It is well known that the human health is adversely affected by toxic air pollutants such as sulfur dioxide, nitrous oxide etc. present in the atmosphere. The removal of such pollutants from the atmosphere is, therefore, very much desirable. In this paper, a nonlinear mathematical model is proposed to study the population density dependent industrial emission of toxic air pollutants in the atmosphere and their removal by spraying liquid (water droplets) and particulate matter. In the modeling process, five variables are considered, namely; the cumulative concentration of toxic air pollutants, the density of human population affected by the toxic pollutants, the density of industrialization which is population density dependent, the number density of liquid droplets sprayed in the environment and the density of particulate matter sprayed in the environment. It is assumed that the emissions of toxic air pollutants are linearly related to the density of industrialization, the growth rate of which is directly proportional to the density of human population. It is also assumed that the growth rate of externally sprayed species in the environment is directly proportional to the concentration of toxic air pollutants in the environment. The model is analyzed using stability theory of nonlinear differential equations and numerical simulations. The model analysis shows that as the rate of spray of external species in the environment increases, the cumulative concentration of toxic air pollutants decreases. It is also found that as the rate of removal of toxic pollutants increases, the cumulative concentration of toxic air pollutants in the environment decreases. The effect of toxic air pollutants is observed to decrease the density of human population. The numerical simulation confirms analytical results.
Various types of toxicants such as 
,
, emitted from human population density dependent sources like manmade thermal power plants etc., affect human health and therefore it is very desirable to remove these toxic pollutants from the atmosphere. It is noted that nitric oxide (
) converts to nitrogen oxide (
) which in turn reacts with available moisture in the atmosphere to form nitric acid. Similarly sulfur dioxide converts into sulfuric acid. It has been shown that resource and hence population dependent on them may lead to extinction as a result of increased industrialization 1. The most important techniques by which toxic air pollutants can be removed from the atmosphere are; the precipitation scavenging in which toxic air pollutants are precipitated by the use of liquid droplets or by using particulate matter such as calcium oxide 
In precipitation scavenging, pollutants are absorbed / trapped in liquid droplets and as such these pollutants are precipitated on earth surface whereas using calcium oxide 
as particulate matter with 
results in forming calcium carbonate, thus removing pollutants from the atmosphere as per reaction below,
 |
Since visibility is increased after rain, the same phenomenon is used artificially to remove pollutants from the atmosphere. Several experimental investigations have been made to study the removal of pollutants by the process of precipitation 2, 3, 4, 5, 16. It can be seen that after rain the visibility always increases and the pollutants are removed from the atmosphere resulting in the enhanced visibility. In some studies around the cities of Kanpur, Varanasi, Pune in India 3, 4, 5 and Sheffield in United Kingdom 2 appreciable decline in the concentration of pollutants after rain is observed.
Many researchers have developed mathematical models and analyzed them to understand the scavenging of pollutants by precipitation 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. A theoretical framework for scavenging of gases in the atmosphere using rain was developed 10. A mathematical model to calculate the redistribution and washout of sulfur dioxide by raindrop spectra characteristic of drizzle and heavy rain was also presented 17. A six dimensional mathematical model has been proposed to study the effect of the density of cloud droplets on the removal of pollutants, gaseous as well as particulate, from the atmosphere 18. Some investigations have also been made to study the phenomenon of removal of gaseous pollutants and particulate matters by precipitation scavenging using nonlinear mathematical models 19, 20, 21.
Thus, in order to reduce the concentration of gaseous pollutants, particulate matters and dust particles which affect our environment considerably in various ways, using liquid droplets and particulate matters can be very significant removal mechanism to keep the environment clean.
From the above, it is observed that no study has been made to remove the pollutants from the atmosphere by using both the liquid droplets and particulate matter calcium oxide (
) associated with some human activity from the atmosphere. Therefore, in this paper, we propose and analyze a nonlinear mathematical model to study the removal of toxic air pollutants from the atmosphere using above concepts.
Consider that in a human habitat toxic air pollutants are emitted by human population dependent industrial sources which affect the human population. Let 
be the cumulative concentration of toxic air pollutants, 
be the density of human population governed by a logistic model, the growth rate of which decreases due to toxic air pollutants. Let 
be the density of industrialization, the growth rate of which is directly proportional to the density of human population. It is further assumed that the growth rate of number densities of liquid droplets 
and particulate matter 
are proportional to the concentration of toxic air pollutants present in the environment. The effect of these externally sprayed species is to reduce the concentration of toxic air pollutants in the atmosphere.
Keeping these considerations in view, the model is proposed as follows.
The emission of toxic air pollutants 
is governed by the equation (1), wherein 
is the constant emission rate of pollutants in the atmosphere. The growth rate of toxic air pollutants is enhanced by population density dependent industrialization and therefore it is assumed that 
is the growth rate coefficient of toxic air pollutants in the atmosphere due to increase in industrialization. The constant 
is natural depletion rate coefficient of toxic air pollutants in the atmosphere. Some of the pollutants are removed from the atmosphere by the use of liquid droplets 
in atmosphere, 
being the depletion rate coefficient of pollutants due to externally introduced liquid species. Particulate matters are also used to remove the toxic air pollutants from the atmosphere and therefore the removal of toxic air pollutants is taken in the direct proportion of number density of external species as well as the concentration of these pollutants as in equation (1), 
being the depletion rate coefficients of toxic air pollutants due to particulate matters. The constant 
is the depletion rate coefficient of toxic air pollutants due to self awareness of human beings about the adverse effects of these pollutants. In equation (2), 
is the population density, the growth rate of which is assumed to follow logistic equation. Let 
be the intrinsic growth rate of 
with carrying capacity
. Since toxic air pollutants emitted from industries have adverse effect on human population and therefore it is reasonable to assume 
as depletion rate coefficient of population due to toxic air pollutants. As population increases, demand and supply equations get changed. Thus, in order to fulfill demand and supply equations, more industries are to be established and therefore the growth of industrialization is assumed to be proportional to the density of human population as shown in equation (3). Therefore, in this equation, the constant 
is assumed to be growth rate coefficient of industrialization and 
is its natural depletion rate coefficient. In equation (4), 
is the growth rate coefficient of number density of liquid droplets used to reduce the concentration of toxic air pollutants in the atmosphere. Since some of the liquid droplets decays themselves and hence 
is taken as natural depletion rate coefficient of liquid droplets. The constant 
is the depletion coefficient of liquid droplets due to toxic air pollutants. In equation (5), 
is the growth rate coefficient of particulate matters. Since some of the particulate matters are depleted itself and hence it is assumed that 
is the natural depletion rate coefficient of particulate mattes. The constant 
is the depletion coefficient of particulate matters due to toxic air pollutants.
Thus, in view of the above, the system is assumed to be governed by the following nonlinear ordinary differential equations,
 | (1) |
 | (2) |
 | (3) |
 | (4) |
 | (5) |
, 
, 
, 
, 
Remark 1:
It is noted from equations (2), (4) and (5) that 
, 
and 
are growth rates of population, liquid droplets and particulate matters respectively and hence must be positive for all time
.
The region of attraction of the model system (1) – (5) is given as follows,
 |
 |
where 
Proof:
From equation (2), we have
 |
implying that 
From equation (3), we note that 
From equation (1) we have,
 |
 |
which gives, 
(say)
where 
From equation (4), we have
 |
 |
implying 
In a similar manner, we can show from equation (5) that
.
The model system has following non-negative equilibria,
1. 
Existence of 
is obvious.
2. 
To show the existence of 
,
Let
 | (6) |
From equation (6), we note that,
(i) 
(ii) 
(iii) 
This implies that there exists a unique positive root (say
) of 
in
. Using this value we can find the value of other variables
and 
,
3. 
Equilibrium values of different variables in 
are given by the following algebraic equations
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 | (11) |
Using equations (8) – (11) in equation (7), we get
 | (12) |
From equation (12), we note that,
(i) 
(ii) 
in view of remark 1.
Since we have 
and 
this implies that equilibrium level can be attained and thus equilibrium 
exists. We also note from equation (12) that
(iii)
, which implies that 
is unique.
Thus, 
has a unique positive root (say
) in 
. Using this value we can find the value of other variables from equations (8) – (11).
with 
Differentiating equation (12) with respect to 
we note that
 |
provided
 | (13) |
This implies that concentration of toxic air pollutants in the atmosphere increases as the growth rate of toxic air pollutants due to population density dependent industrialization increases.
with 
From equation (8) we get
 |
Hence using the relation 
and noting that 
we get
.
This implies that as the growth rate of toxic air pollutants increases, the growth of population density decreases.
with 
Differentiating equation (12) with respect to 
, then in view of (13), we get
 |
This implies that the concentration of toxic air pollutants decreases with increase in the rate of increase of externally introduced liquid droplets.
with 
Differentiating equation (12) with respect to 
, then in view of (13), we get,
 |
which also implies that the concentration of toxic air pollutants decreases with increase in the rate of increase of particulate matters in the atmosphere.
2.5. Stability AnalysisIn order to establish the local stability behavior of equilibrium, we compute the Jacobian matrix 
for the model system (1) – (5)
 |
From the above matrix we note that,
(i) Equilibrium 
is unstable as one eigenvalue 
of the Jacobian matrix 
corresponding to 
is positive.
(ii) Equilibrium 
is unstable as one eigenvalue 
of the Jacobian matrix 
corresponding to 
is positive.
In the following, we state the local and nonlinear stability theorems for the equilibrium 
.
The equilibrium 
is locally asymptotically stable provided the following conditions are satisfied,
 | (14) |
 | (15) |
(See Appendix A for proof).
The equilibrium 
is nonlinearly stable inside the region of attraction 
provided the following conditions are satisfied,
 | (16) |
 | (17) |
(See Appendix B for proof).
It is noted from the above theorems that if population density dependent growth rate coefficient of industrialization 
growth rate coefficient of industrialization (
) and depletion rate coefficient (
) of human population growth tend to zero, the local and nonlinear stability conditions, stated in Theorems 1 and 2, will be satisfied automatically. This implies that 
and 
have destabilizing effect on the model system.
In this section, we perform some numerical simulations to study the local and nonlinear stability behavior of equilibria and feasibility of the model system (1)-(5) numerically using MAPLE 
by choosing the following set of parameter values, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
The equilibrium values of different variables in
corresponding to above data are given as 
,
, 
,
,
The eigenvalues of the Jacobean matrix corresponding to 
for the model system 
are, 
Since all eigenvalues are negative or having negative real part and hence the interior equilibrium 
is locally asymptotically stable. The nonlinear stability behavior of 
is shown in the Figure 1. This figure depicts that the solution trajectories that start at any point within the region of attraction approach to equilibrium
. In Figure 2, the variation of concentration of toxic air pollutants (
) with time 
for different values of 
the growth rate coefficient of toxic air pollutants due to population density dependent industrialization, is plotted. It is observed from the figure that as 
increases, the concentration of toxic air pollutants increases in the atmosphere. In Figure 3, the variation of concentration of toxic air pollutants (
) with time 
for different values of 
the depletion rate coefficient of pollutants due to liquid droplets is plotted. It is seen that as 
increases, the concentration of toxic air pollutants 
decreases. In Figure 4, the variation of toxic pollutants 
with time 
for different values of 
the depletion rate coefficient of pollutants due to particulate matters is plotted. From this figure, it is noted that as 
increases, the concentration of toxic pollutants decreases in the atmosphere. Thus, the level of toxic air pollutants increases with increase in the industrialization level but it decreases when liquid droplets or particulate matters are introduced in the atmosphere. As the depletion rate coefficient 
of toxic pollutants due to human activity increases, the concentration of toxic pollutants 
decreases in the atmosphere, (Figure 5). In Figure 6, the variation of human population 
with time 
for different values of 
, the depletion rate coefficient of population due to toxic air pollutants is shown and it is observed that as 
increases, the growth of population 
decreases. This implies that the abundance of toxic air pollutants in the atmosphere adversely affects the human population. Figure 7 shows the variation of toxic pollutants concentration 
with time 
for different values of 
the growth rate coefficient of liquid droplets in the atmosphere. It is observed that as the rate of introduction of liquid droplets increases, the concentration of toxic air pollutants decreases in the atmosphere. Similar phenomenon of decrease of toxic air pollutants is observed when particulate matters are introduced in the atmosphere with different rates (Figure 8). In Figure 9, the variation of toxic pollutants 
with time 
for different values of 
is plotted. It is found that the rate of industrialization 
increases due to human activities, the concentration of toxic air pollutants increases in the atmosphere.
We have also plotted stability condition with respect to crucial parameters to study the effect of these variables on stability condition. Figure 10 and Figure 11 show the variation of nonlinear stability condition 
and 
with respect to parameters 
and 
respectively. It is apparent from Figure 10 that 
remains positive for 
and negative for 
. This implies that the stability condition is satisfied for 
and for higher values of 
it will not be satisfied. Hence, 
has destabilizing effect on the model system. Likewise, from Figure 11 we infer that 
has destabilizing effect on the model system.
Plane
with time 
for different values of 
with time 
for different values of 
with time 
for different values of 
with time 
for different values of 
with time 
for different values of 
with time 
different values of
with time 
for different values of 
with time 
for different values of 
with 
In this paper, a nonlinear mathematical model has been proposed to study the population density dependent industrial emissions of toxic air pollutants in the atmosphere and their removal by liquid droplets and particulate matters. In the modeling process, the following variables have been considered,
(1) The cumulative concentration of toxic air pollutants which is discharged by population density dependent industrialization in the atmosphere.
(2) The density of human population, the growth rate of which decreases due to cumulative density of toxic air pollutants.
(3) The density of industrialization, the growth rate of which is directly proportional to the density of human population.
(4) The density of liquid droplets sprayed in the atmosphere, the growth rate of which is assumed to be proportional to the cumulative concentration of toxic air pollutants.
(5) The concentration of particulate matter which is assumed to be proportional to the cumulative concentration of toxic air pollutants. It is assumed that the droplets and the particulate phases, formed in the atmosphere due to interaction of toxic air pollutants with spraying liquid droplets and particulate matter, remove the toxic air pollutants in the atmosphere in the same proportion by which their concentration/density get increased.
The model has been analyzed by using the stability theory of differential equations. The existence of interior equilibrium is established and its local as well as nonlinear stability has been studied. It has been shown further that due to population density depended emissions (mainly industries), the cumulative concentration of toxic air pollutants in the atmosphere increases. It has also been shown that cumulative concentration of toxic air pollutants decreases considerably by spraying liquid droplets and particulate matters. The model has also been analyzed using numerical simulation which confirms the above analytical results.
| [1] | Dubey, B., et.al. Modeling the depletion of forestry resources by population and population pressure augmented industrialization, Applied Mathematical Modeling, 33, 3002-3014, 2009. | ||
| In article | View Article | ||
| [2] | Davies, T.D., Precipitation scavenging of Sulfur dioxide in an industrial area, Atmospheric Environment, 10, 879-890, 1976. | ||
| In article | View Article | ||
| [3] | Sharma, V.P., Arora, H.C., Gupta, R.K., Atmospheric pollution studies at Kanpur- suspended particulate matter, Atmospheric Environment, 17, 1307-1314, 1983. | ||
| In article | View Article | ||
| [4] | Pandey, J., Agarwal, M., Khanam, N., Narayan, D., Rao, D.N., Air pollutant concentration in Varanasi, India, Atmospheric Environment, 26 (B), 91-98, 1992. | ||
| In article | |||
| [5] | Pillai, A., Naik, M.S., Momin, G., Rao, P., Ali, K., Rodhe, H., Granet, L., Studies of wet deposition and dust fall at Pune, India, Water, Air, & Soil Pollution, 130 (1-4), 475-480, 2001. | ||
| In article | View Article | ||
| [6] | Chang, T.Y., Rain and snow scavenging of HNO3 vapour in the atmosphere, Atmospheric Environment, 18, 191-197, 1984. | ||
| In article | View Article | ||
| [7] | Chen, W.H., Atmospheric ammonia scavenging mechanisms round a liquid droplet in convective flow, Atmospheric Environment, 38 (8), 1107-1116, 2004. | ||
| In article | View Article | ||
| [8] | Fisher, B.E.A., The transport and removal of sulfur dioxide in a rain system, Atmospheric Environment, 16, 775-783, 1982. | ||
| In article | View Article | ||
| [9] | Goncalves, F.L.T., Ramos, A.M, Freitas, S, Silva Dias, M.A., Massambani, O., In cloud and below cloud numerical simulation of scavenging processes at Serra Do Mar Region S E Brazil, Atmospheric Environment, 36, 5245-5255, 2002. | ||
| In article | View Article | ||
| [10] | Hales, J.M., Fundamentals of the theory of gas scavenging by rain, Atmospheric Environment, 6, 635- 650, 1972. | ||
| In article | View Article | ||
| [11] | Hales, J.M., Wet removal of sulphur compounds from the atmosphere, Atmospheric Environment, 12, 389-400, 1978. | ||
| In article | View Article | ||
| [12] | Koelliker, Y., Totten, L.A., Gigliotti, C.L., Offenberg, H, Reinfelder, J.R., Zhuang, Y, Eisenreich, S.J., Atmospheric wet deposition of total phosphorus in New Jersey, Water, Air, & Soil Pollution, 154 (1-4), 139-150, 2004. | ||
| In article | View Article | ||
| [13] | Kumar, S., An Eulerian model for scavenging of pollutants by rain drops, Atmospheric Environment, 19, 769-778, 1985. | ||
| In article | View Article | ||
| [14] | Levine, S.Z., Schwartz, S.E., In-cloud and below-cloud scavenging of HNO3vapors, Atmospheric Environment, 16, 1725-1734, 1982. | ||
| In article | View Article | ||
| [15] | Pandis, S.N., Seinfeld, J.H., On the interaction between equilibration processes and wet or dry deposition, Atmospheric Environment, 24(A), 2313-2327, 1990. | ||
| In article | |||
| [16] | Rao, P.S.P., Khemani, L.T., Momin, G.A., Safari, P.D., Pillai, A.G., Measurements on wet and dry deposition at an urban location in India, Atmospheric Environment, 26(B), 73-78, 1992. | ||
| In article | |||
| [17] | Bariie L.A., An improved model of reversible SO2 washout by rain, Atmospheric Environment, 12, 407- 412, 1978. | ||
| In article | View Article | ||
| [18] | Sundar, S., Sharma, R.K., Naresh, R., Modeling the role of cloud density on the removal of gaseous pollutants and particulate matters from the atmosphere, Application and Applied Mathematics: An International Journal, 8 (2), 416-435, 2013. | ||
| In article | |||
| [19] | Shukla, J.B., Misra, A.K., Sundar, S, Naresh, R., Effect of rain on removal of a gaseous pollutant and two different particulate matters from the atmosphere of a city, Mathematical and Computer Modeling, 48, 832- 844, 2008. | ||
| In article | View Article | ||
| [20] | Shukla, J.B., Sundar, S., Misra, A.K., Naresh, R., Modeling the removal of gaseous pollutants and particulate matters from the atmosphere of a city by rain: Effect of cloud density, Environmental Model Assessment, 13, 255-263, 2008. | ||
| In article | View Article | ||
| [21] | Naresh, R., Sundar, S., Mathematical modeling and analysis of the removal of gaseous pollutants by precipitation using general nonlinear interaction, International Journal of Applied Mathematics and Computation, 2(2), 45- 56, 2010. | ||
| In article | |||
To establish the local stability of 
let us consider the following positive definite function
 | (A1) |
where
,
, 
,
and 
are the small perturbations about 
described below,
 |
 |
 |
 |
 |
Differentiating equation (A1) with respect to 
we get
 | (A2) |
The linearized system of the model system (1) – (5) corresponding to 
is written as follows
 |
Using the above linearized system in equation (A2) and after simplification we have
 | (A3) |
Now, 
will be negative definite under the following conditions
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
 | (A8) |
After some algebraic manipulations and choosing
,
, 
and 
will be negative definite provided the condition (14) – (15) are satisfied showing that 
is Liaponuv function and hence the theorem.
Consider the following positive definite function about 
 | (B1) |
Differentiating (B1) with respect to 
we get
 |
Now 
will be negative definite under the following conditions
 | (B2) |
 | (B3) |
 | (B4) |
 | (B5) |
 | (B6) |
Maximizing left hand side and minimizing right hand side and taking 
, 
, 
,
, 
will be negative definite provided the condition (16) – (17) are satisfied inside the region of attraction 
showing that 
is Liapunov function hence the theorem.
Published with license by Science and Education Publishing, Copyright © 2019 Shyam Sundar, Niranjan Swaroop and Ram Naresh
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
https://creativecommons.org/licenses/by/4.0/
| [1] | Dubey, B., et.al. Modeling the depletion of forestry resources by population and population pressure augmented industrialization, Applied Mathematical Modeling, 33, 3002-3014, 2009. | ||
| In article | View Article | ||
| [2] | Davies, T.D., Precipitation scavenging of Sulfur dioxide in an industrial area, Atmospheric Environment, 10, 879-890, 1976. | ||
| In article | View Article | ||
| [3] | Sharma, V.P., Arora, H.C., Gupta, R.K., Atmospheric pollution studies at Kanpur- suspended particulate matter, Atmospheric Environment, 17, 1307-1314, 1983. | ||
| In article | View Article | ||
| [4] | Pandey, J., Agarwal, M., Khanam, N., Narayan, D., Rao, D.N., Air pollutant concentration in Varanasi, India, Atmospheric Environment, 26 (B), 91-98, 1992. | ||
| In article | |||
| [5] | Pillai, A., Naik, M.S., Momin, G., Rao, P., Ali, K., Rodhe, H., Granet, L., Studies of wet deposition and dust fall at Pune, India, Water, Air, & Soil Pollution, 130 (1-4), 475-480, 2001. | ||
| In article | View Article | ||
| [6] | Chang, T.Y., Rain and snow scavenging of HNO3 vapour in the atmosphere, Atmospheric Environment, 18, 191-197, 1984. | ||
| In article | View Article | ||
| [7] | Chen, W.H., Atmospheric ammonia scavenging mechanisms round a liquid droplet in convective flow, Atmospheric Environment, 38 (8), 1107-1116, 2004. | ||
| In article | View Article | ||
| [8] | Fisher, B.E.A., The transport and removal of sulfur dioxide in a rain system, Atmospheric Environment, 16, 775-783, 1982. | ||
| In article | View Article | ||
| [9] | Goncalves, F.L.T., Ramos, A.M, Freitas, S, Silva Dias, M.A., Massambani, O., In cloud and below cloud numerical simulation of scavenging processes at Serra Do Mar Region S E Brazil, Atmospheric Environment, 36, 5245-5255, 2002. | ||
| In article | View Article | ||
| [10] | Hales, J.M., Fundamentals of the theory of gas scavenging by rain, Atmospheric Environment, 6, 635- 650, 1972. | ||
| In article | View Article | ||
| [11] | Hales, J.M., Wet removal of sulphur compounds from the atmosphere, Atmospheric Environment, 12, 389-400, 1978. | ||
| In article | View Article | ||
| [12] | Koelliker, Y., Totten, L.A., Gigliotti, C.L., Offenberg, H, Reinfelder, J.R., Zhuang, Y, Eisenreich, S.J., Atmospheric wet deposition of total phosphorus in New Jersey, Water, Air, & Soil Pollution, 154 (1-4), 139-150, 2004. | ||
| In article | View Article | ||
| [13] | Kumar, S., An Eulerian model for scavenging of pollutants by rain drops, Atmospheric Environment, 19, 769-778, 1985. | ||
| In article | View Article | ||
| [14] | Levine, S.Z., Schwartz, S.E., In-cloud and below-cloud scavenging of HNO3vapors, Atmospheric Environment, 16, 1725-1734, 1982. | ||
| In article | View Article | ||
| [15] | Pandis, S.N., Seinfeld, J.H., On the interaction between equilibration processes and wet or dry deposition, Atmospheric Environment, 24(A), 2313-2327, 1990. | ||
| In article | |||
| [16] | Rao, P.S.P., Khemani, L.T., Momin, G.A., Safari, P.D., Pillai, A.G., Measurements on wet and dry deposition at an urban location in India, Atmospheric Environment, 26(B), 73-78, 1992. | ||
| In article | |||
| [17] | Bariie L.A., An improved model of reversible SO2 washout by rain, Atmospheric Environment, 12, 407- 412, 1978. | ||
| In article | View Article | ||
| [18] | Sundar, S., Sharma, R.K., Naresh, R., Modeling the role of cloud density on the removal of gaseous pollutants and particulate matters from the atmosphere, Application and Applied Mathematics: An International Journal, 8 (2), 416-435, 2013. | ||
| In article | |||
| [19] | Shukla, J.B., Misra, A.K., Sundar, S, Naresh, R., Effect of rain on removal of a gaseous pollutant and two different particulate matters from the atmosphere of a city, Mathematical and Computer Modeling, 48, 832- 844, 2008. | ||
| In article | View Article | ||
| [20] | Shukla, J.B., Sundar, S., Misra, A.K., Naresh, R., Modeling the removal of gaseous pollutants and particulate matters from the atmosphere of a city by rain: Effect of cloud density, Environmental Model Assessment, 13, 255-263, 2008. | ||
| In article | View Article | ||
| [21] | Naresh, R., Sundar, S., Mathematical modeling and analysis of the removal of gaseous pollutants by precipitation using general nonlinear interaction, International Journal of Applied Mathematics and Computation, 2(2), 45- 56, 2010. | ||
| In article | |||
with 
