Submit manuscript...
MOJ
eISSN: 2576-4519

Applied Bionics and Biomechanics

Research Article Volume 1 Issue 4

Numerical modeling and simulation to monitor dispersion pressure on the deposition of halobacterium in slight heterogeneous semi confined bed

Eluozo SN,1 Amagbo LG,2 Afiibor BB3

1Department of Civil Engineering, Gregory University Uturu (GUU), Nigeria
2Department of Chem-Petrochemical Engineering, Rivers State University of Science and Technology Port Harcourt, Nigeria
3Department of Statistics, Nigeria

Correspondence: Eluozo SN, Department of Civil Engineering, Gregory University Uturu (GUU) Abia State of Nigeria

Received: June 24, 2017 | Published: December 7, 2017

Citation: Eluozo SN, Amagbo LG, Afiibor B. Numerical modeling and simulation to monitor dispersion pressure on the deposition of halobacterium in slight heterogeneous semi confined bed. MOJ App Bio Biomech. 2017;1(4):153–159. DOI: 10.15406/mojabb.2017.01.00024

Download PDF

Abstract

This paper monitor the rate of Halobacterium deposition in semi confined bed, the study observed the transport of this contaminant to be influenced by high deposition of porosity in the study area. The study has monitored the behavior of Halobacterium expressing the pressure from high degree of porosity in the deltaic formations. Application of numerical modeling and simulation were applied and it generated different concentration that ranged from 0.0334-0.8854, 0.111-0.67824, 4.0200-0.0012, 0.8720-7.400, and0.1256-0.8793few locations that observed higher concentration are definitely influenced by the deposition of high porosity, the study applying numerical simulation has monitored the contaminants more discrete generated through exact deposition of change in concentration with respect to depth, the predictive results were subjected to model validation with experimental values and both parameters developed faviourable fits.

Keywords: numerical modeling, dispersion, halobacterium, heterogeneous, semi confined bed

Abbreviations

EPA, environmental protection agency; LMERS, lake michigan ecological research station; UTI, urinary tract infections

Introduction

Groundwater is considered to be of excellent quality because of the soil barrier providing effective isolation of this high quality source water from surface pollutants. This is true for most groundwater resources although we know that many aquifers all over the world are polluted and/or is being polluted.1-4 Habitats containing only a single kind of microorganism are found only in the laboratory. Natural habitats contain many kinds of organisms which interact in complex ways. The great reservoir of bacteria in nature is the soil, which contains both the largest population and the greatest variety of species. Most bacteria that are found in surface waters are derived from the soil. However, the quality of subsurface waters may be impacted both by naturally occurring processes as well as by actions directly attributable to human activities. The number and variety of the microorganisms in natural waters vary greatly in different places and under different conditions. Bacteria are washed into the water from the air, the soil and from almost every conceivable object. Significant numbers of bacteria can be removing through media even when the percentage retained is very high. The faeces of animals contain vast numbers of bacteria and many enter natural water systems. The sizes of openings in subsurface material can be assumed to be variable and are generally not measured, but porosity and permeability measurements on aquifer sediments indicate that adequate spaces for bacteria exist in many sediment types, even in some rather dense porous rocks.5-8 The interstices of the shallow aquifer sediments can easily accommodate bacteria and probably protozoa and fungi as well. Larger organisms will be excluded from most subsurface formations, except for gravelly and cavernous aquifers2,3,6 Microbiological pollution derived mostly from human and animal activities such as unsewered settlements; on-site sanitation; cemeteries; waste disposal; waste disposal; feedlots; etc. Microorganisms certainly will be the dominant forms of life and, in most cases; they will be the only forms of life present in aquifers. However, with very few exceptions the only waterborne microbial pathogens of man are essentially human bacteria, viruses and protozoa, and in considering the safety of drinking water from the point of view of infectious diseases one can almost completely ignore any source of infectious agents except human excreta. In relation to microbial pollution of groundwater it is therefore only necessary to ensure that at the point of extraction no contamination with human excreta occurs1,3,5,8 bacteria are the bacteria most commonly associated with well water.

The United States environmental protection agency (EPA) standard for drinking water is a total coliforms count of zero. Coliforms bacteria are a large group of various rod-shaped species and strains of bacteria. The group includes bacteria that occur naturally in the intestines of warm-blooded animals (fecal coliforms) and no fecal coliforms. Non-fecal coliforms bacteria are very common and are found virtually everywhere on soil particles, insects, plants, animals, walls and furniture in homes and on your skin and clothes. Fecal coliforms can include disease causing (pathogen species) and non-disease causing species. Over 200 types of non-disease causing bacteria have been found in human digestive tracts. Most arrive on the food and drink we consume. Many yogurt cultures include coliforms bacteria. Lactobacillus acidophilus is the most common bacteria strain used in commercial yogurts and some studies show it creates an acidic environment that inhibits harmful bacteria in the digestive tract. Escherichia coli (E. coli), often listed in water quality analyses, is one species of fecal coliforms bacteria. A single E. coli is 2 microns long and about 0.5 microns in diameter. There are hundreds strains of E. coli bacteria that differ only in the type of toxin or enzyme that they produce. Despite the fact that they originate in the digestive system of a warm-blooded creature, most E. coli strains are not harmful to humans. E. coli can be easily cultured in a laboratory and therefore, they are a good indicator species for bacterial contamination in water tests. Its presence in a water sample indicates that sewage material may be present and that if sewage is present, more harmful disease-causing organisms may also be present, and for example Vibrio cholerae that causes cholera.9-12 Researchers today have discovered that E. coli may not always be an effective indicator of water quality. While it is true that E. coli is found in the intestines of warm blooded animals, scientists have recently revealed that E. coli can also persist and perhaps thrive in many other natural environments.13-15 Take soil for example. Research conducted at the USGS Lake Michigan Ecological Research Station (USGS LMERS) has shown that temperate forest soils in the Indiana Dunes harbor E. coli throughout the entire year (winter included)! The sediments and soil in the watershed of Dunes Creek (a Lake Michigan tributary) contain E. coli, and the persistently high E. coli counts in Dunes Creek itself may be due to rainfall and stream flow eroding the sediment-borne bacteria into the water. In these cases there was no significant human fecal input, yet the E. coli was there.16,17 What about sand? E. coli is found in beach sand as well! Bacteria harbored in sand may even persist longer than in water because the bacteria adhere to sediment particles, unlike bacteria that are free in the water.12,15 Research has shown that E. coli counts were higher in the near shore sand and submerged sand than in the beach water. Additionally, the E. coli counts were typically several orders of magnitude higher in the sand than in the water. The geometric mean of E. coli counted in the foreshore sand in a study on 63rd street beach in Chicago was 4,000 CFU’s/ 100 ml of water, as compared to only 43 CFU’s /100 ml water in the water.12,15 How ironic that by closing the swimming waters that may have 240 colonies/100 ml of water, we may actually be increasing the contact people have with even higher concentrations of E. coli (sometimes as high as 11,000 CFU/100 ml of water) in shallow water and sand.12,14,15 Water samples for bacteria testing are collected and cultured, and then must incubate for 18 hours before the colony growth is visible. Therefore, after a water sample is collected, results are not available until the next day.” By that time, the bacteria levels in our beach waters may have changed significantly. In fact, most studies show little or no correlation between indicator levels from the sampling day to the next day when the results are actually used by the beach managers to make decisions about beach closings.16 Urinary tract infections (UTI) are the most common nosocomial infections which accounts for 40% of hospital acquired infections.10-12 Escherichia coli are the most frequently found bacteria in both community and hospital acquired UTIs.14-16 In recent years antimicrobial resistance has emerged explosively in many diverse bacterial types largely as a consequence of unrestrained antimicrobial use in medicine.7,8,10 This affects the management of UTI by increasing prevalence of multidrug resistant strains of E. coli .6,10 Therefore developing methods for accurate identification of multidrug resistant strains of E .coli is mandatory.6,9,10,13 In recent years several methods have been diffusion agar is a traditional and routine method of antimicrobial sensitivity testing. E-test provides a rapid and convenient means for determining minimal inhibitory concentration (MIC) for a variety of antimicrobial agents. Studies have shown that E-test shows good agreement with reference “agar dilution” susceptibility testing methods.8,9 MIC determining methods like E-test, although provide quantitative measurement of antimicrobial sensitivity11,16 because of their cost and limited availability in developing countries, their application is not as frequent as disk diffusion method.14-16 Although, previous reports have compared E-test with disk diffusion in determining antimicrobial susceptibility, differences in their capabilities for selection of multidrug resistant strains of E. coli in UTI has not been fully encountered. In this study we have compared E-test and disk diffusion results in finding out multidrug resistant strains of E. coli in urinary tract infections.6,7,17

Governing equation

The Implicit Scheme Numerical Solution

     C t = Q A C x +D 2 C x 2 + q L IN A C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyRaamiDaaaacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaWGrbaakeaajugibiaadgeaaaqcfa 4aaSaaaOqaaKqzGeGaeyOaIyRaam4qaaGcbaqcLbsacqGHciITcaWG 4baaaiabgUcaRiaadseajuaGdaWcaaGcbaqcLbsacqGHciITjuaGda ahaaWcbeqaaKqzadGaaGOmaaaajugibiaadoeaaOqaaKqzGeGaeyOa IyRaamiEaSWaaWbaaeqabaqcLbmacaaIYaaaaaaajugibiabgUcaRK qbaoaalaaakeaajugibiaadghacaWGmbqcfa4aaSbaaSqaaKqzadGa amysaiaad6eaaSqabaaakeaajugibiaadgeaaaGaam4qaaaa@5E9C@ (1)

But Q A = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba qcLbsacaWGrbaajuaGbaqcLbsacaWGbbaaaiabg2da9aaa@3AD7@  Velocity, v in meter per second (m/s).

Thus equation (1) becomes:

C t =v C x +D 2 C x 2 + q L IN A C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaam4qaaqaaiabgkGi2kaadshaaaGaeyypa0JaamODamaa laaabaGaeyOaIyRaam4qaaqaaiabgkGi2kaadIhaaaGaey4kaSIaam iramaalaaabaGaeyOaIy7aaWbaaeqabaqcLbmacaaIYaaaaKqbakaa doeaaeaacqGHciITcaWG4bWaaWbaaeqabaqcLbmacaaIYaaaaaaaju aGcqGHRaWkdaWcaaqaaiaadghacaWGmbWaaSbaaeaacaWGjbGaamOt aaqabaaabaGaamyqaaaacaWGdbaaaa@5341@ (2)

Converting the PDE to its algebraic equivalent equation by applying the finite different approximation technique for the implicit scheme, we obtain as follows.

C t = C i j+1 C i j Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaam4qaaqaaiabgkGi2kaadshaaaGaeyypa0ZaaSaaaeaa caWGdbWcdaqhaaqcfayaaKqzadGaamyAaaqcfayaaKqzadGaamOAai abgUcaRiaaigdaaaqcfaOaeyOeI0Iaam4qaSWaa0baaKqbagaajugW aiaadMgaaKqbagaajugWaiaadQgaaaaajuaGbaGaeuiLdqKaamiDaa aaaaa@4EC5@ (3)

C x = C i+1 j+1 C i1 j+1 2Δx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaam4qaaqaaiabgkGi2kaadIhaaaGaeyypa0ZaaSaaaeaa caWGdbWcdaqhaaqcfayaaKqzadGaamyAaiabgUcaRiaaigdaaKqbag aajugWaiaadQgacqGHRaWkcaaIXaaaaKqbakabgkHiTiaadoealmaa DaaajuaGbaqcLbmacaWGPbGaeyOeI0IaaGymaaqcfayaaKqzadGaam OAaiabgUcaRiaaigdaaaaajuaGbaGaaGOmaiabfs5aejaadIhaaaaa aa@546B@ (4)

2 C x 2 = C i+1 j+1 2 C i j+1 + C i1 j+1 Δ x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqabaqcLbmacaaIYaaaaKqbakaadoeaaeaacqGH ciITcaWG4bWcdaahaaqcfayabeaajugWaiaaikdaaaaaaKqbakabg2 da9maalaaabaGaam4qaSWaa0baaKqbagaajugWaiaadMgacqGHRaWk caaIXaaajuaGbaqcLbmacaWGQbGaey4kaSIaaGymaaaajuaGcqGHsi slcaaIYaGaam4qaSWaa0baaKqbagaajugWaiaadMgaaKqbagaajugW aiaadQgacqGHRaWkcaaIXaaaaKqbakabgUcaRiaadoealmaaDaaaju aGbaqcLbmacaWGPbGaeyOeI0IaaGymaaqcfayaaKqzadGaamOAaiab gUcaRiaaigdaaaaajuaGbaGaeuiLdqKaamiEaSWaaWbaaKqbagqaba qcLbmacaaIYaaaaaaaaaa@6634@ (5)

Substituting equation (3) through (5) into (2) gives:

C i j=1 C i j Δt =v[ C i+1 j+1 C i1 j+1 2Δx ]+D[ C i+1 j+1 2 C i j+1 + C i1 j+1 Δ x 2 ]+ q L IN A C i j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qaSWaa0baaKqbagaajugWaiaadMgaaKqbagaajugWaiaadQga cqGH9aqpcaaIXaaaaKqbakabgkHiTiaadoealmaaDaaajuaGbaqcLb macaWGPbaajuaGbaqcLbmacaWGQbaaaaqcfayaaiabfs5aejaadsha aaGaeyypa0JaamODamaadmaabaWaaSaaaeaacaWGdbWcdaqhaaqcfa yaaKqzadGaamyAaiabgUcaRiaaigdaaKqbagaajugWaiaadQgacqGH RaWkcaaIXaaaaKqbakabgkHiTiaadoealmaaDaaajuaGbaqcLbmaca WGPbGaeyOeI0IaaGymaaqcfayaaKqzadGaamOAaiabgUcaRiaaigda aaaajuaGbaGaaGOmaiabfs5aejaadIhaaaaacaGLBbGaayzxaaGaey 4kaSIaamiramaadmaabaWaaSaaaeaacaWGdbWcdaqhaaqcfayaaKqz adGaamyAaiabgUcaRiaaigdaaKqbagaajugWaiaadQgacqGHRaWkca aIXaaaaKqbakabgkHiTiaaikdacaWGdbWcdaqhaaqcfayaaKqzadGa amyAaaqcfayaaKqzadGaamOAaiabgUcaRiaaigdaaaqcfaOaey4kaS Iaam4qaSWaa0baaKqbagaajugWaiaadMgacqGHsislcaaIXaaajuaG baqcLbmacaWGQbGaey4kaSIaaGymaaaaaKqbagaacqqHuoarcaWG4b WcdaahaaqcfayabeaajugWaiaaikdaaaaaaaqcfaOaay5waiaaw2fa aiabgUcaRmaalaaabaGaamyCaiaadYeadaWgaaqaaiaadMeacaWGob aabeaaaeaacaWGbbaaaiaadoealmaaDaaajuaGbaqcLbmacaWGPbaa juaGbaqcLbmacaWGQbGaey4kaSIaaGymaaaaaaa@9B1E@

C i j+1 C i j = Δt 2Δx v[ C i+1 j+1 C i1 j+1 ]+ ΔtD Δ x 2 [ C i+1 j+1 2 C i j+1 + C i1 j+1 ]+ Δtq L IN A C l j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaWGPbaajuaGbaqcLbmacaWGQbGaey4kaSIa aGymaaaajuaGcqGHsislcaWGdbWcdaqhaaqcfayaaKqzadGaamyAaa qcfayaaKqzadGaamOAaaaajuaGcqGH9aqpdaWcaaqaaiabfs5aejaa dshaaeaacaaIYaGaeuiLdqKaamiEaaaacaWG2bWaamWaaeaacaWGdb WcdaqhaaqcfayaaKqzadGaamyAaiabgUcaRiaaigdaaKqbagaajugW aiaadQgacqGHRaWkcaaIXaaaaKqbakabgkHiTiaadoealmaaDaaaju aGbaqcLbmacaWGPbGaeyOeI0IaaGymaaqcfayaaKqzadGaamOAaiab gUcaRiaaigdaaaaajuaGcaGLBbGaayzxaaGaey4kaSYaaSaaaeaacq qHuoarcaWG0bGaamiraaqaaiabfs5aejaadIhalmaaCaaajuaGbeqa aKqzadGaaGOmaaaaaaqcfa4aamWaaeaacaWGdbWcdaqhaaqcfayaaK qzadGaamyAaiabgUcaRiaaigdaaKqbagaajugWaiaadQgacqGHRaWk caaIXaaaaKqbakabgkHiTiaaikdacaWGdbWcdaqhaaqcfayaaKqzad GaamyAaaqcfayaaKqzadGaamOAaiabgUcaRiaaigdaaaqcfaOaey4k aSIaam4qaSWaa0baaKqbagaajugWaiaadMgacqGHsislcaaIXaaaju aGbaqcLbmacaWGQbGaey4kaSIaaGymaaaaaKqbakaawUfacaGLDbaa cqGHRaWkdaWcaaqaaiabfs5aejaadshacaWGXbGaamitamaaBaaaba Gaamysaiaad6eaaeqaaaqaaiaadgeaaaGaam4qaSWaa0baaKqbagaa jugWaiaadYgaaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaaaa@9FAB@

C i j+1 C i j =λ( C i+1 j+1 C i1 j+1 )+K( C i+1 j+1 2 C i j+1 + C i1 j+1 )+α C i j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaWGPbaajuaGbaqcLbmacaWGQbGaey4kaSIa aGymaaaajuaGcqGHsislcaWGdbWcdaqhaaqcfayaaKqzadGaamyAaa qcfayaaKqzadGaamOAaaaajuaGcqGH9aqpcqaH7oaBdaqadaqaaiaa doealmaaDaaajuaGbaqcLbmacaWGPbGaey4kaSIaaGymaaqcfayaaK qzadGaamOAaiabgUcaRiaaigdaaaqcfaOaeyOeI0Iaam4qaSWaa0ba aKqbagaajugWaiaadMgacqGHsislcaaIXaaajuaGbaqcLbmacaWGQb Gaey4kaSIaaGymaaaaaKqbakaawIcacaGLPaaacqGHRaWkcaWGlbWa aeWaaeaacaWGdbWcdaqhaaqcfayaaKqzadGaamyAaiabgUcaRiaaig daaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaKqbakabgkHiTiaa ikdacaWGdbWcdaqhaaqcfayaaKqzadGaamyAaaqcfayaaKqzadGaam OAaiabgUcaRiaaigdaaaqcfaOaey4kaSIaam4qaSWaa0baaKqbagaa jugWaiaadMgacqGHsislcaaIXaaajuaGbaqcLbmacaWGQbGaey4kaS IaaGymaaaaaKqbakaawIcacaGLPaaacqGHRaWkcqaHXoqycaWGdbWc daqhaaqcfayaaKqzadGaamyAaaqcfayaaKqzadGaamOAaiabgUcaRi aaigdaaaaaaa@8CE4@

Or

C i j +( αλ2K1 ) C i j+1 +( λ+K ) C i+1 j+1 +K C i1 j+1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaWGPbaajuaGbaqcLbmacaWGQbaaaKqbakab gUcaRmaabmaabaGaeqySdeMaeyOeI0Iaeq4UdWMaeyOeI0IaaGOmai aadUeacqGHsislcaaIXaaacaGLOaGaayzkaaGaam4qaSWaa0baaKqb agaajugWaiaadMgaaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaK qbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaayjkaiaa wMcaaiaadoealmaaDaaajuaGbaqcLbmacaWGPbGaey4kaSIaaGymaa qcfayaaKqzadGaamOAaiabgUcaRiaaigdaaaqcfaOaey4kaSIaam4s aiaadoealmaaDaaajuaGbaqcLbmacaWGPbGaeyOeI0IaaGymaaqcfa yaaKqzadGaamOAaiabgUcaRiaaigdaaaqcfaOaeyypa0JaaGimaaaa @6E04@ (6)

For cases where the initial and final conditions are given, boundary condition at the first node can be expressed as:

C 0 j+1 = f 0 ( t j+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaIWaaajuaGbaqcLbmacaWGQbGaey4kaSIa aGymaaaajuaGcqGH9aqpcaWGMbWaaSbaaeaajugWaiaaicdaaKqbag qaamaabmaabaGaamiDaSWaaWbaaKqbagqabaqcLbmacaWGQbGaey4k aSIaaGymaaaaaKqbakaawIcacaGLPaaaaaa@4AC7@ (7a)

Hence, first node equation is expressed as:

C i j +( αλ2K1 ) C i j+1 +( λ+K ) C i+1 j+1 =K f 0 ( t j+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaWGPbaajuaGbaqcLbmacaWGQbaaaKqbakab gUcaRmaabmaabaGaeqySdeMaeyOeI0Iaeq4UdWMaeyOeI0IaaGOmai aadUeacqGHsislcaaIXaaacaGLOaGaayzkaaGaam4qaSWaa0baaKqb agaajugWaiaadMgaaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaK qbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaayjkaiaa wMcaaiaadoealmaaDaaajuaGbaqcLbmacaWGPbGaey4kaSIaaGymaa qcfayaaKqzadGaamOAaiabgUcaRiaaigdaaaqcfaOaeyypa0JaeyOe I0Iaam4saiaadAgadaWgaaqaaKqzadGaaGimaaqcfayabaWaaeWaae aacaWG0bWaaWbaaeqabaqcLbmacaWGQbGaey4kaSIaaGymaaaaaKqb akaawIcacaGLPaaaaaa@6DA6@ (7b)

Similarly, the last node boundary condition is:

C l j+1 = f l+1 ( t j+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaWGSbaajuaGbaqcLbmacaWGQbGaey4kaSIa aGymaaaajuaGcqGH9aqpcaWGMbWcdaWgaaqcfayaaKqzadGaamiBai abgUcaRiaaigdaaKqbagqaamaabmaabaGaamiDamaaCaaabeqaaKqz adGaamOAaiabgUcaRiaaigdaaaaajuaGcaGLOaGaayzkaaaaaa@4CD2@ (8a)

C l j +( αλ2K1 ) C l j+1 +K C l1 j+1 =( λ+K ) f l+1 ( t j+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaWGSbaajuaGbaqcLbmacaWGQbaaaKqbakab gUcaRmaabmaabaGaeqySdeMaeyOeI0Iaeq4UdWMaeyOeI0IaaGOmai aadUeacqGHsislcaaIXaaacaGLOaGaayzkaaGaam4qaSWaa0baaKqb agaajugWaiaadYgaaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaK qbakabgUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaamiBaiab gkHiTiaaigdaaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaKqbak abg2da9iabgkHiTmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaayjk aiaawMcaaiaadAgalmaaBaaajuaGbaqcLbmacaWGSbGaey4kaSIaaG ymaaqcfayabaWaaeWaaeaacaWG0bWcdaahaaqcfayabeaajugWaiaa dQgacqGHRaWkcaaIXaaaaaqcfaOaayjkaiaawMcaaaaa@70C0@ (8b)

For 1x9 and 0t4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWG4bGaeyizImQaaGyoauaabeqabeaaaeaacaWGHbGaamOB aiaadsgaaaqbaeqabeqaaaqaaiaaicdacqGHKjYOcaWG0bGaeyizIm QaaGinaaaaaaa@4511@ ; and for the first instance, we obtain as follows:

At time=0(i.e j=0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WG0bqbaeqabeqaaaqaaiaadshacaWGPbGaamyBaiaadwgacqGH9aqp caaIWaGaaiikaiaadMgacaGGUaGaamyzauaabeqabeaaaeaacaWGQb Gaeyypa0JaaGimaiaacMcaaaaaaaaa@4463@ :

i=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaIXaaaaa@3928@ ,

C 1 0 +K C 0 1 +( αλ2K1 ) C 1 1 +( λ+K ) C 2 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGimaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaIYaaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@644B@

C 1 0 +( αλ2K1 ) C 1 1 +( λ+K ) C 2 1 =K f 0 ( t 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRmaabmaabaGaeqySdeMaeyOeI0Iaeq4UdWMaeyOeI0IaaGOmai aadUeacqGHsislcaaIXaaacaGLOaGaayzkaaGaam4qaSWaa0baaKqb agaajugWaiaaigdaaKqbagaajugWaiaaigdaaaqcfaOaey4kaSYaae WaaeaacqaH7oaBcqGHRaWkcaWGlbaacaGLOaGaayzkaaGaam4qaSWa a0baaKqbagaajugWaiaaikdaaKqbagaajugWaiaaigdaaaqcfaOaey ypa0JaeyOeI0Iaam4saiaadAgalmaaBaaajuaGbaqcLbmacaaIWaaa juaGbeaadaqadaqaaiaadshalmaaCaaajuaGbeqaaKqzadGaaGymaa aaaKqbakaawIcacaGLPaaaaaa@66FB@ (9a)

i=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaIYaaaaa@3929@ ,

C 2 0 +K C 1 1 +( αλ2K1 ) C 2 1 +( λ+K ) C 3 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaIYaaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGymaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaIYaaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaIZaaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@644F@ (9b)

i=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzadGaamyAai abg2da9iaaiodaaaa@39CA@ ,

C 3 0 +K C 2 1 +( αλ2K1 ) C 3 1 +( λ+K ) C 4 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaIZaaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGOmaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaIZaaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaI0aaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@6453@ (9c)

i=4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaI0aaaaa@392B@ ,

C 4 0 +K C 3 1 +( αλ2K1 ) C 4 1 +( λ+K ) C 5 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaI0aaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaG4maaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaI0aaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaI1aaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@6457@ (9d)

i=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaI1aaaaa@392C@ ,

C 5 0 +K C 4 1 +( αλ2K1 ) C 5 1 +( λ+K ) C 6 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaI1aaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGinaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaI1aaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaI2aaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@645B@ (9e)

i=6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaI2aaaaa@392D@ ,

C 6 0 +K C 5 1 +( αλ2K1 ) C 6 1 +( λ+K ) C 7 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaI2aaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGynaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaI2aaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaI3aaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@645F@ (9f)

i=7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaI3aaaaa@392E@ ,

C 7 0 +K C 6 1 +( αλ2K1 ) C 7 1 +( λ+K ) C 8 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaI3aaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGOnaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaI3aaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaI4aaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@6463@ (9g)

i=8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaI4aaaaa@392F@ ,

C 8 0 +K C 7 1 +( αλ2K1 ) C 8 1 +( λ+K ) C 9 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaI4aaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaG4naaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaI4aaajuaGbaqcLbmacaaI XaaaaKqbakabgUcaRmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaay jkaiaawMcaaiaadoealmaaDaaajuaGbaqcLbmacaaI5aaajuaGbaqc LbmacaaIXaaaaKqbakabg2da9iaaicdaaaa@6467@ (9h)

i=9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaI5aaaaa@3930@ ,

C 9 0 +K C 8 1 +( αλ2K1 ) C 9 1 =( λ+K ) f 10 ( t 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoealm aaDaaajuaGbaqcLbmacaaI5aaajuaGbaqcLbmacaaIWaaaaKqbakab gUcaRiaadUeacaWGdbWcdaqhaaqcfayaaKqzadGaaGioaaqcfayaaK qzadGaaGymaaaajuaGcqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiab eU7aSjabgkHiTiaaikdacaWGlbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaadoealmaaDaaajuaGbaqcLbmacaaI5aaajuaGbaqcLbmacaaI XaaaaKqbakabg2da9iabgkHiTmaabmaabaGaeq4UdWMaey4kaSIaam 4saaGaayjkaiaawMcaaiaadAgadaWgaaqaaKqzadGaaGymaiaaicda aKqbagqaamaabmaabaGaamiDaSWaaWbaaKqbagqabaqcLbmacaaIXa aaaaqcfaOaayjkaiaawMcaaaaa@6733@ (9i)

Atime,t=0, C 1 0 = C 2 0 = C 3 0 = C 4 0 = C 5 0 = C 6 0 = C 7 0 = C 8 0 = C 9 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WG0bGaamyAaiaad2gacaWGLbGaaiilaiaadshacqGH9aqpcaaIWaGa aiilaiaadoealmaaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmaca aIWaaaaKqbakabg2da9iaadoealmaaDaaajuaGbaqcLbmacaaIYaaa juaGbaqcLbmacaaIWaaaaKqbakabg2da9iaadoealmaaDaaajuaGba qcLbmacaaIZaaajuaGbaqcLbmacaaIWaaaaKqbakabg2da9iaadoea lmaaDaaajuaGbaqcLbmacaaI0aaajuaGbaqcLbmacaaIWaaaaKqbak abg2da9iaadoealmaaDaaajuaGbaqcLbmacaaI1aaajuaGbaqcLbma caaIWaaaaKqbakabg2da9iaadoealmaaDaaajuaGbaqcLbmacaaI2a aajuaGbaqcLbmacaaIWaaaaKqbakabg2da9iaadoealmaaDaaajuaG baqcLbmacaaI3aaajuaGbaqcLbmacaaIWaaaaKqbakabg2da9iaado ealmaaDaaajuaGbaqcLbmacaaI4aaajuaGbaqcLbmacaaIWaaaaKqb akabg2da9iaadoealmaaDaaajuaGbaqcLbmacaaI5aaajuaGbaqcLb macaaIWaaaaKqbakabg2da9iaaicdaaaa@831F@

Arranging equations (6a) through (6i) in vector matrix gives:

[ ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω ]{ C 1 1 C 2 1 C 3 1 C 4 1 C 5 1 C 6 1 C 7 1 C 8 1 C 9 1 }={ K f 0 ( t 1 ) 0 0 0 0 0 0 0 ( λ+K ) f 10 ( t 1 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabKqcaaaaaaaabaGaeqyYdChabaGaeq4UdWMaey4kaSIaam4s aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdChabaGaeq4U dWMaey4kaSIaam4saaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGa eqyYdChabaGaeq4UdWMaey4kaSIaam4saaqaaiaaicdaaeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaWGlbaabaGaeqyYdChabaGaeq4UdWMaey4kaSIaam4saaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa aGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdChabaGaeq4UdWMaey 4kaSIaam4saaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdC habaGaeq4UdWMaey4kaSIaam4saaqaaiaaicdaaeaacaaIWaaabaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca WGlbaabaGaeqyYdChabaGaeq4UdWMaey4kaSIaam4saaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaWGlbaabaGaeqyYdChabaGaeq4UdWMaey4kaSIa am4saaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdChaaaGa ay5waiaaw2faamaacmaabaqbaeqabKqaaaaaaeaacaWGdbWcdaqhaa qcfayaaKqzadGaaGymaaqcfayaaKqzadGaaGymaaaaaKqbagaacaWG dbWcdaqhaaqcfayaaKqzadGaaGOmaaqcfayaaKqzadGaaGymaaaaaK qbagaacaWGdbWcdaqhaaqcfayaaKqzadGaaG4maaqcfayaaKqzadGa aGymaaaaaKqbagaacaWGdbWcdaqhaaqcfayaaKqzadGaaGinaaqcfa yaaKqzadGaaGymaaaaaKqbagaacaWGdbWcdaqhaaqcfayaaKqzadGa aGynaaqcfayaaKqzadGaaGymaaaaaKqbagaacaWGdbWcdaqhaaqcfa yaaKqzadGaaGOnaaqcfayaaKqzadGaaGymaaaaaKqbagaacaWGdbWc daqhaaqcfayaaKqzadGaaG4naaqcfayaaKqzadGaaGymaaaaaKqbag aacaWGdbWcdaqhaaqcfayaaKqzadGaaGioaaqcfayaaKqzadGaaGym aaaaaKqbagaacaWGdbWcdaqhaaqcfayaaKqzadGaaGyoaaqcfayaaK qzadGaaGymaaaaaaaajuaGcaGL7bGaayzFaaGaeyypa0ZaaiWaaeaa faqabeqcbaaaaaqaaiabgkHiTiaadUeacaWGMbWcdaWgaaqcfayaaK qzadGaaGimaaqcfayabaWaaeWaaeaacaWG0bWcdaahaaqcfayabeaa jugWaiaaigdaaaaajuaGcaGLOaGaayzkaaaabaGaaGimaaqaaiaaic daaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGim aaqaaiabgkHiTmaabmaabaGaeq4UdWMaey4kaSIaam4saaGaayjkai aawMcaaiaadAgadaWgaaqaaKqzadGaaGymaiaaicdaaKqbagqaamaa bmaabaGaamiDaSWaaWbaaKqbagqabaqcLbmacaaIXaaaaaqcfaOaay jkaiaawMcaaaaaaiaawUhacaGL9baaaaa@F395@

Where:

ω=( αλ2K1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3j abg2da9maabmaabaGaeqySdeMaeyOeI0Iaeq4UdWMaeyOeI0IaaGOm aiaadUeacqGHsislcaaIXaaacaGLOaGaayzkaaaaaa@4336@

Hence, at any point with time, the general form of the above equation is presented as:

[ ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω λ+K 0 0 0 0 0 0 0 K ω ]{ C 1 j+1 C 2 j+1 C 3 j+1 C 4 j+1 C 5 j+1 C 6 j+1 C 7 j+1 C 8 j+1 C 9 j+1 }={ K f 0 ( t j+1 ) 0 0 0 0 0 0 0 ( λ+K ) f l+1 ( t j+1 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabKqcaaaaaaaabaGaeqyYdChabaGaeq4UdWMaey4kaSIaam4s aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdChabaGaeq4U dWMaey4kaSIaam4saaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGa eqyYdChabaGaeq4UdWMaey4kaSIaam4saaqaaiaaicdaaeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaWGlbaabaGaeqyYdChabaGaeq4UdWMaey4kaSIaam4saaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa aGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdChabaGaeq4UdWMaey 4kaSIaam4saaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdC habaGaeq4UdWMaey4kaSIaam4saaqaaiaaicdaaeaacaaIWaaabaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca WGlbaabaGaeqyYdChabaGaeq4UdWMaey4kaSIaam4saaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaWGlbaabaGaeqyYdChabaGaeq4UdWMaey4kaSIa am4saaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiaaicdaaeaacaWGlbaabaGaeqyYdChaaaGa ay5waiaaw2faamaacmaabaqbaeqabKqaaaaaaeaacaWGdbWcdaqhaa qcfayaaKqzadGaaGymaaqcfayaaKqzadGaamOAaiabgUcaRiaaigda aaaajuaGbaGaam4qaSWaa0baaKqbagaajugWaiaaikdaaKqbagaaju gWaiaadQgacqGHRaWkcaaIXaaaaaqcfayaaiaadoealmaaDaaajuaG baqcLbmacaaIZaaajuaGbaqcLbmacaWGQbGaey4kaSIaaGymaaaaaK qbagaacaWGdbWcdaqhaaqcfayaaKqzadGaaGinaaqcfayaaKqzadGa amOAaiabgUcaRiaaigdaaaaajuaGbaGaam4qaSWaa0baaKqbagaaju gWaiaaiwdaaKqbagaajugWaiaadQgacqGHRaWkcaaIXaaaaaqcfaya aiaadoealmaaDaaajuaGbaqcLbmacaaI2aaajuaGbaqcLbmacaWGQb Gaey4kaSIaaGymaaaaaKqbagaacaWGdbWcdaqhaaqcfayaaKqzadGa aG4naaqcfayaaKqzadGaamOAaiabgUcaRiaaigdaaaaajuaGbaGaam 4qaSWaa0baaKqbagaajugWaiaaiIdaaKqbagaajugWaiaadQgacqGH RaWkcaaIXaaaaaqcfayaaiaadoealmaaDaaajuaGbaqcLbmacaaI5a aajuaGbaqcLbmacaWGQbGaey4kaSIaaGymaaaaaaaajuaGcaGL7bGa ayzFaaGaeyypa0ZaaiWaaeaafaqabeqcbaaaaaqaaiabgkHiTiaadU eacaWGMbWcdaWgaaqcfayaaKqzadGaaGimaaqcfayabaWaaeWaaeaa caWG0bWcdaahaaqcfayabeaajugWaiaadQgacqGHRaWkcaaIXaaaaa qcfaOaayjkaiaawMcaaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqa aiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacqGHsislda qadaqaaiabeU7aSjabgUcaRiaadUeaaiaawIcacaGLPaaacaWGMbWc daWgaaqcfayaaKqzadGaamiBaiabgUcaRiaaigdaaKqbagqaamaabm aabaGaamiDaSWaaWbaaKqbagqabaqcLbmacaWGQbGaey4kaSIaaGym aaaaaKqbakaawIcacaGLPaaaaaaacaGL7bGaayzFaaaaaa@0942@

Method of application

numerical Method were applied through the developed system to generate the governing equations, derived solution generated the derived model solution, this were simulated to monitor the contaminants at different depth, values of contaminant known as concentration at different depth were generated, this results are within the values of concentration from other experimental values from the same contaminant by other experts, validation of the developed through is concept is for monitoring such microbes in deltaic environment.

Results and discussion

Results and discussion are presented in tables including graphical representation for Acetobacter stated below. The study express these values through graphical representation as it monitor the Halobacterium at different deposition. Figure 1 shows that the behavior of the microbes migrates under exponential phase with sudden slight decrease between 10-15m thus rapidly increase to the optimum values at 30m, Figure 2 developed rapid exponential migration to the optimum level recorded at 30m, Figure 3 observed the migration experiencing degradation with respect to depth, high to low concentration of Halobacterium were observed in the study location, Figure 4 experiences sudden increase from initial Concentration and rapidly experienced increase were it observed slight decrease between 20-25m, fluctuating to maximum rate at 30m, Figure 5 observed Halobacterium with linear homogeneous increase to the optimum level recorded at 30m while Figure 6-10 where compared with experimental values for model validation and bother parameters developed faviourable fits (Tables 1-10).

Figure 1  Simulation Values from Halobacterium Concentration at Different Depth.

Figure 2  Simulation Values from Halobacterium Concentration at Different Depth.

Figure 3  Simulation Values from Halobacterium Concentration at Different Depth.

Figure 4  Simulation Values from Halobacterium Concentration at Different Depth.

Figure 5  Simulation Values from Halobacterium Concentration at Different Depth.

Figure 6  Predictive and Experimental Values from Halobacterium Concentration at Different Depth.

Figure 7  Predictive and Experimental Values from Halobacterium Concentration at Different Depth.

Figure 8  Predictive and Experimental Values from Halobacterium Concentration at Different Depth.

Figure 9  Predictive and Experimental Values from Halobacterium Concentration at Different Depth.

Figure 10 Predictive and Experimental Values from Halobacterium Concentration at Different Depth.

Depth (m)

Concentration(g/ml)

0

0.0334

3

0.2106

6

0.3034

9

0.3359

12

0.3319

15

0.3154

18

0.3105

21

0.341

24

0.431

27

0.6045

30

0.8854

Table 1 Simulation Values from Halobacterium Concentration at Different Depth

Depth (m)

Concentration(g/ml)

0

0.111

3

0.115074

6

0.130848

9

0.158322

12

0.197496

15

0.24837

18

0.310944

21

0.385218

24

0.471192

27

0.568866

30

0.67824

Table 2 Simulation Values from Halobacterium Concentration at Different Depth

Depth (m)

Concentration(g/ml)

0

4.0200

3

2.1723

6

1.1738

9

0.6342

12

0.3426

15

0.1849

18

0.0995

21

0.0529

24

0.0271

27

0.0118

30

0.0012

Table 3 Simulation Values from Halobacterium Concentration at Different Depth

Depth (m)

Concentration(g/ml)

0

0

3

0.8720

6

2.7634

9

4.9094

12

6.7594

15

7.9763

18

8.4370

21

8.2322

24

7.6666

27

7.2584

30

7.7400

Table 4 Simulation Values from Halobacterium Concentration at Different Depth

Depth (m)

Concentration(g/ml)

0

0

3

0.1256

6

0.2487

9

0.3668

12

0.4776

15

0.5788

18

0.6684

21

0.7446

24

0.8060

27

0.8512

30

0.8793

Table 5 Simulation Values from Halobacterium Concentration at Different Depth

Depth (m)

Predictive Conc. (g/ml)

Experimental Values[ Conc.(g/ml)

0

0.0334

0.032

3

0.2106

0.244

6

0.3034

0.398

9

0.3359

0.352

12

0.3319

0.306

15

0.3154

0.362

18

0.3105

0.414

21

0.341

0.468

24

0.431

0.522

27

0.6045

0.676

30

0.8854

0.831

Table 6 Predictive and Experimental Values from Halobacterium Concentration at Different Depth

Depth (m)

Predictive Conc. (g/ml)

Experimental Values[Conc.(g/ml)

0

0.111

0.023

3

0.115074

0.077

6

0.130848

0.131

9

0.158322

0.185

12

0.197496

0.239

15

0.24837

0.293

18

0.310944

0.247

21

0.385218

0.401

24

0.471192

0.455

27

0.568866

0.509

30

0.67824

0.563

Table 7 Predictive and Experimental Values from Halobacterium Concentration at Different Depth

Depth (m)

Predictive Conc. (g/ml)

Experimental Values Conc.(g/ml)

0

4.02

3.403

3

2.1723

2.45

6

1.1738

1.637

9

0.6342

0.968

12

0.3426

0.443

15

0.1849

0.062

18

0.0995

-0.175

21

0.0529

-0.268

24

0.0271

-0.217

27

0.0118

-0.022

30

0.0012

0.317

Table 8 Predictive and Experimental Values from Halobacterium Concentration at Different Depth

Depth (m)

Predictive Conc. (g/ml)

Experimental Values Conc.(g/ml)

0

0

-0.738

3

0.872

1.536

6

2.7634

3.486

9

4.9094

5.122

12

6.7594

6.414

15

7.9763

7.392

18

8.437

8.046

21

8.2322

8.376

24

7.6666

8.382

27

7.2584

8.064

30

7.74

7.422

Table 9 Predictive and Experimental Values from Halobacterium Concentration at Different Depth

Depth (m)

Predictive Conc. (g/ml)

Experimental Values[ Conc.(g/ml)

0

0

0.071

3

0.1256

0.161

6

0.2487

0.251

9

0.3668

0.341

12

0.4776

0.431

15

0.5788

0.521

18

0.6684

0.611

21

0.7446

0.701

24

0.806

0.791

27

0.8512

0.881

30

0.8793

0.971

Table 10 Predictive and Experimental Values from Halobacterium Concentration at Different Depth

Conclusion

The study of Halobacterium has been monitored at different simulation values expressed through graphical representation, the deposition of the Halobacterium were observed to be influences by various depositional structure of the formation, these conditions explained the rates of Halobacterium depositions in the study locations, the transport process were thoroughly observed to migrates through the behavior of the soil in semi confined bed. The results from these locations explain the depositions of the formation in semi confined under the pressure of high rate of porosity, this were observed to deposit at different structure of semi confined aquifers. The simulation values were subjected to model validation and both parameters developed faviourable fits.

Acknowledgements

None.

Conflict of interest

Author declares that there is no conflict of interest.

References

  1. Whitman B, Shively, F. Growth Potential of Indicator Bacteria, E.coli and Enterococci, in Natural Temperate and Tropical Soils. 5th International Symposium on the Sediment Quality Assessment. Aquatic Ecosystem and Health Management Society. USA; 2002. p. 16‒18.
  2. Engelbrecht JFP, Tredoux. G bacteria in “unpolluted” groundwater Presented at the WISA 2000 Biennial Conference. Sun City, South Africa; 2000. p. 1‒2.
  3. Engelbrecht JFP. An assessment of health aspects of the impact of domestic and industrial waste disposal activities on groundwater resources. USA; 1993.
  4. Ghiores WC, Wilson JT. Microbial ecology of the Terrestrial Subsurface. Advances in Applied Microbiology. 1988;33:107‒172.
  5. American Ground Water Trust. Coliform and E.Coli bacteria. USA: well owner; 2002.
  6. Whitman RL, Nevers MB. Foreshore Sand as a Source of Escherichia coliin Near shore Water of a Lake Michigan Beach. Applied and Environmental Microbiol. 2003;69(9):5555‒5562.
  7. Whitman RL, Nevers MB. Escherichia coli Sampling Reliability at a Frequently Closed Chicago Beach: Monitoring and Management Implications. Environ Sci Technol. 2004;38(16):4241‒4246.
  8. Whitman RL, Shively DA, Pawlik H, et al. Occurrence of Escherichia coli and Enterococci in Cladophora (Chlorophyta) in Near shore Water and Beach Sand of Lake Michigan. Appl Environ Microbiol. 2003;69(8):4714‒4719.
  9. Rabinovici SJ, Bernknopf RL, Wein AM, et al. Economic Trade-Offs of Swim Closures at a Lake Michigan Beach. Environ Sci Technol. 2004;38(10):2737‒2745.
  10. Erfani Y, Rasti A, Mirsalehian SM, et al. E-test versus disk diffusion method in determining multidrug resistant strains of Escherichia coliin urinary tract infection. African Journal of Microbiology Research. 2011;5(6):608‒611.
  11. Daza R, Gutierrez J, Piedrola G. Antibiotic susceptibility of bacterial strains isolated from patients with community-acquired urinary tract infections. Int J Antimicrob Agents. 2001;18(3):211‒215.
  12. Erfani Y, Choobineh H, Safdari R, et al. Comparison of E-test and Disk Diffusion Agar in amtibiotic Suseceptibility of E. coliIsolated from patients with urinary tract infections in Shariati Hospital(Iran). Res J Biol Sci. 2008;15(2):27‒31.
  13. Farrell DJ, Morrissey I, Robbins M, et al. A UK mulicenre sud of The Antimicrobial Susceptibility of Bacterial Pathogens Causing Urinary tract Infections. J Infect. 2003;46(2):94‒100.
  14. Giamarellou H, Poulakou G. Multidrug-resistant Gramnegative infections: what are the treatment options? Drugs. 2009;69(14):1879‒1901.
  15. Rahbar M, Yaghoobi M, Fattahi A. Comparison of different laboratory methods for detection of Methicillin Resistant Staphylococcus aureus. Pak J Med Sci. 2006;22(4):442‒445.
  16. Katz OT, Peled N, Yagupsky P. Evaluation of the current NCCLS guidelines for creening and confirming extended-spectrum beta-lactamase production in isolates of E. Coli and Klebsiellaspecies from bacteremic patients. Eur J Clin Microbiol Infect Dis. 2004;23(11):813‒817.
  17. Eluozo SN. Predictive model to monitor the transport of E. Coliin homogeneous aquifers in port harcourt Niger delta of Nigeria. Scientific Journal of Pure and Applied Sciences 2013;2(3):140‒150.
Creative Commons Attribution License

©2017 Eluozo, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.