Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method

Abstract

Pollution has become an intense danger to our environment. The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. The main feature of the NIM is that the procedure is very simple, and it does not need to calculate any special type of polynomial or multipliers such as Adomian polynomials and Lagrange’s multipliers. Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.

1. Introduction

Over the years, differential equations are employed by researchers that care about what is going on in the surroundings, for example, one can see the following ref. [1–10]. Therefore, differential equations play a prominent role to solve physical problems using numerical or analytical methods. There has recently been much attention devoted to the search for better and more efficient solution methods for determining a solution, approximate or exact, analytical, or numerical, to physical models. Finding exact or approximate solutions of these nonlinear equations is interesting and important. Several semianalytic methods including the Adomian decomposition method (ADM) [11], homotopy perturbation method (HPM) [12], and variational iteration method (VIM) [13] have been used for solving mathematical models of differential equations.

Finding accurate and analytical procedures for solving lake pollution model has been a vital research undertaking. The system of lake pollution models has been solved first time by Biazar et al. [14] using a semianalytic technique called the Adomian decomposition method (ADM). So often, it is very hard to ascertain the Adomian polynomials involved with ADM. To overcome the shortcomings of ADM, many analytical techniques have been successfully implemented by various groups of researchers. In particular, the lake pollution system was solved by Ganji et al. [15] via homotopy-perturbation method (HPM). Recently, Ahmed and Khan [16] have investigated dynamics of system of polluted lakes through the numerical procedure fractional derivative approach. Numerous authors used some of the approximate numerical techniques to obtain approximate analytical solutions such as the Bessel matrix method [17], Haar wavelet collocation method [18], Differential Transform method (DTM) [19], Laplace-Padé Differential transform method (LPDTM) [20], and variational iteration method (VIM) [21].

Another prominent semianalytical technique which has been demonstrated to be a lot more straightforward and efficient than the abovementioned methods is called the new iterative method (NIM), first proposed by Varsha Daftardar-Gejji and Jafari [22] . Very recently, NIM was employed for solving Fokker-Planck equations [23], chemical kinetic equations [24], and Klein-Gordon equations [25]. Some other applications of the new iterative method (NIM) on nonlinear problems can be found in [26–29]. In this paper, the NIM motivates us to solve the lake pollution systems.

2. Mathematical Modeling of the Lake Pollution System

Pollutants have different kinds of characteristic property, and nonpolar pollutants never mix with lake water, whereas polar one makes a reaction with lake water; as a result, mild to severe contamination takes place [30]. Figure 1 represents the three lakes with interconnecting passage with each other. We contemplate every lake to be a big compartment and interconnecting passage as pipes between the compartments. In the figure, each arrow pointed out the flow direction of the pollutant through the channels and represents the rate of the pollutant that enters the lake per unit time. The volume of water in each lake remains constants, and amount of the pollutant is denoted by , constantly distributed in each lake at any time where . Here, the concentration of the contaminant in each lake is represented by . We consider the constant flow rate is and the motion of the flow is in the direction from lake to lake , and there is no reverse flow. The flux rate is symbolized by and can be defined as .

Figure 1 

Interconnecting channels of the system of three lakes [14].

Now, the rate of change of in every lake is the output rate in every lake. For the volume of every lake to stay steady, the stream rate into every lake must adjust the progression of the lake. So, we acquire the accompanying conditions between (1), (2), and (3):

The following system of differential equations yielded as the lake pollution model:

With initial conditions,

The parametric values are

3. Analysis of the Method

3.1. New Iterative Method (NIM)

In this section, we have discussed the new iterative method (NIM) [31, 32] as follows. Let us consider the general equation:where is a nonlinear operator from a Banach space , and is a known function.

Let be a solution for equation (7) having the series form.

The nonlinear operator can be written as

Then, series solution becomes

We define the consecutive relation:

Therefore,

3.2. Stability Analysis

Let be Banach spaces and a map. denotes the set of all linear maps from to . is also a Banach space.

Definition 1 (see [33, 34]). is said to be Fréchet differentiable at if there exists a continuous linear map such thatwhere is called the Fréchet derivative of at and is also denoted by Its value at is denoted by Note that is a linear map from to

Definition 2 (see [33, 34]). is said to be twice differentiable if the map is Fréchet differentiable. The second derivative of is denoted by and is a linear map from to that is isomorphic to

Theorem 3 (see [33, 34]). The map is symmetric, that where

In this manner, is inductively defined, and is multilinear and symmetric map.

Theorem 4 (see [33, 34]). [Taylor’s theorem]). Suppose that where is an open subset of containing the line segment from to , and thenwhere is such that

Since is symmetric, we denote by.

Using Theorem 4,

In general,

In view of (9)-(15) and (18)-(21),

Equation (22) is Taylor series expansion of around . Thus, NIM is equivalent to Taylor series expansion around .

3.3. Error Bound

The ^th degree NIM polynomial at is

Since the NIM approximation becomes more accurate as more terms are included, the polynomial must be more accurate than .

Since the difference between and is just that last term, the error of can be no larger than that term. In other words, the error is

Since and are constant in this formula, terms depending only on those constants and are unaffected by the max operator and can be pulled outside:

The largest value obtainable by could not possibly exceed the maximum value of that derivative between and . Call the value that provides that maximum value , and the error becomes

3.4. Convergence Analysis

Theorem 5. Let and be the elements in a Banach Space , and is nonlinear contraction from such that and . Then, according to the principle of Banach fixed point theorem [35], we can prove that

In view of (11)-(14), we have

According to the Banach contraction principle, these are the necessary and sufficient conditions for convergence of the solutions of the NIM procedure. Therefore, the series is absolutely and uniformly convergent to the solution of equation (7).

4. Application of NIM

Here, we monitor the pollutants for three different types of input models. According to the values of , these three models are impulse input, step input, and sinusoidal input [14, 30];

4.1. Impulse Input

This model is considered when there is a spike in concentration of dumped contaminant. Considering the value of is 100 units, the system becomes

Integrating the system of equations from 0 to and using the initial conditions (33), we get

We let the nonlinear terms as

In view of (11)–(14), we obtain first few components of the new iterative solution for the considered problem as follows:and so on. In the same manner, the successive approximations can be obtained.