SCIFED. Publishers. Keywords Antibodies; Richards Equation; Soil Moisture;

Transcription

1 Reserch Article SCIFED Publishers Bin Zho,, 7, : SciFed Journl of Applied Microbiology Open Access Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties * Bin Zho, Lichun Ling, Aibing Li, Jinming Co * College of Science, Northwest A&F University, Yngling, Shnxi, Chin School of Informtion nd Mthemtics, Yngtze University, Jingzhou, Hubei, Chin Abstrct In this pper, effects of mnure composition emissions from soil wter fter poultry mnure ppliction were investigted, nd the vectors representing the reltionship between soil wter chrcteristics nd ntibodies were generted. A Semi-empiricl Richrds eqution is shown for describing the motion lw of soil moisture, nd the - expnsion method nd the homogeneous blnce method. Then model, which involves severl vribles including soil moisture content, soil depth nd time, is ssumed ccording to this exct solution of Semi-empiricl Richrds eqution. At lest one ntibody ws detected in ll the soil wter nd niml mnure smples. Dt used in this model were collected during n internl dringe experiment, which ws crried out on sndy-lom Red Yellow Ltosol (Typic Hpludox) of the county of Pircicb nd it is worth noting tht ll of prmeters or constnts in this function expression re determined by these dt. exct solution of Richrds s eqution is obtined by using the Keywords Antibodies; Richrds Eqution; Soil Moisture; Exct Solution; The - Expnsion Method Introduction Fertiliztion by poultry mnure hs shown n importnt vrince in soil wter chemicl chrcteristics. The stte for regionl soil moisture reserve is the strtegic storge of wter resources in the district. The distribution of soil moisture directly ffects the supply of groundwter resources, determines the mount of wter, which is bsorbed from the soil nd evported by the erths surfce plnts, plys decisive fctor for plnts productivity, nd lso is regrded s strtegic fctors influencing the ecologicl environment security, the economic development nd the people s lives in rid nd semi-rid res. Presence of ntibodies nd soil wter physicochemicl properties plyed key roles in degrdtion of numerous molecules nd other processing. Fertiliztion is the commonest mnging griculturl soils, nd for long time, intensive frming ppeled to fertilizer to increse yields. * Corresponding uthor: Bin Zho, College of Science, Northwest A&F University, Yngling, Shnxi, Chin. E-mil: zhobin835@nwsuf.edu.cn Received November 7, 7; Accepted Jnury 5, 8; Published Februry 5, 8 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. :. Copyright: 7 Bin Zho. This is n open-ccess rticle distributed under the terms of the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl uthor nd source re credited. Volume Issue 7 pge of 7

2 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. SF J Appl Microbiol :. Drcy lw is fundmentl theoreticl method to describe the motion lw of soil moisture, therefore vriety of Richrds equtions re deduced. For the nonliner prtil differentil equtions, the previous reserch method is to discuss their definite solutions nd we commonly cn cquire their numericl solution through the numericl method. Whether the nlyticl solution of Richrds s equtions, which describes the chnge of soil moisture content with the chnge of time nd spce position in Drcys lw, hs been the expecttion. If we substitute some empiricl representtions of hydrulic conductivity nd wter diffusivity into Richrds, equtions, the exct solution of Richrds s eqution on soil moisture content, soil depth nd time is of gret significnce. Furthermore, the greter prts of ntibodies were found in the soil wter by poultry mnure. In the pst few yers, mny powerful methods to construct exct solutions of nonliner evolution equtions hve been estblished nd developed such s the homogeneous blnce method [-3], the ( G / G) -expnsion method [4, 5], the exp-function method [6, 7] nd so on. One of the most effective nd direct methods for constructing exct solutions of nonliner differentil equtions is the - expnsion method. The - expnsion method, first introduced by Wng et l [4], hs been widely used to serch for vrious exct - expnsion method is bsed on the explicit lineriztion of nonliner differentil equtions for trveling wves with certin substitution, which leds to second-order differentil eqution with constnt coefficients [-3]. Finding n exct solution for Richrds eqution, by using the solutions of NLEEs [8-]. The - expnsion method, is our one gol. Mthemticl Models nd Explntions First, we introduce form of Richrds s equtions s follows: ( ) θ θ θ θ = D( θ) + D( θ) + D( θ) + t x x y y z z z Where ( ) D θ K ( θ ), (.) denotes wter diffusivity; K θ denotes hydrulic conductivity; t denotes time; θ denotes soil moisture content; xyz,, denote coordinte xes. If the soil moisture content is lower thn the sturted (unsturted) moisture content with little chnge, θ =, where is constnt. Mny reserchers hve committed themselves to estimting soil hydrulic conductivity, s result, vrious empiricl representtions of hydrulic conductivity re proposed. We ssume tht [4, 5] unsturted hydrulic conductivity is clculted by using the Librdi method, tht is we tke s D( ) (.) Where β is constnt; K ndθ re the vlues K of ndθ during stedy-stte infiltrtion, respectively. Next, we hve intend to simplify eqution (.), in other words, here we only consider the cse tht soil moisture flows in the verticl direction, nd therefore we hve θ θ K ( θ ) = D( θ ) + t z z z (.3) θ = nd eqution (.) into eqution (.3), hence the following semi-empiricl Richrds eqution is obtined: By substituting D( ) (.4) In this section, by mke use of the ( G / G) -expnsion method, we obtin n exct solution for the eqution (.4), however, we omit the description of the -expnsion method. If you re interested in this method, you cn refer to the reference [4]. Let K θ = θ t ( θ) = K exp β( θ θ ) θ = θ z t, z { } exp { ( )} t z z θ = θ + K θ β β θ θ nd, α = K exp{ βθ } then the eqution (.4) cn be equivlently chnged into θ + αβe βθ θ θ = zz z t (.5) Using the trvelling wve vrible θ( zt, ) = θ( ξ) nd ξ = z ωt crries out the eqution (.5) into n ordinry differentil eqution for θ = θ( ξ) Volume Issue 7 pge of 7

3 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. SF J Appl Microbiol :. In order to pply the method, we use the Pinlevé trnsformtion v (.6) -expnsion = e βθ, or equivlently θ = ln v, hence the eqution (.6) cn be written s β (.7) Suppose tht the solution of ordinry differentil eqution (.7) cn be expressed by polynomil in s follows: Where G G( ξ ) form n v( ξ ) = i +, n (.8) i= = stisfies the second order LODE in the dg Where G =, G dξ dg constnts to be determined lter. According to the =, n dξ,, λµ, re rel (.9) - expnsion method, considering the homogeneous blnce between vv nd vv in the eqution (.7), we get 3n+ = n+ n=, hence we cn write (.8) s θ + αβe βθ θ + ωθ = ( ) vv vv αβvv ωvv + + = i G λ G µ G + + = v( ξ ) = +, (.) Substituting (.) long with (.9) into (.7) nd collecting ll terms with the sme order of together, the left-hnd side of (.7) re converted into polynomil in we derive set of lgebric equtions for λµω,,,, s follows: Solving the lgebric equtions bove yields αβ =, ω =, λ =, µ =, (.) αβ Where is rbitrry constnt By using (.), (.) cn be written s,, (.) Where ξ = z. Substituting the generl solutions of eqution (.9) into (.), we hve n exct solution of the eqution (.7) s follows: v ( ξ ) Where ξ = z, λ 4µ = αβ 4 : 3 αβ 3 : ω 3 :3λ αβ ( + µ + λ ) ω ( + λ), : ( µ + λ ) λµ αβ ( λ + µ ) ω ( λ + µ ), : ( λµ µ ) αβ ( µ ) ωµ, v ( ξ ) (.3), α = K { } exp βθ K, θ, C, C,, β, nd re rbitrry constnts. = + αβ λ µ = + αβ λ 4µ λ 4µ Ccosh ξ + Csinh ξ λ 4µ λ 4µ 4 Csinh ξ + Ccosh ξ. Setting ech coefficient of ech polynomil to zero, Volume Issue 7 pge 3 of 7

4 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. SF J Appl Microbiol :. Therefore, by θ = ln v, we hve n exct β solution of the eqution (.4) s follows: λ µ θ = ln + β αβ λ 4µ λ 4µ Ccosh ξ + Csinh ξ λ 4µ λ 4µ 4 Csinh ξ + Ccosh ξ (.4) According to wht hs been discussed bove, we ssume tht soil moisture content, soil depth nd time stisfy λ µ θ = ln + β αβ λ 4µ λ 4µ Ccosh ζ + Csinh ζ λ 4µ λ 4µ 4 Csinh ζ + Ccosh ζ Where ζ = δz εt, δ, ε re rbitrry constnts. (.5) 3. Results nd Discussions In this section, we use the dt [6] to determine the prmeters in the identity (.5), nd therefore the predicted vlues of soil moisture content re clculted from (.5). From dt in Tble, the prmeters of eqution (.5) for ech time nd soil depth were determined by using dt fitting method. These prmeters re shown in Tble. From these prmeters, we obtin function express in which soil wter content is independent vrible wheres time nd soil depth re two dependent vribles. Thus, the predicted vlues of soil moisture content re presented in Tble 3 ccording to eqution (.5). Compred with the ctul vlues, the men squred error (MSE) is.7743, which suggests tht eqution (.5) describes the chnges of soil wter content with the time nd soil depth very well. According to Tble, the following figure cn be presented. Tble : Volumetric Soil Wter contents θ (m 3 /m 3 ), for different Redistribution Times (t) nd different Depths (z) Time Soil depth (m) (h) Figure : Originl Figure Soil Wter Contents With Times And Depths Soil Wter Contents θ Depths Z Times T Volume Issue 7 pge 4 of 7

5 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. SF J Appl Microbiol :. The Figure indictes tht the oil wter contents is decresing with the increse of times, while is Tble : Prmeters of Eqution (.5) lmost the sme contents t the rnge of -4 m. K β A C C ϵ ϴ δ Time Tble 3: The Predicted Vlues Soil depth (m) (h) Figure. The smoothed figure Soil WterCcontents With Times And Depths Soil Wter Contents θ Times t Depths z The Figure shows tht eqution (.5) hs the sme tendency with the figure. Although the contents re not equl t the rnge of -4 m, it cn be recorded s the sme contents duo to little chnge in this rnge. Since the hypothesis tht the wter diffusivity is constnt, figure obviously conforms to the ctul sitution. Conclusion We propose new model eqution (.5), which describes the chnges of soil wter content with the time nd soil depth, by finding out n exct solution of eqution -expnsion method. Even though eqution (.5) is estblished with the ssumptions tht the hydrulic conductivity stisfies exponentil function nd wter diffusivity is constnt, to some extent, the very low men squred error (MSE) indictes tht eqution (.5) is gret reflection in chnge (.4) through currently prevlent Volume Issue 7 pge 5 of 7

6 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. SF J Appl Microbiol :. of soil wter content with the time nd soil depth. However, the simple ssumption of wter diffusivity might incur lrger error in term of ctul vlues of soil wter content; thus, precise empiricl representtion of wter diffusivity is expected to be proposed. With the rpid development of science nd technology, the solutions to nonliner differentil equtions would be enriched, nd more excellent results on Drcys lw might be cquired. All these results lso suggest tht soil wter ply considerble role in the fte of ntibody in the environment. Further reserch is needed to give the detection of ntibodies from niml mnure in soil wter by the regression equtions constructed in the present study. Conflict of Interest We hve no conflict of interests to disclose nd the mnuscript hs been red nd pproved by ll nmed uthors. Acknowledgements This work ws supported by the Fundmentl Reserch Funds for the Centrl Universities (4YB3), Ministry of Eduction nd Stte Administrtion of Foreign Experts Affirs "Overses Techer" project (MSXBNL57), the Key Construction Progrm (5SD8) of Interntionl Coopertion Bse in S&T, Shnxi Province, Chin. References. Wng ML (996) Exct solution for compound KdV- Burgers eqution. Phys Lett A 3: Wng ML (995) Solitry wve solutions for vrint Boussinesq equtions. Phys Lett A 99: Wng ML, Zhou YB, Li ZB (996) Appliction of homogeneous blnce method to exct solutions of nonliner evolution equtions in mthemticl physics. Phys Lett A 6: Wng ML, Li X, Zhng J (8) The-expnsion method nd trveling wve solutions of nonliner evolution equtions in mthemticl physics. Phys Lett A 37: Borhnifr A () Appliction of the-expnsion method for the Zhiber-shbt eqution nd other relted equtions. Mth. Comput Model 54: Borhnifr A, Kbir MM, Mrym Vhdt L (9) New periodic nd soliton wve solutions for the generlized Zkhrov system nd (+)-dimensionl Nizhnik- Novikov-Veselov system. Chos Solitons Frctls 4: Borhnifr A, Kbir M M (9) New periodic nd soliton solutions by ppliction of exp-function method for nonliner evolution equtions. Comput Appl Mth 9: Asln İ (9) Exct nd explicit solutions to some nonliner evolution equtions by utilizing the-expnsion method. Appl Mth Comput 5: Bekir A (8) Appliction of the-expnsion method for nonliner evolution equtions. Phys Lett A 37: Zhng J, Wei X, Lu Y (8) A generlized-expnsion method nd its pplictions. Phys Lett A 37: Librdi PL, Reichrdt K, Nielsen, et l. (98) Simple field methods for estimting soil hydrulic conductivity. Soil Physics.. Prijono S, Lksmn MTS, Supryogo D (6) Effects of hedgerow systems on soil moisture nd unsturted hydrulics conductivity mesured by the Librdi method. Journl of degrded nd mining lnds mngement 3: Zyed EME (9) New trveling wve solutions for higher dimensionl nonliner evolution equtions using generlized-expnsion method. J Phys A 4: Li LX, Li EQ, Wng ML () The-expnsion method nd its ppliction to trveling wve solutions of the Zkhrov eqution. Appl Mth J Chinese Univ 5: Zhng J, Jing FL, Zho XY () An improvedexpnsion method for solving nonliner evolution equtions. Int J Comput Mth 87: Librdi PL, Reichrdt K () Librdi s method refinement for soil hydrulic conductivity mesurement. Aust J Soil Res 39: Volume Issue 7 pge 6 of 7

7 Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. SF J Appl Microbiol :. Cittion: Bin Zho (7) Bio mthemticl Modeling on the Assocition between Presence of Antibodies nd Soil Wter Physicochemicl Properties. :. Volume Issue 7 pge 7 of 7