1. Introduction
Recently, there has been extensive research on the long-distance transportation of atmospheric pollutants, particularly in the context of chemical and nuclear tracers. The models used in these studies aim to predict the
concentrations of contaminants in the atmosphere based on the distribution of emission sources. To accurately predict the qualitative properties of water reservoir exploitation, it is necessary to periodically specify the model parameters based on specific measurements as stated by [
1].
One type of mathematical model, discussed by [
2], addresses the increasing demand for groundwater as a source of drinking water. The damage caused by contaminants to groundwater varies depending on the migration of the contaminants through the water and soil as noted by [
3,
4]. The concentration distribution of a contaminant inside of a
large domain as a result of an unknown point source is considered as a simple parabolic convection–diffusion with a point overposed data condition. Inverse problems for determination of the time-dependent right-hand side of parabolic equations from the interior domain and final-time measurements using Green’s function have been studied in the papers [
5,
6,
7,
8,
9].
Applications of inverse problems in groundwater contamination are discussed in [
10,
11,
12]. In this work, we solve inverse initial-value parabolic convection–diffusion problems of estimation of the time-dependent right-hand side using point measurements into two stages. On the first stage, we reduce the parabolic Cauchy problem to an equivalent one on a bounded domain. Following this approach, we apply two methods for identification of the right side. The first one is the least-squares method and the second is the decomposition of the solution on the base of the point observation. Applications to water and air pollution models are presented.
The remainder of the paper is structured as follows. In
Section 2, we formulate and discuss the direct and inverse problems. In
Section 3 we propose a method for reducing the Cauchy problem with a general variable coefficient elliptic part of the differential operator, see [
13], to a Dirichlet problem on a bounded domain. A finite difference solution of the problem (
1), (
2) on a rectangle is given in
Section 4. The algorithm of a numerical solution to the inverse problem is described in
Section 5. Numerical results for examples from the water and air pollution modeling are presented in the following section.
3. Localization of the Parabolic Problem
A few methods have been developed for numerical solutions to PDEs defined on unbounded domains, such as artificial boundary conditions [
15], transparent boundary conditions [
16,
17,
18] and independent and/or dependent variable transformations [
19]. Such approaches are required since the numerical treatment of the problem requires the computations to be performed on a finite domain. In the following, we adopt the methodology that was developed in [
13].
In this section, instead of problem (
4), (
5), we solve an initial-boundary value problem for Equation (
4) on a bounded domain
with
,
. For this, it is necessary to pose boundary conditions for function
on the boundary
. We follow the methodology of the fundamental solutions developed in the papers [
5,
6,
13].
Let
be the fundamental solution of Equation (
4). Then, the solution of the Cauchy problem (
4), (
5) can be presented in the form [
20]
Assume that, outside the bounded domain
, and
. Then, the integral representation (
8) takes the form
i.e., the integration is performed on a bounded domain for the localized initial condition and right-hand side. Therefore, a transition of the Cauchy problem (
4), (
5) to an equivalent one on a bounded domain is performed. Thus, if we numerically solve problem (
4), (
5) on a finite domain, we can use formula (
9) to take appropriate boundary conditions. However, this procedure is expensive, and we construct a more economical one below.
We start with the introduction of the auxiliary problems,
Let
be the adjoint to the
operator [
21], i.e.,
Then, with respect to arguments
, the fundamental solution
satisfies the equation
which means that
G is the fundamental solution of the adjoint equation with respect to the variables
.
Next, we multiply CDE (
14) with
and integrate the result on
, where
We let
,
and apply, to (
15), a variant of the second Green’s formula, which is based on the equality
Considering the boundary condition (
12) and the initial condition (
11), we find
where
is the co-normal derivative
and
is the unit outward normal at
.
Taking the limit
in (
16) and regarding (
5), we find
Next, using (
12), we find that, on
,
Now, we can formulate the following Algorithm 1 for solving the Cauchy problem (
4), (
5) (respectively, (
1), (
2)) on a finite domain.
Algorithm 1 Solution to the Cauchy problem on a truncated domain |
Step 1. Solve the problems ( 10)–( 12). Step 2. Calculate the boundary conditions of on from ( 17). Step 3. Solve the boundary value problem ( 4), ( 5), ( 17). |
If the initial condition
(respectively,
) and the right-hand side
are localized, (i.e., outside the domain
is equal to zero), then the boundary condition is exact. Thus, for a small time value, the initial boundary value problems (
10)–(
12) can be used for a first approximation of the original problem (
4), (
5) (respectively, (
1), (
2)), while the problem (
4), (
5), (
17) can be used as a second approximation.
4. Numerical Solution to Problem (1), (2) on a Rectangle
We use the change
to write the problem (
4), (
5) in the form
It is known that the Cauchy problem (
18), (
19) is solved by [
21],
where
Let us take
. Then, the solution to the initial boundary problem
is given by [
21]
where
Now, we can find Dirichlet boundary conditions for the solution of problem (
18), (
19) from formula (
17), which now takes the form
Once the boundary conditions are derived, the problem (
4), (
5), which is imposed on an infinite domain, is transformed to a problem posed on a finite domain. This action has several advantages. First of all, now the computations become feasible and even efficient, since the domain could be arbitrarily small. Secondly, the transformation is exact, i.e., the transformed problem is equivalent to the original one on the bounded domain, which means that there is no truncation error. This is in contrast to the classical methods, which require the truncated domain to be large, thus, introducing both truncation error and heavy computational load.
5. Numerical Solution to the Inverse Problem
When the four boundary conditions are chosen, we have
where
is the solution to problem (
18), (
19) when
and
—when
but all boundary conditions and initial condition are also zero. Then,
takes the form
Now, when (
4), (
5) (respectively, (
1), (
2)) is a problem with unknown time-dependent strength
, it is referred to as an inverse problem. To reconstruct the unknown function in (
21), the overposed data condition (
7) is applied:
Thus, we obtain a Volterra integral equation for
:
Since the function
is known, the problem (
22) has a unique solution
; see, e.g., [
22].
We use a collocation numerical method in order to derive the solution to the first-kind Volterra Equation (
22), with the convolution kernel
We introduce a time uniform mesh
. Then, we seek the approximate solution
where
is the
m-th base function defined by
We note that
is an orthonormal set in
. By placing (
23) in (
22), at successive time
,
, we obtain
where
and the respective integrals are calculated with a suitable numerical procedure [
23].
Now, let us consider the linear algebraic system of equations
which is obtained by (
23) and (
24), such that
is a lower-triangular Toeplitz matrix given by
In order to solve (
24), we implement the Tikhonov regularization method, and for this, we define the discrete functional
where
is a predefined regularization parameter and
L is a matrix of type
A instead of identity one
I in the sequential Tikhonov regularization algorithm. To estimate the unknown parameter
f using the least-squares method, we minimize the function
as shown in (
25). We can accomplish this by differentiating
with respect to the unknown parameter
for any
and then setting the resulting expression equal to zero. By following the approach outlined in [
22], we can calculate the vector
f. Assuming that we have already found the values of
through
, we can obtain the value of
by performing certain calculations.
with
we determine
by finding the vector
from the minimization of
in the form
Substituting
f in (
23),
will be estimated for
.
6. Numerical Simulations
In this section, we provide computational experiments to verify the robustness and the applicability of the proposed algorithm. In order to gather measurements, we first solve the direct problem. Then, we use these observations to solve the inverse problem. In this quasi-real framework, we are able to calculate the accuracy of the solution to the inverse problem. We solve the problems using the finite difference method [
24].
Let us consider the following counterpart of (
1).
Example 1 (One-dimensional advection-diffusion solute transport equation).
with the initial conditionswhere the time-varying concentration source is located at . Again, by substituting
where
then the problem (
26), (
27) is transformed to
where
Further, the unbounded problem is localized accordingly to .
Now, we will briefly explain the application of the finite difference method on the model (
29). Let us introduce the meshes
where
h and
are the spatial and temporal steps, respectively.
Let the approximation of
and
,
,
, then the fully implicit finite difference scheme is written in canonical form as
where
The tridiagonal system can be solved by means of the Thomas algorithm [
24].
Let
m,
s,
m
2s
−1,
ms
−1,
s
−1,
m and the concentration source be
If
mgm
−3, where
is the PDF of the normal distribution with mean
and standard deviation
, then the solution to the localized problem (
26) is plotted on
Figure 1 (left), and the boundary conditions are given in
Figure 1 (right).
Currently, we are able to start solving the inverse problem.
Let us take the measurements
and, respectively,
for
.
The recovered function
(
31) is presented in
Figure 2. It relatively closely follows the true value (
31).
Let us try another experiment with a more fluctuating value of
:
The identified source is given in
Figure 3. Although lagging slightly, the recovered function follows the true trend.
The same is true for the source, similar to [
8]; see
Figure 4.
The suggested approach could also be successfully used as follows.
Example 2 (An instantaneous injection at the origin into a uniform steady in an infinite plane).
Let an amount M of pollutant be released instantaneously over a river with depth θ at . The model is mathematically formulated as the following governing CDE,with initial conditionsand boundary conditionswhere , , , and θ are constants, and Let us localize the domain as and gm−2, m, m2s−1, m2s−1 and ms−1. The measurements are taken at the point .
The concentration is plotted at
s and
s in
Figure 5.
The boundary conditions at
and
are given in
Figure 6. The boundary conditions for
are almost zero for all
and are not interesting to plot.
7. Conclusions
In this paper, we proposed efficient algorithms for solving Cauchy inverse problems with unknown sources. In the present study, using Green’s function method, we developed a procedure for constructing appropriate boundary conditions for a square domain such that the new problem preserves the properties of the original one, defined on an unbounded domain. Then, the time-dependent unknown source strength was reconstructed from point measurements inside the bounded domain.
We successfully applied the Tikhonov regularization method along with a first Fredholm integral equation to the inverse source problem. Test examples demonstrate that the proposed numerical method is efficient and accurate for the estimation of the unknown source on the base of point observations.