VIM for Solving the Pollution Problem of a System of Lakes

Abstract

Pollution has become a very serious threat to our environment. Monitoring pollution is the first step toward planning to save the environment. The use of differential equations of monitoring pollution has become possible. In this paper the pollution problem of three lakes with interconnecting channels has been studied. The variational iteration method has been applied to compute an approximate solution of the system of differential equations, governing on the problem. Three different types of input models: sinusoidal, impulse, and step will be considered for monitoring the pollution in the lakes. The results are compared with those obtained by Adomian decomposition method. This comparison reveals that the variational iteration method is easier to be implemented.

1. Introduction

He’s variational iteration method plays an important role in recent researches in different fields of sciences and engineering. This method is proposed by the Chinese mathematician, He, [1–3] as a modification of the general Lagrange multiplier method. It has been shown to solve efficiently, easily, and accurately a large class of nonlinear problems with approximation converging to accurate solutions. This method has been used to study many problems in different disciplines, hyperbolic equation [4], wave equation [5], nonlinear chemistry problems [6], and quadratic Riccati differential equation [7].

To illustrate its basic idea of the method, we consider the following nonlinear equation: 𝐿𝑢(𝑡)+𝑁𝑢(𝑡)=𝑔(𝑡), (1) where L is a linear operator, N is a nonlinear operator, and 𝑔(𝑡) is a known analytical function.

According to VIM, we can construct the following correction functional: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀𝜆{𝐿𝑢_𝑛(𝑠)+𝑁𝑢_𝑛(𝑠)−𝑔(𝑠)}𝑑𝑠, (2) where 𝜆 is a Lagrange multiplier [8], which can be identified optimally via variational theory [9, 10], 𝑢_𝑛 is the 𝑛th approximate solution, and 𝑢_𝑛 denotes a restricted variation [10], that is, 𝛿𝑢_𝑛=0.

2. Modeling of the Pollution Problem

Figure 1 shows the system of three lakes that are modeled in [11]. Each lake is considered to be a large compartment and the interconnecting channel as pipes between the compartments.

Figure 1 

System of three lakes with interconnecting channels.

The direction of flow in the channels or pipes is indicated by the arrows in the figure. A pollutant is introduced into the first lake and 𝑝(𝑡) denotes the rate at which the pollutant enters the lake per unit time. The function 𝑝(𝑡) may be constant or may vary with time. We are interested in knowing the level of pollution in each lake at any time.

Let 𝑥_𝑖(𝑡) denotes the amount of the pollution in lake 𝑖 at any time 𝑡≥0, where 𝑖=1,2,3. We assume the pollutant in each lake to be uniformly distributed throughout the lake by some mixing process, and the volume of water 𝑉_𝑖 in lake 𝑖 remains constant for each of the lakes. Also we assume that the type of pollution is persistent and not degrading to other forms. Then the concentration of the pollutant in lake 𝑖 at any time is given by 𝑐_𝑖𝑥(𝑡)=^𝑖(𝑡)𝑉_𝑖. (3) Each lake initially is assumed to be free of any contaminant, so 𝑥_𝑖(0)=0 for each 𝑖=1,2,3. To model the dynamic behavior of the system of lakes, we let constant 𝐹_𝑗𝑖 denote the flow rate from lake 𝑖 to lake j. The flux of pollutant flowing from lake i into lake j at any time t, denoted by 𝑟_𝑗𝑖(𝑡), is defined by 𝑟_𝑗𝑖(𝑡)=𝐹_𝑗𝑖𝑐_𝑖𝐹(𝑡)=^𝑗𝑖𝑉_𝑖𝑥_𝑖(𝑡). (4) Thus 𝑟_𝑗𝑖(𝑡) measures the rate at which the concentration of pollutant in lake 𝑖 flows into lake j at time t. We will observe that rateofchangeofpollutant=Inputrate−outputrate. (5) Applying this principle to each lake results in the following system of first-order equations: 𝑑𝑥₁=𝐹𝑑𝑡𝑉¹³₃𝑥₃𝐹(𝑡)+𝑝(𝑡)−³¹𝑉₁𝑥₁𝐹(𝑡)−²¹𝑉₁𝑥₁(𝑡),𝑑𝑥₂=𝐹𝑑𝑡𝑉²¹₁𝑥₁𝐹(𝑡)−³²𝑉₂𝑥₂(𝑡),𝑑𝑥₃=𝐹𝑑𝑡𝑉³¹₁𝑥₁𝐹(𝑡)+³²𝑉₂𝑥₂𝐹(𝑡)−¹³𝑉₃𝑥₃(𝑡). (6) To remain constant the volume of each lake, the flow rate into each lake must balance the flow out of the lake. So we earn the following conditions: Lake1:𝐹₁₃=𝐹₂₁+𝐹₃₁,Lake2:𝐹₂₁=𝐹₃₂,Lake3:𝐹₃₁+𝐹₃₂=𝐹₁₃. (7)

3. Numerical Results and Comparison with the ADM

There are three different types of input models used to monitor pollutants in a lake. The input models are the sinusoidal, impulse, and step input.

3.1. Sinusoidal Input

The Sinusoidal input model is used for pollutants that are introduced to the lake periodically. Sinusoidal input changes 𝑝(𝑡) to a more useful one 𝑝(𝑡)=𝑝_𝑖∗(1+𝑎∗sin(2𝜋∗𝑡𝑇)). (8)

The additional variables for the sinusoidal model are defined as follows:𝑎:the normalize amplitude,𝑇:the period of fluctuations,𝑝_𝑖:the average input concentration of pollutant.

The actual amplitude of the term is 𝑝_𝑖(1+𝑎) for the peaks and 𝑝_𝑖(1−𝑎) for the valleys.

An example would be a manufacturing plant dumping waste producing more output during the day than at night because of hour of operation, hence a periodic input. The concentration of the lake eventually converges to the average input concentration of the contaminant.

Now we assume 𝑝(𝑡)=1+sin(𝑡), and for comparison with the results found by Biazar [11], the following values, for variables, are considered: 𝑉₁=2900mi³,𝑉₂=850mi³,𝑉₃=1180mi³,𝐹₂₁=18mi³/year,𝐹₃₂=18mi³𝐹/year,₃₁=20mi³/year,𝐹₁₃=38mi³/year. (9)

So system (6) will be 𝑥^′₁(𝑡)=38𝑥1180³(𝑡)+(1+sin(𝑡))−38𝑥2900¹𝑥(𝑡),_′2(𝑡)=18𝑥2900¹(𝑡)−18𝑥850²𝑥(𝑡),_′3(𝑡)=20𝑥2900¹(𝑡)+18𝑥850²(𝑡)−38𝑥1180³𝑥(𝑡),₁(0)=0,𝑥₂(0)=0,𝑥₃(0)=0. (10)

According to VIM, we can construct the correction functionals of system (10) as follows: 𝑥_1𝑛+1(𝑡)=𝑥_1𝑛(𝑡)+^𝑡₀𝜆₁{𝑥^′_1𝑛(𝜉)−38𝑥1180^3𝑛(𝜉)−1−sin(𝜉)+38𝑥2900^1𝑛𝑥(𝜉)}𝑑𝜉,_2𝑛+1(𝑡)=𝑥_2𝑛(𝑡)+^𝑡₀𝜆₂{𝑥^′_2𝑛(𝜉)−18𝑥2900^1𝑛+(𝜉)18𝑥850^2𝑛𝑥(𝜉)}𝑑𝜉,_3𝑛+1(𝑡)=𝑥_3𝑛(𝑡)+^𝑡₀𝜆₃{𝑥^′_3𝑛(𝜉)−20𝑥2900^1𝑛−(𝜉)18𝑥850^2𝑛(𝜉)+38𝑥1180^3𝑛(𝜉)}𝑑𝜉, (11) where 𝜆₁, 𝜆₂, and 𝜆₃ are general Lagrange multipliers, 𝑥_1𝑛(𝜉), 𝑥_2𝑛(𝜉), and 𝑥_3𝑛(𝜉) denote restricted variations, that is, 𝛿𝑥_1𝑛(𝜉)=𝛿𝑥_2𝑛(𝜉)=𝛿𝑥_3𝑛(𝜉)=0.

Making the above correction functional stationary, we have 𝛿𝑥_1𝑛+1(𝑡)=𝛿𝑥_1𝑛(𝑡)+𝜆₁(𝜉)𝛿𝑥_1𝑛(𝜉)|^𝑡₀−^𝑡₀𝜆^′₁(𝜉)𝛿𝑥_1𝑛(𝜉)𝑑𝜉=0,𝛿𝑥_2𝑛+1(𝑡)=𝛿𝑥_2𝑛(𝑡)+𝜆₂(𝜉)𝛿𝑥_2𝑛(𝜉)|^𝑡₀−^𝑡₀𝜆^′₂(𝜉)𝛿𝑥_2𝑛(𝜉)𝑑𝜉=0,𝛿𝑥_3𝑛+1(𝑡)=𝛿𝑥_3𝑛(𝑡)+𝜆₃(𝜉)𝛿𝑥_3𝑛(𝜉)|^𝑡₀−^𝑡₀𝜆^′₃(𝜉)𝛿𝑥_3𝑛(𝜉)𝑑𝜉=0. (12)

The stationary conditions can be obtained as follows: 𝛿𝑥_1𝑛∶1+𝜆₁(𝜉)|_𝜉=𝑡𝜆=0,_′1(𝜉)=0,𝛿𝑥_2𝑛∶1+𝜆₂(𝜉)|_𝜉=𝑡𝜆=0,_′2(𝜉)=0,𝛿𝑥_3𝑛∶1+𝜆₃(𝜉)|_𝜉=𝑡𝜆=0,_′3(𝜉)=0. (13)

Therefore, the Lagrange multipliers can be identified as 𝜆₁=𝜆₂=𝜆₃=−1. (14)

Substituting (14) into the correction functionals (11) results in the following iteration formula: 𝑥_1𝑛+1(𝑡)=𝑥_1𝑛(𝑡)−^𝑡₀{𝑥^′_1𝑛(𝜉)−38𝑥1180^3𝑛+(𝜉)−(1+sin(𝜉))38𝑥2900^1𝑛𝑥(𝜉)}𝑑𝜉,_2𝑛+1(𝑡)=𝑥_2𝑛(𝑡)−^𝑡₀{𝑥^′_2𝑛(𝜉)−18𝑥2900^1𝑛+(𝜉)18𝑥850^2𝑛𝑥(𝜉)}𝑑𝜉,_3𝑛+1(𝑡)=𝑥_3𝑛(𝑡)−^𝑡₀{𝑥^′_3𝑛(𝜉)−20𝑥2900^1𝑛−(𝜉)18𝑥850^2𝑛(𝜉)+38𝑥1180^3𝑛(𝜉)}𝑑𝜉. (15)

We start with initial approximations 𝑥₁₀=𝑥₁(0)=0,𝑥₂₀=𝑥₂(0)=0,𝑥₃₀=𝑥₃(0)=0.

By the above iteration formula, we can obtain a few first terms being calculated: 𝑥₁₁𝑥=1+𝑡−cos(𝑡),₂₁𝑥=0,₃₁𝑥=0,₁₂=1+14311450𝑡−cos(𝑡)−19𝑡2900²+19𝑥1450sin(𝑡),₂₂=991450𝑡+𝑡2900²−9𝑥1450sin(𝑡),₃₂=11145𝑡+𝑡290²−1𝑥145sin(𝑡),₁₃=123998651+12404750014311450𝑡−123998651−124047500cos(𝑡)1576601𝑡248095000²+191450sin(𝑡)+16283𝑡248095000³,𝑥₂₃=91450𝑡+42849𝑡14297000²−9+1450sin(𝑡)1521−7148500507𝑡14297000³−1521𝑥7148500cos(𝑡),₃₃=1145𝑡+7080881𝑡2108807500²−1+145sin(𝑡)190869−105440375063623𝑡2108807500³−190869⋮1054403750cos(𝑡). (16)

Continuing in this manner, we can find the rest of components.

A few 𝑚 terms approximation to the solutions are considered as 𝑥₁(𝑡)≈𝑥_1𝑚,𝑥₂(𝑡)≈𝑥_2𝑚,𝑥₃(𝑡)≈𝑥_3𝑚. (17)

Some numerical values of solutions are presented in Table 1 with 𝑛 being the order of iteration formula of VIM.

Table 1 

Result for sinusoidal model.

3.2. Impulse Input

The impulse input model is used for pollutants that have been released into the lake immediately. Impulse input functions have a spike, and everywhere else the function is zero. The spike represents the time at which the contaminant was dumped.

An example would be dumping one barrel of gasoline into a lake at time zero. In one short heavy dumping of waste, with no reaction with the lake, input 𝑝(𝑡) is equal to 100 units for a duration of 10 units of time.

Now we assume 𝑝(𝑡)=100, and so the system (6) with parameters that are defined in Section 3.1 will be 𝑥^′₁(𝑡)=38𝑥1180³(𝑡)+100−38𝑥2900¹𝑥(𝑡),_′2(𝑡)=18𝑥2900¹(𝑡)−18𝑥850²𝑥(𝑡),_′3(𝑡)=20𝑥2900¹(𝑡)+18𝑥850²(𝑡)−38𝑥1180³𝑥(𝑡),₁(0)=0,𝑥₂(0)=0,𝑥₃(0)=0. (18)

Therefore, by solving system (18) with VIM, as have been shown in Sinusoidal input model, the following results are earned: 𝑥₁₁=100𝑡,𝑥₂₁=0,𝑥₃₁𝑥=0,₁₂=100𝑡−19𝑡29²,𝑥₂₂=9𝑡29²,𝑥₃₂=10𝑡29²,𝑥₁₃=100𝑡−19𝑡29²+16283𝑡2480950³,𝑥₂₃=9𝑡29²−507𝑡142970³,𝑥₃₃=10𝑡29²−63623𝑡21088075³.⋮ (19)

Some numerical values of solutions are presented in Table 2 with 𝑛 being the order of iteration formula of VIM.

Table 2 

Result for impulse model.

3.3. Step Model

The step input model is used for pollutants that enter the lake at steady concentration and rate and continue the same way indefinitely. The pollutant enters the system at time zero, and the concentration previous to time zero was nothing. The significant points for step input are that the input increases suddenly at time zero and that the input remains essentially constant after time zero.

An example would be when a manufacturing plant starts production at time zero and starts dumping raw sewage at a constant rate and concentration.

Now we assume 𝑝(𝑡)=100𝑡, so system (6) with parameters that are defined in Section 3.1 would be 𝑥^′₁(𝑡)=38𝑥1180³(𝑡)+100𝑡−38𝑥2900¹𝑥(𝑡),_′2(𝑡)=18𝑥2900¹(𝑡)−18𝑥850²𝑥(𝑡),_′3(𝑡)=20𝑥2900¹(𝑡)+18𝑥850²(𝑡)−38𝑥1180³𝑥(𝑡),₁(0)=0,𝑥₂(0)=0,𝑥₃(0)=0. (20)

Now by using VIM for system (20), the following approximations for the solution are earned: 𝑥₁₁=50𝑡²,𝑥₂₁𝑥=0,₃₁𝑥=0,₁₂=50𝑡²−19𝑡87³,𝑥₂₂=3𝑡29³,𝑥₃₂=10𝑡87³,𝑥₁₃=50𝑡²−19𝑡87³+16283𝑡9923800⁴,𝑥₂₃=3𝑡29³−507𝑡571880⁴,𝑥₃₃=10𝑡87³−63623𝑡84352300⁴.⋮ (21)

Some numerical values of solutions are presented in Table 3 with 𝑛 being the order of iteration formula of VIM.

Table 3 

Result for step model.

4. Conclusion and Discussion

In this paper, the problem of pollutions of three lakes with interconnecting channels has been considered. Different input models have been used for monitoring the pollution in three lakes. The variational iteration method as a powerful device has been used to solve the system of differential equations, governing on the problem. For different input models, the results of applying 𝑛 times iteration formula in VIM procedure for different times t (year) are shown in suitable tables. The results are compared with those obtained by ADM in [11]. This comparison shows that the results are the same, but the solution procedure of the VIM is simpler than of ADM and the amounts of computations required in VIM is much lesser than ADM.

Acknowledgment

The authors are grateful to the anonymous reviewer for his (her) useful comments which have improved the quality of the paper.