adomian decomposition method
TRANSCRIPT
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Mathematical Sciences Vol. 2, No. 2 (2008) 159-180
Adomian Decomposition Method for the Solution of
Optimal Control of Waste Water Model
F. B. Agusto1
Department of Mathematical Sciences, Federal University of Technology Akure, Nigeria.
Abstract
Most often, our waterways are being polluted by municipal, agricultural and
industrial wastes. When this happens, much of the available dissolved oxygen is
consumed by aerobic bacteria in decomposing the waste, thereby robbing other
aquatic organisms of the oxygen they need to live. In this paper, we consider the
solution of the optimal control problem for the modified Dissolve Oxygen(DO)/
Biological Oxygen Demand(BOD) system by the method of Adomain Decomposi-
tion.
Keywords: Optimal Control, dissolve oxygen, biological oxygen demand, waterpollution, Adomian decomposition method.
c 2008 Published by Islamic Azad University-Karaj Branch.
1 Introduction
Most often, our waterways are being polluted by municipal, agricultural and industrial
wastes, including many toxic synthetic chemicals which cannot be broken down at all
by natural processes. Even in tiny amounts, some of these substances can cause serious
harm.
Oxygen is required to support aquatic life and maintain water quality, it is the most
important dissolved gas in water. Fish and zooplankton breath dissolved oxygen, and
1E-mail Address: [email protected].
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160 Mathematical Sciences Vol. 2, No. 2 (2008)
without sufficient oxygen mortality will occur. Oxygen consuming wastes include de-
composing organic matter or chemicals which reduces Dissolved Oxygen (DO) in the
water. Maintaining a sufficient level of DO in water is critical to most forms of aquatic
life.
Microorganisms such as bacteria are responsible for decomposing organic waste. When
organic matter such as dead plants, leaves, grass clippings, manure, sewage, or even food
waste is present in a water supply, the bacteria will begin the process of breaking down
this waste. When this happens, much of the available dissolved oxygen is consumed by
aerobic bacteria, robbing other aquatic organisms of the oxygen they need to live [25].
DO concentrations are affected by a number of factors. Higher DO is produced by
turbulent actions such as waves, which mix air and water. Lower water temperatures
also allows for retention of higher DO concentrations. Low DO levels tend to occur
more often in warmer, slow moving waters. In general, low DO levels occur during the
warmest summer months and particularly during low flow periods. Water depth is also
a factor. In deep slow moving waters DO concentrations may be high near the surface
due to wind action and plant photosynthesis, but may be entirely depleted (anoxic) at
the bottom.
Biological oxygen demand (BOD) is an indicator for the concentration of biodegradable
organic matter present in a sample of water. It can be used to infer the general quality
of the water and its degree of pollution. BOD measures the rate of uptake of oxygen by
micro-organisms in the sample of water at a fixed temperature and over a given period
of time.
If the pollution level is not too high this need can be satisfied by the dissolved oxygen.
Notice that the oxygen is very sensitive to wastewater discharges namely the thermal
ones. Indeed, at high temperature solubility of oxygen decreases while activity of mi-
croorganisms which are oxygen consuming increases. If the quantity of organic matter
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F. B. Agusto 161
increases beyond a maximum value the dissolved oxygen is not enough to decompose
it leading to modification in the ecosystem.
In recent times a number of researchers have worked on the BOD/DO parabolic mod-
els. Streeter and Phelps [24] gave the classical model for BOD/DO. Bermudez [12]
considered Streeter model, and in [13] gave an optimal location for wastewater outfall
for steady case parabolic equation. Martinez [21] gave the theoretical analysis for the
optimal control problem related to wastewater treatment resulting in pointwise control
for both the objective functional and state constraint stating the existence of unique
solution. Alvarez-Vazquez [10] threated the case of evolution parabolic equation of [13]
for an optimal location of wastewater outfall. Piasecki [22] and [23] in his work modified
the Streeters model to include Sediment Oxygen Demand(SOD).
This paper concerns the application of Adomian decomposition method to the solu-
tion of a diffusive-convective population problem, whose growth is governed by logistic
terms. The growth of microorganisms population have been shown to follow logistic
growth pattern [15].
The Adomian decomposition method is useful for obtaining both a closed form and
the explicit solution and numerical approximations of linear or nonlinear differential
equations, and it is also quite straightforward to write computer codes. The scheme is a
method for solving a wide range of problems whose mathematical models yield equation
or system of ordinary or partial differential equations (see [1, 8, 27]). This method has
been applied to obtain formal solution to a wide class of stochastic and deterministic
problems in science and engineering involving algebraic, differential, integrodifferential,
differential delay, integral and partial differential equations.
This method was proposed by the American mathematician G. Adomian (1923 - 1996).
It is based on the search for a solution in the form of a series and on decomposing
the nonlinear operator into a series in which the terms are calculated recursively using
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162 Mathematical Sciences Vol. 2, No. 2 (2008)
Adomian polynomials [6].
The decomposition method was proven by many authors to be reliable and promising.
It can be used for all types of differential equations, linear or nonlinear, homogeneous
or inhomogeneous [2, 3, 4, 5, 6]. The technique has many advantages over the clas-
sical techniques, it avoids perturbation in order to find solutions of given nonlinear
equations. The decomposition approach was used to handle a variety of linear and
nonlinear problems and provides an immediate and convergent solution without any
need for linearization or discretization.
Generally this method is useful for problems that can be written in the following form
which appears in the large number of problems in applied sciences:
u (u) = g, (1)
where u is unknown, usually is a nonlinear operator, and g is given. Depending
on the nonlinear form , we can consider the Adomian decomposition method as an
efficient method.
The Adomian decomposition method which accurately computes the series solution is
of great interest to applied science, engineering, physics, biology, and so forth. The
method provides the solution in a rapidly convergent series with components that can
be elegantly computed [11, 16, 17, 18, 26]. The present work is aimed at producing
approximate solutions which are obtained in rapidly convergent series with elegantly
computable components by the Adomian decomposition technique. It is well known in
the literature that the decomposition method provides the solution in a rapidly conver-
gent series where the series may lead to the solution in a closed form if it exists. The
rapid convergence of the solution is guaranteed by the work conducted by Cherruault
[14].
Mathematical models play a major role in predicting the pollution level in the regions
under consideration. Obviously, the knowledge of mathematical models for the evolu-
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F. B. Agusto 163
tion of pollutant concentration is an unavoidable first step if one wants to solve this
problem. So, the first part of this work is devoted to the study of Biological Oxygen
Demand (BOD) and Dissolved Oxygen (DO), which is frequently used in the case of
domestic discharges. Next, in section 3 we give the analysis of the system. while in
section 4, we discuss the optimal control for the nonlinear system of equations, stating
the optimality conditions in section 5. Section 6 is devoted to solution for the coupled
system by Adomian decomposition method, while in section 7, we discuss the result
obtained and conclude.
2 Pollutant Dispersion: The BOD-DO Model
We consider a domain occupied by shallow water, where polluting wastewater are dis-
charged. Firstly, in order to simulate the water quality in the domain,we have to choose
some indicators of pollution levels. Two of the most important (especially in the case
of domestic discharges) are the Dissolved Oxygen (DO) and the organic matter, which
can be measured in terms of the need of oxygen to decompose it, the so called Biological
Oxygen Demand (BOD). If the pollution level is not too high the BOD can be satisfied
by the DO. However, if the organic matter increases beyond a maximum value the
DO is not enough for its decomposition, leading to important modifications (anaerobic
processes) in the ecosystem. To avoid this situation a threshold value of BOD may not
be exceeded and a minimum level of DO must be guaranteed.
The evolution of the BOD and the DO in the domain IR2 is governed by a system
of partial differential equations (see Streeter and Phelps [24], A. Bermudez, [12]). We
give a modification of the model given by Bermudez, [12] to include a logistic growth
for BOD. Let us denote by 1(x, t) and 2(x, t) the concentrations of BOD and DO
at point x and at time t [0, T], respectively. Then, these concentrations are
obtained as the solution of the following two initial-boundary value problems:
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1
t + v.1 11 = 1(a b1) k11 in Q
1(x, 0) = 10(x) in
1n
= 0 on
2t
+ v.2 22 = k11 +1h
k2(ds 2) in Q
2(x, 0) = 20(x) in
2n
= 0 on
(2)
where u and h can be obtained from the shallow water equation, positive parameters
1 and 2 (horizontal viscosity coefficients), k1, k2(kinetic coefficients related to BOD
elimination and oxygen transfer through the surface, respectively) and ds (oxygen sat-
uration density) can be obtained from experimental measurements.
3 Analysis of the State System
Let IR2 be a bounded domain with boundary smooth enough and (0 , T) an open
interval. Q denotes the cylindrical domain (0, T), while = (0, T) which is
the lateral boundary of Q. We make the following assumptions of the problem data
adapted from Martinez, Rodriguez, Vazquez-Mendez [21])
v [L( [0, T])]2, h C( [0, T]) h(x, t) > 0 (x, t) [0, T])
10, 20 C2(), a [L()]2, b [L()]2
Theorem 3.1 [9] There exists a unique pair = (1, 2) [L2(0, T; H1())]2
[L2(0, T; L2())]2 with t [L2(0, T; H1())]2 of the state equation (2) satisfying the
following
||||2[L2(0,T;H1())]2 + ||||2[L2(0,T;L2())]2 + ||t||
2[L2(0,T;H1())]2 C
||10||C() + ||20||C()
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4 Optimal Control
In this section, we prove the existence and uniqueness of optimal solution to the state
equations (2), given an appropriate objective functional stated below.
4.1 The Optimization Problem
We state the optimal control problem. We look for a (, m) H1() Uad such that
the cost functional:
J(, v) =
Q
||2 dxdt + m2L(Q) (3)
is minimized subject to the constraints
1t
+ v.1 11 = 1(a b1) k11 + m in Q
1(x, 0) = 10(x) in
1n
= 0 on
2t
+ v.2 22 = k11 +1h
k2(ds 2) in Q
2(x, 0) = 20(x) in
2n
= 0 on
(4)
Now, let Uad, the admissible space of control be defined as
Uad = {m : J(m) < and (4) are satisfied}.
The control problem is to find the values of m(t) > 0 in such a way that they satisfy
(4) and they minimize the objective function, i.e.,
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J(, m) J(, m) (, m) H1() Uad (5)
Any element m Uad, satisfying (3) is called an optimal control and the corresponding
state, denoted by is called an optimal state.
4.2 The Existence of an Optimal Solution
We now show the existence of an optimal solution and give the following theorem
Theorem 4.1 [9] If there exists a feasible control m Uad then the optimal problem
has, at least, a solution.
5 Derivation of the Optimality System
Now, the optimality system (OS) is derived be differentiating the cost functional with
respect to the control. Since the state variable , i.e. the solution of (2), is contained
in the functional, we first need to show that depends on the control in a differentiable
way. then we will characterize the optimal control in terms of the solution of the OS,
which consists of the state equations coupled with the adjoint equations, following the
procedure in Lenhart [20].
Proposition 5.1 The mapping
m Uad = (m) [L2(0, T; H1())]2
is differentiable in the following sense:
(m + ) (m)
in [L2(0, T; H1())]2
as 0 for any m Uad, and L(Q) such that m + Uad for small. Also
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is a solution of the following problem:
1t
= v.1 + 11 + 1[a 2b1 k1] in Q
1(x, 0) = 10(x) in 1n
= 0 on
2t
= v.2 + 22 + k11 1
hk22 in Q
2(x, 0) = 20(x) in 2n
= 0 on
(6)
Proof: Let m Uad, m+ Uad for small, be the optimal control and let = (m)
and = (m + ), be the corresponding state. The quotient ( )/ satisfies
11t = v.
1+1
+ 111
+11
a 2b(
1 + 1) k1
in Q
11
(x, 0) = 10(x) in
11
n
= 0 on
22
t
= v.22
+ 2
22
k1
11
1
hk222
in Q
22 (x, 0) = 20(x) in
22
n
= 0 on
Since the a + 2b(1 + 1) + k1 is bounded independent of , we obtained
L2(Q)
+
L2(Q)
C()
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168 Mathematical Sciences Vol. 2, No. 2 (2008)
and from Ladyzenskaya [19], it simplifies to
t
L2(Q)
+n
i,j=1
xixj
L2(Q)
C.
thus obtaining the weak convergence,
(m + ) (m)
in W1,02 ()
Notice that strongly in L2(Q), as such, we have satisfies (6).
Proposition 5.2 For any optimal control m and corresponding solution = (m),
there exists in [L2(0, T; H1())]2 [L2(0, T; L2())]2 satisfying the adjoint equation
1t
+ v.1 11 = 1[a 2b1 k1] + 1 in Q
1(x, T) = 0 in 1n
= 0 on
2t
+ v.2 22 = 1
hk22 + 2 in Q
2(x, T) = 0 in 2n
= 0 on
(7)
Proof: Suppose that m is an optimal control and is its corresponding solution of (2)
whose existence have been shown above. Consider control m+ Uad with associated
solution = (m + ), let P = , and P = P(m + ). Since the minimum of the
cost functional J is achieved at m, hence,
0 lim0
J(m + ) J(m)
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F. B. Agusto 169
= lim
0
1
Q
[(P)2 P2]dxdt + Q
[m2 + 2m + ()2 m2]dxdt= lim
0
1
Q
[(P)2 P2]dxdt +
Q
[2m + 22]dxdt
= lim0
Q
P P
(P + P)dt +
Q
[2m + 2]dxdt
=
Q
(P)dxdt +
Q
2mdxdt
(8)
where
P P
. By standard linear parabolic results in Ladyzenskaya [19],
there exists a solution in [L2(0, T; H1())]2 [L2(0, T; L2())]2 satisfying the adjoint
equation (7). Substituting from the adjoint equation into the above inequality (8), we
obtain
0
Q
1[
1t
+ v.1 11 a1 + 2b11 + k11]
+ 2[2t
+ v.2 22 +1
Hk22]
dxdt +
T0
2mdt
Integrating by parts, yields
0 Q
(1[1t
+ v.1 11 a1 + 2b11 + k11]
+2[2t
+ v.2 22 +1
Hk22])dxdt +
Q
2mdxdt
From (6) we have;
0
Q
1dxdt +
Q
2mdxdt
Hence, by Propositions 5.1 and 5.2 we have the following theorem
Theorem 5.3 Given that, there exists a unique optimal solution (, m) to equation
(4). The cost function being given by (3) a necessary and sufficient condition for
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m Uad to be an optimal control is that the following equations be satisfied.
1t
+ v.1 11 = 1(a b1) k11 + m in Q
1(x, 0) = 10(x) in 1n
= 0 on
2t
+ v.2 22 = k11 +1
hk2(ds 2) in Q
2(x, 0) = 20(x) in
2n
= 0 on
1t
+ v.1 11 = 1[a 2b1 k1] + 21 in Q
1(x, T) = 0 in 1
n = 0 on
2t
+ v.2 22 = 1
hk22 + 22 in Q
2(x, T) = 0 in 2n
= 0 on
Q
(1 + 2m,n m)dxdt 0 n Uad
(9)
Proof: The proof follows from Propositions 5.1 and 5.2 and their proofs
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6 Adomian Decomposition Method
In this section we solve the optimal control problem of the previous sections by the
Adomian decomposition method.
Adomian (see [6, 7] for example) asserts that the decomposition method provides an
efficient and computationally convenient method for generating approximate series so-
lutions to a wide class of equations. In order that this paper will be reasonably self-
contained, we describe here how this method is applied.
Consider the operator form of an ordinary differential equation in the following form
Lu + Ru + Nu = g(x) (10)
where L is the highest order derivative which is assumed to be easily invertible, R the
linear differential operator of less order than L, Nu represents the nonlinear terms,
and g is the source term. Applying the inverse operator L1 to both sides of (10), and
using the given conditions we have
u = f(x) L1(Ru) L1(Nu) (11)
where the function f(x) represents the terms arising from integrating the source
term g(x) and from using the given conditions.
The nonlinear operator Nu is usually represented by an infinite series of the Adomian
polynomials given by
Nu =
n=0
An (12)
where the component An is an appropriate Adomian polynomial. Adomian polynomials
are found by calculating the nonlinear operator An in the following form:
An =1
n!
dn
dn
N
k=0
kuk
=0
, n 0 (13)
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The solution u(x) is defined as a series
u(x) =n=0
un
where the components u0, u1, u2, are usually determined recursively by
u0 = f(x)
un+1 = L1(Run) L
1(N un), n 0
(14)
Supoose the series
n=0 un and
n=0 An are convergent [14, 1], substituting (13) into
(14) leads to the determination of the components of u.
6.1 Application to the BOD/DO Model
We now employ the ADM to solve BOD/DO coupled model.
Applying the operatort
1
to both sides of the state equation (9) we obtain:
1
(t) = 1
(0) v t0
x1
dt + 1t0
2
x21
dt + t0
(a k1
)1
dt b t0
1
1
dt
+
t0
mdt
2(t) = 2(0) +1
hk2dst v
t0
x2dt + 2
t0
2
x22dt
1
hk2
t0
2dt
k1
t0
1dt
(15)
As usual the solution of (15) are considered to be as the sum of the following series
1(t) =n=0
n1 2(t) =n=0
n2 (16)
Then we approximate the nonlinear term as:
11 =n=0
An (17)
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substituting (16) and (17) into (15) results in:
n=0
n1 = 1(0) + mt v
t0
n=0
xn1dt + 1
t0
n=0
2
x2n1dt +
t0
n=0
(a k1)n1dt
b
t0
n=0
Andt
n=0
n2 = 2(0) +1
hk2dst v
t0
n=0
xn2dt + 2
t0
n=0
2
x2n2dt
1
hk2
t0
n=0
n2dt
k1
t0
n=0
n1dt
(18)From (18) we define the following scheme:
01 = 1(0) + mt, 02 = 2(0) +
1
hk2dst
n+11 = v
t0
xn1dt + 1
t0
2
x2n1dt +
t0
(a k1)n1dt b
t0
Andt
n+12 = vt0
x n2dt + 2t0
2
x2n2dt 1
h k2t0
n2dt k1t0
n1dt
(19)
A0 = 01
01 A2 = 2
01
21 +
11
11 A4 = 2
01
41 + 2
11
31 +
21
21
A6 = 201
61 + 2
11
51 + 2
21
41 +
31
31 A8 = 2
01
81 + 2
11
71 + 2
21
61 + 2
31
51 +
41
41
(20)
Similarly applying the operator t1
to both sides of the adjoint equation (9) we
obtain:
1(t) = 1(T) 1t + v
tT
x1dt 1
tT
2
x21dt
tT
1[a 2b1 k1]dt
2(t) = 2(T) 2t + v
tT
x2dt 2
tT
2
x22dt +
1
hk2
tT
2dt
(21)
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As usual the solution of (15) are considered to be as the sum of the following series
1(t) =n=0
n1 2(t) =n=0
n2 (22)
substituting (22) and (17) into (21) results in:
n=0
n1 = 1(T) 1t + vtT
xn1dt 1
tT
2
x2n1dt
tT
n1 [a 2b1 k1]dt
n=0
n2 = 2(T) 2t + v t
T
x
n2dt 2 t
T
2
x2
n2dt +1
h
k2 t
T
n2dt
(23)
From (23) we define the following scheme:
01 = 1(T) 1t, 02 = 2(T) 2t
n+11 = v
tT
xn1dt 1
tT
2
x2n1dt
tT
n1 [a 2b1 k1]dt
n+12 = vtT
x
n2dt 2tT
2
x2n2dt + 1h
k2tT
n2dt
(24)
Substituting (20), (19) into (18) and (24) into (23) and using Maple we obtained a few
terms approximation to the solution as
N1 =Nn=0
n1 , N2 =
Nn=0
n2 and N1 =
Nn=0
n1 , N2 =
Nn=0
n2 (25)
where
1(t) = limtn
1 , 2(t) = limtn
2 and 1(t) = limtn
1 , 2(t) = limtn
2 (26)
and the control m(t) is obtained as
m(t) = 1
21(t) =
1
2N1 (27)
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7 Conclusion
We have solved the BOD/DO pollution model by implementing the Adomian Decom-
position method. The optimality system of (9) was solved using ADM to obtain the
optimal control m(t), this in turn produces the optimal BOD/DO of Figure 2. And we
observed, comparing Figures 1 and 2, that the level of dissolved oxygen concentration
has greatly been improved while the BOD concentration has greatly been reduced.
Figure 1: The uncontrolled BOD/DO profile
We have established in the preceding sections the uniqueness and existence of opti-
mal solution for the nonlinear system of equations for the BOD/DO model and stated
the optimality conditions. And we have solved the state equations as well as the ad-
joint equations, for obtaining the optimum control by using the Adomian decomposition
method.
Acknowledgment. The author (FBA) acknowledges, with thanks, the support in part
of the African Mathematics Millennium Science Initiative (AMMSI) research scholar-
ship award 2007.
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176 Mathematical Sciences Vol. 2, No. 2 (2008)
Figure 2: The controlled contaminant profile
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