needed to prove convergence of suggested approximation schemes. The existence of a solution was again not addressed. A compact difference scheme for the multi-point boundary-value problem of the heat equation has also been presented in [
15].
The problem in this paper describes a transient semilinear heat equation in
with two controllers located at the interior points
, where
The physical application is controlled cooling of a rod. The role of both controllers is to adjust the boundary data to the measured temperature, i.e.,
We assume perfect contact conditions at controller points
c and
d, i.e.,
Here the denotes the usual jump operator of the quantity at the position .
The main difficulty by stability analysis is the fact that one cannot prove that the governing (steady-state) differential operator is elliptic – due to the nonlocal BCs. Unfortunately, most solution methods rely on the ellipticity of the operator. That is why we first developed a new solution method for the steady-state differential problem. This is based on the principle of linear superposition. Secondly, we showed that the operator remains sectorial in an appropriate function space.
After that, we designed a numerical scheme for approximation of the solution to a semilinear parabolic equation accompanied with the four-point BCs (
1), which is based on semi-discretization in time method. The convergence of approximations towards the exact solution is shown under much weaker regularity assumptions than it was done in [
13,
14]. Therefore, we conclude that the semilinear parabolic equation accompanied with the four-point BCs (
1) is well-posed. Finally, we carried out some numerical experiments to support our results.
2. Linear Steady-State Case
In the study of the existence of solutions to ODEs major advancements have been made thanks to the so-called “Bernstein-Nagumo” conditions [
16,
17]. In this section, we derive a simple construction method for the solution of linear second order ODE-problems with nonlocal BCs. We will study the resolvent operator associated with this problem. This will be later applied to parabolic settings.
Let us consider the following nonlocal problem in
for
with
and
unknown. We look for a classic solution of (
3). Under a classic solution we understand
function, for which the interface condition (
2) is naturally valid. The concept of a weak solution in not appropriate in this situation, because a weak solution may have jumps of the first derivative at the interface points
. We show a very simple constructive way for solving this problem, which is based on the principle of linear superposition.
Let us consider the following seven auxiliary problems
Further we set
,
,
, where
There is a perfect contact at controller points, so we have to force
which implies
The values of
in (
3) are the solution of (
12).
In this way we can see that the solvability of the nonlocal problem (
3) is linked to the solvability of the classical Dirichlet BVPs (
4)–(
8) and the solvability of the algebraic system (
12). We have to check if the matrix
M is invertible. To do this, we derive the formulas for solutions to particular problems, first. The general solution to
has the following form
with the constants
depending on the boundary conditions under consideration. One can find the exact forms of the particular solutions, namely
Involving these relations into (
12) we get
The matrix
M is regular if its determinant is different from 0. To check this, we rewrite
into a more suitable form for our purposes. To obtain this, we use basic functional relations between trigonometric and hyperbolic functions, namely
Using the relations (
17) and
for
, we can easily derive the following estimates
Let
. A simple calculation yields
If
then
which implies
Thus, we have proved that
Analogously we deduce that
for
. Using (
19) we see that
Thus, we have proved that
We can clearly see that
if and only if
or
, thus
This concludes the proof that the nonlocal problem (
3) can be solved using the classical Dirichlet BVPs (
4)–(
8) in the way described above if
.
3. Estimates for
According to (
12) we may write
where
We may rewrite the matrix in (
23) in a more suitable form, namely
Applying (
18), we can estimate the particular entries of the last matrix as follows
and
According to (
23) and the considerations above we have
Now, we characterize all r for which .
Using
we have
We see that if
the complex numbers
r lie on a horizontal parabola (see
Figure 1)
The top of the parabola is at the point
. From this we conclude that if
lies on the right-hand side of the horizontal parabola, then
. Clearly, there exists a sector
in the complex plane with
in which
Our next concern is to derive estimates of
at the points
for
. This will be done in the next section, cf. (
28).
4. Resolvent Estimate
Consider the problem (
3). Let us denote
. We see
A as an operator from
into
, where
The norm in is induced by and denoted by . We see that .
The goal of this section is to derive some uniform estimates (with respect to r) of the resolvent operator in an appropriate function space. We show that this can be achieved in the sector .
First, we address the closedness of
A. Let
. Due to the fact that
is convergent, it is bounded. Applying the embedding theorem for continuous functions
we have the boundedness and equi-continuity of
, i.e.,
Using the Arzela–Ascolli theorem (Thm. 1.5.3 in [
18]), we get the relative compactness of
, i.e., there exists a subsequence of
that converges uniformly to some
g. We know that
converges pointwise to
u. Thus,
u is differentiable and
. Further we may write for any smooth function
which implies
, i.e., the operator
A is closed. The aim of this section is to prove that
A, together with the nonlocal BCs (
3), is a sectorial operator in a suitable function space, cf. [
19,
20,
21].
According to (
11) we have
One can prove that a strongly elliptic partial differential operator of second order with continuous coefficients in a smooth bounded domain
together with the homogeneous Dirichlet boundary condition generates an analytic semigroup in
for
. This is based on the resolvent estimate with respect to the
-norm, see [
21] (Chapter 7.3). In the analysis of our subject, we need to have a
-bound rather than an
-bound. The resolvent estimate in the
-norm can be found in [
22] and the
-bound in [
21] (Chapter 7.3).
The particular problem (
4) is a classical homogeneous Dirichlet setting. The spectrum of the operator
A is real and strict positive. In this standard case we have the following estimate, cf. [
22]
and
Let us note that a similar estimate is also valid for
and
using the same argument. The situation for
and
is analogous. First, we have to get rid of the non-homogeneous BC by shifting the solution by an appropriate linear function
g to get (e.g.,
)
along with the homogeneous Dirichlet BCs. Then, using the same argumentation as for
, we get
such that
Analogously we arrive at the same kind of estimates for , and .
Let
. Then
is a linear bounded operator and
First, we note that for a bounded operator
B and for
we have
Using the inequality
we obtain (skipping the function space)
For the particular choice
we have
In light of this we may write
The same estimates are valid for and .
Wrapping up the considerations above, we successively deduce that
Finally, we conclude that
Thus,
A is a sectorial operator in
. For the definition of a sectorial operator we refer the reader to [
19,
20,
21]. In our special situation it means that the spectrum is real, lies in a half plane and the resolvent operator obeys the inequality above.