CN111667116A - Underground water pollution source identification method and system - Google Patents

Underground water pollution source identification method and system Download PDF

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CN111667116A
CN111667116A CN202010511223.XA CN202010511223A CN111667116A CN 111667116 A CN111667116 A CN 111667116A CN 202010511223 A CN202010511223 A CN 202010511223A CN 111667116 A CN111667116 A CN 111667116A
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张新明
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Shenzhen Graduate School Harbin Institute of Technology
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Abstract

The invention relates to a method and a system for identifying an underground water pollution source. The method comprises the steps of establishing a mathematical model according to the pollution diffusion phenomenon of underground water; the mathematical model includes a contaminant concentration function; according to the mathematical model, converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method, and solving to obtain the pollution concentration; determining additional conditions according to the mathematical model and the pollution concentration; converting a source item inversion problem in the model into a minimized functional based on a mathematical physics inverse problem framework according to the obtained additional conditions; and solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source. The method is researched aiming at the pollution diffusion of the underground water, and the cubic B-spline is applied to numerical solution of a time-fractional order differential equation and source item inversion, so that the position of a pollution source can be accurately predicted.

Description

Underground water pollution source identification method and system
Technical Field
The invention relates to the field of underground water pollution, in particular to an underground water pollution source identification method and system.
Background
Researchers have introduced mathematical models to study groundwater quality as early as the late 60 s of the 20 th century. The phenomenon of hydrodynamic diffusion in interstitial media was initially studied in greater detail by Bel of the former soviet union and indicates that hydrodynamic diffusion systems can be characterized by longitudinal and transverse diffusion coefficients. Fried then further investigated the classical model and the mathematical model of hydrodynamic diffusion that were relevant at the time. In 1977 Wills and Neumman proposed a general model for the dynamic management of underground water quality within a decentralized parameter system. With the continuous and deep research of groundwater, foreign random mathematical models of pollutant migration and transformation are also widely researched, and new research results are continuously developed, wherein the convection diffusion mathematical model is the mathematical model which is mentioned and most used in the research.
However, in the case of an impure or irregular fractal porous medium, the actually measured data of the penetration curve shows an asymmetric off-state distribution, and phenomena of early penetration and tailing exist. Research shows that at the moment, the motion transfer process of pollutants is changed from local small-range Brownian motion to non-local Levy process, and the scholars find that the fractional order differential process and the anomalous diffusion process are similar in nature after considering the fractional order differential process and the anomalous diffusion process, so that the fractional order differential equation is introduced into a convection diffusion model of groundwater pollution to replace the conventional commonly used integral order differential equation.
The analytic method, the semi-analytic method and the numerical solution method are three methods for solving the migration model of the underground water pollutants. We can only find complex analytical solutions under ideal conditions. The semi-analytical solution is an approximate solution to the effects of convection and adsorbate on contaminant migration and is only applicable to stable flow fields of homogeneous media. The numerical solution does not require so many constraints as compared to the first two methods. The numerical solution is obtained by discretizing a target equation, replacing continuous variables with a linear equation set containing a limited number of unknowns, and solving the corresponding linear equation set to obtain a numerical solution as a numerical approximation solution of the problem. In practical problems, solving the migration model of groundwater pollutants by a numerical solution is the most common method. How to obtain a more accurate numerical solution by selecting and improving the numerical solution becomes a very important research content.
At present, in existing documents at home and abroad, most of researches on fractional order differential equations are focused on solving numerical solutions, and specifically, a numerical calculation method is proposed or improved to solve the numerical solutions for a certain type of specific differential equations. The existing resolved fractional order differential equations include: a time-fractional order burgers equation, a time-fractional order gas dynamics equation, a fractional order diffusion problem, a space-fractional order convective diffusion equation, and the like. And the examples for solving the numerical solution of the time fractional order convection diffusion equation are few, and in addition, the related documents and researches for numerical solution of the inverse problem of the fractional order differential equation are few. The source term inversion is an important direction of numerical inversion of fractional order differential equations, and can effectively solve a plurality of practical problems. Therefore, numerical solution based on the one-dimensional time fractional order convection diffusion differential equation has important theoretical value and practical significance, and a large number of problems are to be further researched.
The spline method is a numerical solving method widely applied to solving the convection diffusion equation, and has high feasibility and accuracy, but the method is not used for solving a fractional order differential equation system at present. Meanwhile, most of the prior art schemes are concentrated in forward numerical solution, error analysis is rarely performed, and pollution is not identified through the thought of the inverse problem of mathematics and physics, so that the prior art schemes are low in precision, lack of complete theoretical support and high in complexity.
Disclosure of Invention
The invention aims to provide a method and a system for identifying an underground water pollution source, which are used for quickly and accurately identifying the underground water pollution source.
In order to achieve the purpose, the invention provides the following scheme:
a method of identifying a source of groundwater contamination, the method comprising:
establishing a mathematical model according to the pollution diffusion phenomenon of underground water; the mathematical model includes a contaminant concentration function;
according to the mathematical model, converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method, and solving to obtain the pollution concentration;
determining additional conditions according to the mathematical model and the pollution concentration;
converting a source item inversion problem in the model into a minimized functional based on a mathematical physics inverse problem framework according to the obtained additional conditions;
and solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source.
Optionally, the establishing of the mathematical model according to the diffusion phenomenon of the pollution of the underground water specifically includes:
assuming that the diffusion system is one-dimensional spatial diffusion, the time fractional order convection diffusion equation is in the form:
Figure BDA0002528366050000031
initial conditions:
U(x,0)＝h(x) (2)
boundary conditions:
U(1,t)＝φ(t)，U(0,t)＝ψ(t) (3)
where x denotes spatial position, t denotes time, and 0 ≦ x, t ≦ 1, f (x, t) denotes source term, p (x, t) is convection term coefficient, q is diffusion term coefficient, α is known time fractional order parameter, U (x, t) is unknown function to be determined, representing contaminant concentration, h (x) denotes initial contaminant concentration, and Φ (t) and ψ (t) denote contaminant concentration at solution area boundary x ≦ 0 and x ≦ 1, respectively.
Optionally, the converting the numerical solution problem of the fractional order differential equation into the solution problem of the linear equation set according to the mathematical model based on a cubic B-spline method, and solving to obtain the pollution concentration specifically includes:
determining an approximate function of the pollutant concentration function based on a cubic B-spline method according to the mathematical model;
and converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set according to the approximate function, and solving to obtain the pollution concentration.
Optionally, determining additional conditions according to the mathematical model and the contamination concentration specifically includes:
based on the mathematical model, the additional conditions are:
Figure BDA0002528366050000032
where x denotes spatial position, t denotes time, U (x, t) denotes contaminant concentration, and g (t) denotes additional conditions at time t.
Optionally, the established minimization functional is solved by using a regularized cuckoo algorithm, where the solution represents a position of the pollution source, and the method specifically includes:
giving an initial value of a source item;
based on the initial value f of the source item0Via operator equation A (f)0)＝g0Calculating an additional condition g corresponding to the initial value of the source item at time t0(t), calculate g | |0(t)-g(t)||；f0Denotes the initial value of the source item, g0Indicating an initial value f corresponding to the source item0Additional conditions of (a);
determining the next source item iteration value f by a cuckoo algorithmi(i＝1,2,…)；
Based on the iterative value f of the source termiVia operator equation A (f)i)＝giCalculating the iteration value f of the source item corresponding to the time tiAdditional condition gi(t), calculate g | |i(t)-g(t)||，giRepresenting the iteration value f corresponding to the source termiAdditional conditions of (a);
if g | | |i(t) -g (t) if less than a given error limit, the calculation is stopped.
The invention also provides a groundwater pollution source identification system, which comprises:
the modeling module is used for establishing a mathematical model according to the pollution diffusion phenomenon of the underground water; the mathematical model includes a contaminant concentration function;
the first conversion module is used for converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method according to the mathematical model, and solving to obtain the pollution concentration;
an additional condition determining module for determining an additional condition based on the mathematical model and the contamination concentration;
the second conversion module is used for converting the source item inversion problem in the model into a minimization functional based on a mathematical physics inverse problem frame according to the obtained additional conditions;
and the solving module is used for solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source.
Optionally, the mathematical model specifically includes:
assuming that the diffusion system is one-dimensional spatial diffusion, the time fractional order convection diffusion equation is in the form:
Figure BDA0002528366050000041
initial conditions:
U(x,0)＝h(x) (2)
boundary conditions:
U(1,t)＝φ(t)，U(0,t)＝ψ(t) (3)
where x denotes spatial position, t denotes time, and 0 ≦ x, t ≦ 1, f (x, t) denotes source term, p (x, t) is convection term coefficient, q is diffusion term coefficient, α is known time fractional order parameter, U (x, t) is unknown function to be determined, representing contaminant concentration, h (x) denotes initial contaminant concentration, and Φ (t) and ψ (t) denote contaminant concentration at solution area boundary x ≦ 0 and x ≦ 1, respectively.
Optionally, the first conversion module specifically includes:
the approximate function determining unit is used for determining an approximate function of the pollutant concentration function based on a cubic B spline method according to the mathematical model;
and the pollution concentration solving unit is used for converting the numerical solving problem of the fractional order differential equation into a linear equation solving problem according to the approximate function, and solving to obtain the pollution concentration.
Optionally, the additional condition specifically includes:
using the mathematical modelBased on type, the additional conditions are:
Figure BDA0002528366050000051
where x denotes spatial position, t denotes time, U (x, t) denotes contaminant concentration, and g (t) denotes additional conditions at time t.
Optionally, the solving module specifically includes:
a given unit, configured to give an initial value of a source item;
a first calculation unit for calculating the initial value f based on the source item0Via operator equation A (f)0)＝g0Calculating an additional condition g corresponding to the initial value of the source item at time t0(t), calculate g | |0(t)-g(t)||；f0Denotes the initial value of the source item, g0Indicating an initial value f corresponding to the source item0Additional conditions of (a);
a source item iteration value determining unit for determining the next source item iteration value f by the cuckoo algorithmi(i＝1,2,…)；
A second calculation unit for iterating the value f based on the source termiVia operator equation A (f)i)＝giCalculating the iteration value f of the source item corresponding to the time tiAdditional condition gi(t), calculate g | |i(t)-g(t)||，giRepresenting the iteration value f corresponding to the source termiAdditional conditions of (a);
a judging unit for judging gi(t) -g (t) if | is less than a given error limit.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects: the method is characterized in that a mathematical model is established according to the pollution diffusion phenomenon of underground water; the mathematical model includes a contaminant concentration function; according to the mathematical model, converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method, and solving to obtain the pollution concentration; determining additional conditions according to the mathematical model and the pollution concentration; converting a source item inversion problem in the model into a minimized functional based on a mathematical physics inverse problem framework according to the obtained additional conditions; and solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source. The method is researched aiming at the pollution diffusion of the underground water, and the cubic B-spline is applied to numerical solution of a time-fractional order differential equation and source item inversion, so that the position of a pollution source can be accurately predicted.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a flow chart of a groundwater contamination source identification method according to an embodiment of the present invention;
fig. 2 is a block diagram of an underground water pollution source identification system according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a method and a system for identifying an underground water pollution source, which are used for quickly and accurately identifying the underground water pollution source.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
As shown in fig. 1, a method for identifying a groundwater pollution source includes the following steps:
step 101: establishing a mathematical model according to the pollution diffusion phenomenon of underground water; the mathematical model includes a contaminant concentration function.
Assuming that the diffusion system is one-dimensional spatial diffusion, the time fractional order convection diffusion equation is in the form:
Figure BDA0002528366050000061
initial conditions:
U(x,0)＝h(x) (2)
boundary conditions:
U(1,t)＝φ(t)，U(0,t)＝ψ(t) (3)
where x denotes spatial position, t denotes time, and 0 ≦ x, t ≦ 1, f (x, t) denotes source term, p (x, t) is convection term coefficient, q is diffusion term coefficient, α is known time fractional order parameter, U (x, t) is unknown function to be determined, representing contaminant concentration, h (x) denotes initial contaminant concentration, and Φ (t) and ψ (t) denote contaminant concentration at solution area boundary x ≦ 0 and x ≦ 1, respectively.
Step 102: and converting the numerical solving problem of the fractional order differential equation into a linear equation solving problem based on a cubic B-spline method according to the mathematical model, and solving to obtain the pollution concentration.
Firstly, the continuous variable x is positioned in [ a, b]The interval n is equally divided, and for convenience here the interval is set to [0,1 ]]Then x can be orderediI/n, i is 0,1, n, Δ x is 1/n. Cubic B-spline basis function
Figure BDA0002528366050000071
(i-1, 0.., n, n +1) is defined as:
Figure BDA0002528366050000072
for the function U (x, t), we can approximate the expression U with a cubic B-spline basis functionn(x, t) approximately represents U (x, t), i.e.:
Figure BDA0002528366050000073
whereini(t) unknown. For any point x in the divided setiWe have:
Figure BDA0002528366050000074
then we can get, according to the definition of cubic B-spline basis function:
Un(xi,t)＝i-1(t)+4i(t)+i+1(t)，
similarly, we can also get an approximate expression of the first and second order spatial differentials of the unknown function:
Figure BDA0002528366050000075
Figure BDA0002528366050000081
according to the approximate function, converting the numerical solving problem of the fractional order differential equation into a linear equation set to be solved to obtain the pollution concentration U (x, t);
the first step is as follows: the time fractional order derivative in process equation (1) is defined based on the Caputo fractional order derivative, i.e. (1) can be rewritten as follows:
Figure BDA0002528366050000082
let [ a, b ] of continuous variable t]The interval m is equally divided, and for convenience here the interval is set to [0,1 ]]Then let t bejJ/m, j 0,1, m, Δ t 1/m. Arbitrarily take t as tj，x＝xiThen, the above formula (4) can be rewritten as the following form:
Figure BDA0002528366050000083
using forward differential format pairs
Figure BDA0002528366050000084
Discretization is performed for τ ∈ [ t ]k,tk+1]Comprises the following steps:
Figure BDA0002528366050000085
formula (6) is substituted for formula (5) and integrated to convert:
Figure BDA0002528366050000086
wherein
Figure BDA0002528366050000087
The second step is that: based on the approximation formula of the unknown function U (x, t) and its first and second spatial partial derivatives in the first step, equation (7) above can be transformed into:
Figure BDA0002528366050000091
wherein
Figure BDA0002528366050000092
Figure BDA0002528366050000093
Figure BDA0002528366050000094
Figure BDA0002528366050000095
For i ═ 0., n, equation (8) holds.
Let dj＝[-1(tj),0(tj),...,n(tj),n+1(tj)]TAt the same timeConsidering the boundary condition (3), the fractional order differential equation (1) can be discretized to obtain a linear equation set:
Figure BDA0002528366050000096
wherein
Figure BDA0002528366050000097
Figure BDA0002528366050000098
Figure BDA0002528366050000101
The third step: an initial state equation set is established by the initial condition (2), that is, the formula (9) can be solved to obtain the unknown function U (x, t).
Step 103: from the mathematical model and the contamination concentration, additional conditions (typically measurable time-varying contamination concentration at a certain location) are determined.
Based on the mathematical model, the additional conditions are:
Figure BDA0002528366050000102
where x denotes spatial position, t denotes time, U (x, t) denotes contaminant concentration, and g (t) denotes additional conditions at time t.
The procedure for finding additional conditions from the above procedure can be summarized as the following operator equation in the case of known source terms:
A(f)＝g(t)
the operator a can be seen as a mapping from the source term f to an additional condition g (t).
Step 104: and converting the source term inversion problem in the model into a minimized functional based on a mathematical physics inverse problem framework according to the obtained additional conditions.
Under the condition that a source item is unknown, the source item inversion problem in the model can be converted into a functional minimization problem according to the obtained additional conditions and based on a mathematical physics inverse problem framework, namely, by a method for minimizing the error of the new additional conditions and the known additional conditions;
Figure BDA0002528366050000103
therein
Figure BDA0002528366050000104
Is the iteration value of the unknown source item.
The solution of the corresponding inverse problem is usually ill-posed, and in order to eliminate the ill-posed nature, a Tikhonov regularization method is adopted to obtain the Tikhonov regularization functional extreme value problem:
Figure BDA0002528366050000111
wherein, α is a regularization parameter, and is usually obtained by an L-curve method.
Step 105: and solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source.
Solving the problem of the Tikhonov regularization functional extreme value by adopting a cuckoo algorithm to obtain a solution of the functional minimum value, namely an unknown pollution source item.
The method comprises the following specific steps:
the first step is as follows: given the initial value of the source item, the selection range of the initial value can be larger due to the adoption of the regularized cuckoo algorithm;
the second step is that: based on the initial value f of the source item0Via operator equation A (f)0)＝g0Calculating an additional condition g corresponding to the initial value of the source item at time t0(t), calculate g | |0(t)-g(t)||；f0Denotes the initial value of the source item, g0Indicating an initial value f corresponding to the source item0Is additionally provided with(ii) an additional condition;
the third step: determining the next source item iteration value f by a cuckoo algorithmi(i ═ 1,2, …), iterating the value f based on the source termiVia operator equation A (f)i)＝giCalculating the iteration value f of the source item corresponding to the time tiAdditional condition gi(t), calculate g | |i(t) -g (t) i, gi representing the iteration value f corresponding to the source termiAdditional conditions of (a);
the fourth step: if g | | |i(t) -g (t) if | is less than a given error limit, stopping calculation; otherwise, the third step is carried out.
As shown in fig. 2, the present invention also provides a groundwater pollution source identification system, including:
the modeling module 201 is used for establishing a mathematical model according to the pollution diffusion phenomenon of the underground water; the mathematical model includes a contaminant concentration function. The mathematical model specifically comprises:
assuming that the diffusion system is one-dimensional spatial diffusion, the time fractional order convection diffusion equation is in the form:
Figure BDA0002528366050000112
initial conditions:
U(x,0)＝h(x) (2)
boundary conditions:
U(1,t)＝φ(t)，U(0,t)＝ψ(t) (3)
where x denotes spatial position, t denotes time, and 0 ≦ x, t ≦ 1, f (x, t) denotes source term, p (x, t) is convection term coefficient, q is diffusion term coefficient, α is known time fractional order parameter, U (x, t) is unknown function to be determined, representing contaminant concentration, h (x) denotes initial contaminant concentration, and Φ (t) and ψ (t) denote contaminant concentration at solution area boundary x ≦ 0 and x ≦ 1, respectively.
And the first conversion module 202 is used for converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method according to the mathematical model, and solving to obtain the pollution concentration.
The first conversion module 202 specifically includes:
the approximate function determining unit is used for determining an approximate function of the pollutant concentration function based on a cubic B spline method according to the mathematical model;
and the pollution concentration solving unit is used for converting the numerical solving problem of the fractional order differential equation into a linear equation solving problem according to the approximate function, and solving to obtain the pollution concentration.
An additional condition determining module 203 for determining an additional condition based on the mathematical model and the contamination concentration. The additional conditions specifically include:
based on the mathematical model, the additional conditions are:
Figure BDA0002528366050000121
where x denotes spatial position, t denotes time, U (x, t) denotes contaminant concentration, and g (t) denotes additional conditions at time t.
And the second conversion module 204 is used for converting the source term inversion problem in the model into a minimization functional based on the mathematical physics inverse problem framework according to the obtained additional conditions.
And the solving module 205 is configured to solve the established minimization functional by using a regularized cuckoo algorithm, where the solution represents a location of the pollution source.
The solving module 205 specifically includes:
a given unit, configured to give an initial value of a source item;
a first calculation unit for calculating the initial value f based on the source item0Via operator equation A (f)0)＝g0Calculating an additional condition g corresponding to the initial value of the source item at time t0(t), calculate g | |0(t)-g(t)||；f0Denotes the initial value of the source item, g0Indicating an initial value f corresponding to the source item0Additional conditions of (a);
a source item iteration value determining unit for determining the next source item iteration value f by the cuckoo algorithmi(i＝1,2,…)；
A second calculation unit for iterating the value f based on the source termiVia operator equation A (f)i)＝giCalculating the iteration value f of the source item corresponding to the time tiAdditional condition gi(t), calculate g | |i(t)-g(t)||，giRepresenting the iteration value f corresponding to the source termiAdditional conditions of (a);
a judging unit for judging gi(t) -g (t) if | is less than a given error limit.
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.

Claims (10)

1. A method of identifying a source of groundwater pollution, the method comprising:
establishing a mathematical model according to the pollution diffusion phenomenon of underground water; the mathematical model includes a contaminant concentration function;
according to the mathematical model, converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method, and solving to obtain the pollution concentration;
determining additional conditions according to the mathematical model and the pollution concentration;
converting a source item inversion problem in the model into a minimized functional based on a mathematical physics inverse problem framework according to the obtained additional conditions;
and solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source.
2. The method for identifying the groundwater pollution source according to claim 1, wherein the establishing of the mathematical model according to the diffusion phenomenon of groundwater pollution specifically comprises:
assuming that the diffusion system is one-dimensional spatial diffusion, the time fractional order convection diffusion equation is in the form:
Figure FDA0002528366040000011
initial conditions:
U(x,0)＝h(x) (2)
boundary conditions:
U(1,t)＝φ(t)，U(0,t)＝ψ(t) (3)
where x denotes spatial position, t denotes time, and 0 ≦ x, t ≦ 1, f (x, t) denotes source term, p (x, t) is convection term coefficient, q is diffusion term coefficient, α is known time fractional order parameter, U (x, t) is unknown function to be determined, representing contaminant concentration, h (x) denotes initial contaminant concentration, and Φ (t) and ψ (t) denote contaminant concentration at solution area boundary x ≦ 0 and x ≦ 1, respectively.
3. The method for identifying the pollution source of the groundwater according to claim 1, wherein the converting a numerical solution problem of fractional order differential equations into a solution problem of a linear equation system according to the mathematical model based on a cubic B-spline method, and solving to obtain the pollution concentration specifically comprises:
determining an approximate function of the pollutant concentration function based on a cubic B-spline method according to the mathematical model;
and converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set according to the approximate function, and solving to obtain the pollution concentration.
4. The method for identifying a source of groundwater pollution according to claim 1, wherein the determining additional conditions based on the mathematical model and the concentration of pollution comprises:
on the basis of the mathematical model, assuming that the source and sink terms are only related to time, the convection term coefficient is zero, and the diffusion term coefficient is only related to space;
the additional conditions are:
Figure FDA0002528366040000021
where x denotes spatial position, t denotes time, U (x, t) denotes contaminant concentration, and g (t) denotes additional conditions at time t.
5. The method for identifying the pollution source of the groundwater according to claim 1, wherein the established minimization functional is solved by a regularized cuckoo algorithm, and the solution represents the position of the pollution source, and specifically comprises:
giving an initial value of a source item;
based on the initial value f of the source item0Via operator equation A (f)0)＝g0Calculating an additional condition g corresponding to the initial value of the source item at time t0(t), calculate g | |0(t)-g(t)||；f0Denotes the initial value of the source item, g0Indicating an initial value f corresponding to the source item0Additional conditions of (a);
determining the next source item iteration value f by a cuckoo algorithmi(i＝1,2,…)；
Based on the iterative value f of the source termiVia operator equation A (f)i)＝giCalculating the iteration value f of the source item corresponding to the time tiAdditional condition gi(t), calculate g | |i(t)-g(t)||，giRepresenting the iteration value f corresponding to the source termiAdditional conditions of (a);
if g | | |i(t) -g (t) if less than a given error limit, the calculation is stopped.
6. A groundwater contamination source identification system, the system comprising:
the modeling module is used for establishing a mathematical model according to the pollution diffusion phenomenon of the underground water; the mathematical model includes a contaminant concentration function;
the first conversion module is used for converting the numerical solving problem of the fractional order differential equation into a solving problem of a linear equation set based on a cubic B-spline method according to the mathematical model, and solving to obtain the pollution concentration;
an additional condition determining module for determining an additional condition based on the mathematical model and the contamination concentration;
the second conversion module is used for converting the source item inversion problem in the model into a minimization functional based on a mathematical physics inverse problem frame according to the obtained additional conditions;
and the solving module is used for solving the established minimization functional by adopting a regularized cuckoo algorithm, wherein the solution represents the position of the pollution source.
7. The groundwater contamination source identification system of claim 6, wherein the mathematical model specifically comprises:
assuming that the diffusion system is one-dimensional spatial diffusion, the time fractional order convection diffusion equation is in the form:
Figure FDA0002528366040000031
initial conditions:
U(x,0)＝h(x) (2)
boundary conditions:
U(1,t)＝φ(t)，U(0,t)＝ψ(t) (3)
where x denotes spatial position, t denotes time, and 0 ≦ x, t ≦ 1, f (x, t) denotes source term, p (x, t) is convection term coefficient, q is diffusion term coefficient, α is known time fractional order parameter, U (x, t) is unknown function to be determined, representing contaminant concentration, h (x) denotes initial contaminant concentration, and Φ (t) and ψ (t) denote contaminant concentration at solution area boundary x ≦ 0 and x ≦ 1, respectively.
8. The groundwater contamination source identification system of claim 6, wherein the first conversion module specifically comprises:
the approximate function determining unit is used for determining an approximate function of the pollutant concentration function based on a cubic B spline method according to the mathematical model;
and the pollution concentration solving unit is used for converting the numerical solving problem of the fractional order differential equation into a linear equation solving problem according to the approximate function, and solving to obtain the pollution concentration.
9. The groundwater contamination source identification system according to claim 6, wherein the additional condition specifically comprises:
on the basis of the mathematical model, assuming that the source and sink terms are only related to time, the convection term coefficient is zero, and the diffusion term coefficient is only related to space;
the additional conditions are:
Figure FDA0002528366040000041
where x denotes spatial position, t denotes time, U (x, t) denotes contaminant concentration, and g (t) denotes additional conditions at time t.
10. The groundwater contamination source identification system of claim 6, wherein the solving module specifically comprises:
a given unit, configured to give an initial value of a source item;
a first calculation unit for calculating the initial value f based on the source item0Via operator equation A (f)0)＝g0Calculating an additional condition g corresponding to the initial value of the source item at time t0(t), calculate g | |0(t)-g(t)||；f0Denotes the initial value of the source item, g0Indicating an initial value f corresponding to the source item0Additional conditions of (a);
a source item iteration value determining unit for determining the next source item iteration value f by the cuckoo algorithmi(i＝1,2,…)；
A second calculation unit for iterating the value f based on the source termiVia operator equation A (f)i)＝giCalculating the iteration value f of the source item corresponding to the time tiAdditional condition gi(t), calculate g | |i(t)-g(t)||，giRepresenting the iteration value f corresponding to the source termiAdditional conditions of (a);
a judging unit for judging gi(t) -g (t) if | is less than a given error limit.
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