MOJ MOJES

Ecology & Environmental Sciences
Review Article
Volume 1 Issue 1 - 2016
Software for Pollutants Transport in Rivers and for Identification of Excessive Pollution Sources
Kachiashvili KJ1,2* and Melikdzhanian DI2
1Georgian Technical University, Georgia
2Tbilisi State University, Georgia
Received: August 26, 2016 | Published: November 16, 2016
*Corresponding author: Kachiashvili KJ, Georgian Technical University, St. Kostava, Vekua Institute of Applied Mathematics, Tbilisi State University, Tbilisi, 0175, Georgia, Email:
Citation: Kachiashvili KJ, Melikdzhanian DI (2016) Software for Pollutants Transport in Rivers and for Identification of Excessive Pollution Sources. MOJ Eco Environ Sci 1(1): 00006. DOI: 10.15406/mojes.2016.01.00006

Abstract

The program packages of realization of mathematical models of pollutants transport in rivers and for identification of river water excessive pollution sources located between two controlled cross-sections of the river will be considered and demonstrated. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses test­ing methods. The identification al­go­rithms were elaborated with the supposition that the pollution sources discharge dif­fe­rent compositions of pollutants or (at the identical com­po­sition) different propor­tions of pollutants into the rivers. One-, two-, and three-dimensional advection-diffusion mathematical models of river water quality formation both under classical and new, original boundary conditions are realized in the package. New finite-difference schemes of calculation have been developed and the known ones have been improved for these mathematical models. At the same time, a number of important problems which provide practical realization, high accuracy and short time of obtaining the solution by computer have been solved. Classical and new constrained Bayesian methods of hypotheses testing for identification of river water excessive pollution sources are realized in the appropriate software. The packages are designed as a up-to-date convenient, reliable tools for specialists of various areas of knowledge such as ecology, hydrology, building, agriculture, biology, ichthyology and so on. They allow us to calculate pollutant concentrations at any point of the river depending on the quan­­tity and the conditions of discharging from several pollution sources and to identify river water excessive pollution sources when such necessity arise.

Keywords: Software; Pollutants transport; Identification; Excessive pollution sources; Mathematical models

Introduction

For solving the problems of analysis and control of the quality of the environmental objects, it is necessary to process big volume of measurement information about physical, chemical and biological parameters of these objects. To process big data with guaranteed quality in acceptable period of time is possible by means of wide application of mathematical methods and computers. For collecting, storage and processing data of the environment, there are developed the automated systems of the quality control of the environmental objects and universal software packages [1]. Among them the most widespread are environmental water and air pollution levels control systems. Among the most topical problems of control of environmental water quality, the task of modeling of polluting substances propagation in water objects should be emphasized.

Mathematical Models Describing the Pollutants Transport in Rivers

Transport and diffusion equation

The diffusion of polluting substances in rivers is mostly described by the three-dimensional equation of turbulent diffusion of non-conservative substances [1-4]:

t Φ(t,r)( K(r) )Φ(t,r)+( ν(r) )Φ(t,r)+ζ(r)Φ(t,r)=f(t,r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba qcLbsacqGHciITaKqbagaajugibiabgkGi2kaadshaaaGaeuOPdyKa aiikaiaadshacaGGSaGaaCOCaiaacMcacqGHsisljuaGdaqadaqaaK qzGeGaey4bIeTamGjGgwSixlaahUeacaGGOaGaaCOCaiaacMcacqGH flY1cqGHhis0aKqbakaawIcacaGLPaaajugibiabfA6agjaacIcaca WG0bGaaiilaiaahkhacaGGPaGaey4kaSscfa4aaeWaaeaajugibiaa h27acaGGOaGaaCOCaiaacMcacqGHhis0aKqbakaawIcacaGLPaaaju gibiabfA6agjaacIcacaWG0bGaaiilaiaahkhacaGGPaGaey4kaSIa eqOTdONaaiikaiaahkhacaGGPaGaeuOPdyKaaiikaiaadshacaGGSa GaaCOCaiaacMcacqGH9aqpcaWGMbGaaiikaiaadshacaGGSaGaaCOC aiaacMcaaaa@7717@ ,

Where Φ(t,r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHMo GrcaGGOaGaamiDaiaacYcaieWacaWFYbGaaiykaaaa@3C01@  is the time-averaged concentration of non-conservative pollutant; t is the time;  r=[x,y,z] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFYbGaeyypa0Jaai4waiaadIhacaGGSaGaamyEaiaacYcacaWG6bGa aiyxaaaa@3EA5@  is the radius vector; it’s components  x,y,z are the spatial coordinates (the axis x  is horizontal and its direction coincides with the direction of averaged current of the flow, the axis  y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b aaaa@3784@  is perpendicular to the free surface and is directed downwards; the axis  z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEaa aa@3784@  is directed across the flow);  K(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaC4saK qzadGaaiikaiaahkhacaGGPaaaaa@3AD0@ is the tensor of turbulent diffusion,  v(r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaOaa8 NDaiaacIcacaWFYbGaaiykaaaa@39D4@  is the vector of time-averaged speed of the river flow;  ζ(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA7a6L qbGiaacIcacaWHYbGaaiykaaaa@3AB8@  is the coefficient of non-conservatism of pollutants,  f(t,r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzaK qbGiaacIcacaWG0bGaaiilaGqadiaa=jhacaGGPaaaaa@3B9E@  is the power of polluting sources.

For the engineering practice very often the two-dimensional or one-dimensional models are considered instead of the three-dimensional one [2,5]. If the non-uniformity of distribution of concentrations of pollutants on the depth of the watercourse is not taken into account, then it is possible to obtain the two-dimensional turbulent diffusion equation. The equation of one-dimensional turbulent diffusion is applied when the distribution of the concentration of the pollutant across the stream is homogeneous. It is also used when average indicators are used to represent pollution across the river [1].

Initial and boundary conditions

The unknown function  Φ(t,r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuOPdy ucfaIaaiikaiaadshacaGGSaacbmGaa8NCaiaacMcaaaa@3C2E@  is defined at  t>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg6da+iaaicdaaaa@3940@  and  rG MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFYbGaeyicI4Saam4raaaa@39F4@ , where  G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4raa aa@3751@  is some region with the boundary  G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHciITcaWGhbaaaa@38D7@ , or  r=x[0,L] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFYbGaeyypa0JaamiEaiabgIGiolaacUfacaaIWaGa aiilaiaadYeacaGGDbaaaa@3F26@  (in one-dimensional model).

The additional conditions are specified in the form of

Φ(0,r)= S 0 ,  Φ(t,r) | x=0 =σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHMoGrcaGGOaGaaGimaiaacYcaieWacaWFYbGaaiykaKqb Giabg2da9iaadofajuaGpaWaaSbaaKqbGeaapeGaaGimaaqcfa4dae qaaiaaysW7peGaaiila8aacaaMg8+dbiabfA6agjaacIcacaWG0bGa aiilaiaa=jhacaGGPaGaaiiFa8aadaWgaaqaaKqbG8qacaWG4bGaey ypa0JaaGimaaqcfa4daeqaaKqbG8qacqGH9aqpcqaHdpWCaaa@5178@

( S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb WcdaWgaaqaaKqzadGaaGimaaWcbeaaaaa@397D@ σ=const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo8aZjabg2da9iaabogacaqGVbGaaeOBaiaabohacaqG 0baaaa@3E24@ ). The boundary conditions in the lower extremity of the river section may be classical or non-classical. The classical conditions (condition of complete mixing) look like

x Φ(t,r) | x=L =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWG4baa a8aacaaMh8+dbiabfA6agjaacIcacaWG0bGaaiilaGqadiaa=jhaca GGPaGaaiiFa8aadaWgaaqcfasaa8qacaWG4bGaeyypa0Jaamitaaqc fa4daeqaaKqbG8qacqGH9aqpcaaIWaaaaa@48C3@ ; (condition of full mixing)

non-classical conditions –

Φ(t,r) | x=L =q·Φ(t,r) | x=Ll MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHMoGrcaGGOaGaamiDaiaacYcaieWacaWFYbGaaiykaiaa cYhapaWaaSbaaKqbGeaapeGaamiEaiabg2da9iaadYeaaKqba+aabe aapeGaeyypa0JaamyCaiaacElacqqHMoGrcaGGOaGaamiDaiaacYca caWFYbGaaiykaiaacYhapaWaaSbaaKqbGeaapeGaamiEaiabg2da9i aadYeacqGHsislcaWGSbaapaqabaaaaa@4FE4@ . (not local boundary condition)

Here  q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCaa aa@377B@  is the coefficient of self-cleaning of the river on the considered section,  l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBaa aa@3776@  is the length of the section [5,6].

At using of two- or three-dimensional models, the following boundary conditions on the other part of the line or on the surface  G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHciITcaWGhbaaaa@38D7@  (Neumann condition), are set also:

(ν· )Φ(t,r) | rG =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaacceGae8xVd4Maai4Ta8aacayGGaWdbiab=DGirlaa cMcapaGaaG5bV=qacqqHMoGrcaGGOaGaamiDaiaacYcaieWacaGFYb GaaiykaiaacYhapaWaaSbaaeaapeGaa4NCaiabgIGiolabgkGi2kaa dEeaa8aabeaapeGaeyypa0JaaGimaaaa@4CA8@ ,

where  ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacceqcfaieaa aaaaaaa8qacqWF9oGBaaa@3863@  is unit vector of external normal to the border  G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHciITcaWGhbaaaa@38D7@ . In particular, at m=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gacq GH9aqpcaaIZaaaaa@392E@ , it should be

z Φ(t,r)| z=0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaeiaaba WaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaiabfA6agjaacIca caWG0bGaaiilaiaadkhacaGGPaaacaGLiWoadaWgaaqcfasaaiaadQ hacqGH9aqpcaaIWaaabeaacqGH9aqpcaaIWaaaaa@462B@ .

Algorithms of solving of the initial-boundary-value problems

For solving the considered initial-boundary-value problems one of the following three algorithms is offered to the user at choice in the considered software [5]:

  1. Using the classical explicit different scheme;
  2. Using the classical implicit different scheme;
  3. Using the method of decomposition of the operator.

The problems of optimum choosing of the algorithm parameters on which depend the accuracy, the time and the possibility of practical realization of the equation solution are realized in these algorithms [5,8].

The method of decomposition of the operator is the original algorithm of solving of the diffusion equation use in multidimensional problems. In this algorithm solution of multidimensional diffusion equation is represented in the form of a linear combination of solutions of some one-dimensional diffusion equations [5,7,8]. Thus, the multidimensional problem is reduced to the one-dimensional one. As the experience shows, the algorithm of decomposition of the operator can be realized by computer much more quickly than the classical explicit and implicit different schemes.

Description of river banks by splines

The problem of the analytical description of plane or spatial region for which the diffusion equations and the boundary conditions are investigated is solved [5,9]. This region represents the part of the channel of the river filled with water in the section for which we are modeling water pollution. The interpolation by splines is used for the analytical description of the plane curve, represented in the form of a sequence of distinct points with given Cartesian coordinates. Such a sequence, in particular, could be one of the river bank lines. The construction algorithms for splines of two types of the simplest explicit form which ensure continuous dependence of the tangent vector of the curve on its parameter are realized in the package. In particular, the river bottom is described by polynomial splines, and the river banks are described by trigonometrical splines [9,10].

Mathematical Models Describing the Pollutants Transport in Rivers

The problem of optimum choice of the steps of digitization of the algorithms of all realized different schemes is solved in the considered software [5,10,11]. The criteria of optimality are the requirements of reduction of the time and the errors of calculation as much as possible. The algorithms are proper for the functions describing the polluting substances transport in water having the derivatives up to the fourth order inclusive.

The total number of nodal points  N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtaa aa@3758@  in the difference schemes determines the time necessary for realization of the algorithm; the accuracy of the obtained result depends on it. Naturally, there arises the question: how to select the numbers n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGUbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaaaaa@395E@ , ..., n m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGUbWdamaaBaaajuaibaWdbiaad2gaaKqba+aabeaaaaa@3995@  at the given value of  N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtaa aa@3758@  so that the algorithm should be somewhat optimum.

This problem is solved in the following way [11]: at the given values of total nodal points  N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtaa aa@3758@ , there are determined such values of spatial steps of the grid along coordinate axes  h 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaajuaibaGaaGymaaqabaaaaa@387B@ , ...,  h m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaajuaibaGaamyBaaqcfayabaaaaa@3941@  for which the upper bound of the module of the residual takes on the minimum value:

k=1 m ϱ k h k 2 min, k=1 m h k 2 =H=const, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbmaaqahapaqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbst HrhAG8KBLbacfaWdbiab=f=aX=aadaWgaaqaa8qacaWGRbaapaqaba aajuaibaWdbiaadUgacqGH9aqpcaaIXaaapaqaa8qacaWGTbaajuaG cqGHris5a8aacaaMh8+dbiaadIgapaWaa0baaKqbGeaapeGaam4Aaa WdaeaapeGaaGOmaaaajuaGcqGHsgIRciGGTbGaaiyAaiaac6gapaGa aGjbV=qacaGGSaaakeaajuaGdaqeWbWdaeaapeGaamiAa8aadaqhaa qcfasaa8qacaWGRbaapaqaa8qacaaIYaaaaaWdaeaapeGaam4Aaiab g2da9iaaigdaa8aabaWdbiaad2gaaKqbakabg+GivdGaeyypa0Jaam isaiabg2da9iaabogacaqGVbGaaeOBaiaabohacaqG0bWdaiaaysW7 peGaaiilaaaaaa@6B86@

Where  ϱ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajuaGqaaaaaaaaaWdbiab =f=aX=aadaWgaaqcfasaa8qacaWGRbaapaqabaaaaa@4406@  is the error of approximation of the equation along the  k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3775@ th coordinate ( k=1,...,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6ca caGGSaGaamyBaaaa@3DBE@ ).

The solution of this optimization problem is determined by the formula:

h k 2 = 1 ϱ k ( H k=1 m ϱ k ) 1/m   (k=1,...,m) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGObWdamaaDaaajuaibaWdbiaadUgaa8aabaWdbiaaikda aaqcfaOaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaapeGae8x8de=damaaBaaa baWdbiaadUgaa8aabeaaaaWdbmaabmaapaqaa8qacaWGibWdaiaayE W7peWaaebCa8aabaWdbiab=f=aX=aadaWgaaqaa8qacaWGRbaapaqa baaajuaibaWdbiaadUgacqGH9aqpcaaIXaaapaqaa8qacaWGTbaaju aGcqGHpis1aaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaaGym aiaac+cacaWGTbaaaKqba+aacaaMg8+dbiaacIcacaWGRbGaeyypa0 JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamyBaiaacMca aaa@65BB@

The problem of optimum choice of the value of the parameter  τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDaaa@386A@  at given spatial steps of the grid is solved in the following way: at the given value of upper bound of the module of the residual of the equation which is equal toε, there are determined such values of  τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDaaa@386A@  and  N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtaa aa@3758@  for which the time of calculation takes on the minimum value:

N k /τmin MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eada ahaaqcfasabeaacaWGRbaaaiaac+cacqaHepaDjuaGcqGHsgIRciGG TbGaaiyAaiaac6gaaaa@4051@ ,

c τ τ 2 +P N 2/m =ε=const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogada Wgaaqaaiabes8a0bqabaGaeqiXdq3aaWbaaKqbGeqabaGaaGOmaaaa juaGcqGHRaWkcaWGqbGaamOtamaaCaaajuaibeqaaiabgkHiTiaaik dacaGGVaGaamyBaaaajuaGcqGH9aqpcqaH1oqzcqGH9aqpcaWGJbGa am4Baiaad6gacaWGZbGaamiDaaaa@4BCF@ .

Where  ϱ τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajuaGqaaaaaaaaaWdbiab =f=aX=aadaWgaaqcfasaa8qacqaHepaDaKqba+aabeaaaaa@4569@  is the error of approximation of the equation by the time coordinate;  k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E7@  is the constant dependent on the algorithm of solution of the sets of linear equations, usually  k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacq GHijYUcaaIXaaaaa@39D5@ ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH1oqzaaa@384C@  is the given value of the upper bound of the module of the equation residual.

The solution of this optimization problem is determined by the formula:

N= ( P( k+1/m ) kε ) m/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eacq GH9aqpdaqadaqaamaalaaabaGaamiuamaabmaabaGaam4AaiabgUca RiaaigdacaGGVaGaamyBaaGaayjkaiaawMcaaaqaaiaadUgacqaH1o qzaaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamyBaiaac+cacaaI Yaaaaaaa@45C3@ ; τ 2 = 1 c τ ε 1+km MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes8a0n aaCaaabeqcfasaaiaaikdaaaqcfaOaeyypa0ZaimjGlaaabGWKakac tciIXaaabGWKakactc4GJbWaimjGBaaajuaibGWKakadtciHepaDaK qbagqctciaaaWaaSaaaeaacqaH1oqzaeaacaaIXaGaey4kaSIaam4A aiaad2gaaaaaaa@4E5C@ ;

c 1 h 1 2 =...= c m h m 2 =k c τ τ 2 = kε 1+km MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogada WgaaqcfasaaiaaigdaaeqaaKqbakaadIgadaqhaaqcfasaaiaaigda aeaacaaIYaaaaKqbakabg2da9iaac6cacaGGUaGaaiOlaiabg2da9i aadogadaWgaaqcfasaaiaad2gaaKqbagqaaiaadIgadaqhaaqcfasa aiaad2gaaeaacaaIYaaaaKqbakabg2da9iaadUgacaWGJbWaaSbaaK qbGeaacqaHepaDaKqbagqaaiabes8a0naaCaaajuaibeqaaiaaikda aaqcfaOaeyypa0ZaaSaaaeaacaWGRbGaeqyTdugabaGaaGymaiabgU caRiaadUgacaWGTbaaaaaa@5698@ .

The main difficulty for practical realization of the described scheme of optimization is connected with the necessity in estimation of the upper bound of partial derivatives of the function  Φ(t,r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuOPdy KaaiikaiaadshacaGGSaacbmGaa8NCaiaacMcaaaa@3C00@  with respect to the spatial coordinates in terms of which parameters  ϱ τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajuaGqaaaaaaaaaWdbiab =f=aX=aadaWgaaqcfasaa8qacqaHepaDa8aabeaaaaa@44DB@  and  ϱ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajuaGqaaaaaaaaaWdbiab =f=aX=aadaWgaaqcfasaa8qacaWGRbaajuaGpaqabaaaaa@4494@  are expressed without solving the diffusion equation.

 Estimation of derivatives of the unknown function

For practical realization of the described optimization schemes, it is necessary to estimate somehow the upper bounds of the modules of partial derivatives of the unknown function with respect to independent variables without solving the initial task. It should be taken into account that the derivatives of these functions can be estimated with the accuracy of the common constant multiplier.

One way of solution of this problem consists in the following: for some values of the parameters h k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaajuaibaGaam4Aaaqcfayabaaaaa@393E@  and  τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDaaa@386A@  (at solving the diffusion equation), the values of the function  Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHMoGraaa@381E@  are determined as the first approximation. Then, using the numerical differentiation operations, the required derivatives are determined and the values of the desired parameters are calculated. Using these values, new values of the function  Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHMoGraaa@381E@  are determined and so on until the difference between the neighboring calculated values of the function are less than the given value [12].

At solving the diffusion equation, it is possible to use the explicit scheme at the first stage. Then the determination of the values of the sought for function with higher accuracy is necessary. In this case, for calculation of the unknown function values, the iteration method similar to the one used in work [1] for calculation of the multidimensional integral by the Monte-Carlo method is applied. It is known that the operations of numerical differentiation are not stable, but this should not prevent the realization of the offered method since it requires only rough estimates of unknown derivatives. An obvious drawback of this method is the necessity in performance of a plenty of additional actions. Though, due to the capabilities of modern personal computers, it is negligible at solving the particular tasks.

The effect of smoothness of the inhomogeneous part of the diffusion equation on the accuracy of the results

The inhomogeneous parts of diffusion equation (1) realized in the package contain the impulse functions, which are linear combinations of Dirac delta-functions. These functions and their derivatives are not bounded. Therefore the difference schemes of the solution of diffusion equations are not correct in the vicinities of pollution source localization points.

For elimination of this drawback, it is necessary to consider a model, in which the delta function  δ(xa) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH0oazcaGGOaGaamiEaiabgkHiTiaadggacaGGPaaaaa@3C73@  is replaced by the bounded numerical function  D(u,xa) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb GaaiikaiaadwhacaGGSaGaamiEaiabgkHiTiaadggacaGGPaaaaa@3D22@ , where  u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDaa aa@377F@  is the additional parameter. It means that the point sources are replaced by extended ones, the capacity of each of which has the maximum corresponding to the capacity of the source at the point of its location. Such sources can be named quasi-point sources.

The function  D(u,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWG1bGaaiilaiaadIhacaGGPaaaaa@3B4E@  should satisfy the following requirements: it reaches the maximum value proportional to  1/u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aac+cacaWG1baaaa@38ED@  at the point  x=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg2da9iaaicdaaaa@3942@ ; it tends to zero at  x± MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaeyOKH4QaeyySaeRaeyOhIukaaa@3CEE@ ; its integral between the limits MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcqGHEisPaaa@3903@  and  + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHRaWkcqGHEisPaaa@38F8@  is equal to unit; its plot represents a bell-shaped curve. The width of such curve is naturally defined as the width of the rectangle which height is equal to the ordinate of the peak of the curve, and its area is equal to the area of the figure limited by the curve and the axis of abscissas. Thus, the parameter  u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDaa aa@377F@  characterizes the width of the plot of the function D(u,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWG1bGaaiilaiaadIhacaGGPaaaaa@3B4E@ , i.e. the sizes of the source. At  u0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bGaeyOKH4QaaGimaaaa@3A46@ , the function  D(u,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWG1bGaaiilaiaadIhacaGGPaaaaa@3B4E@  tends to  δ(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH0oazjuaicaGGOaGaamiEaiaacMcaaaa@3ACE@ .

One of the elementary continuous functional dependences which may be used for setting the function  D(u,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWG1bGaaiilaiaadIhacaGGPaaaaa@3B4E@  is the trapezoid dependence:

D(u,x)={ 1/u at t<1s, (1+st)/(2us) at 1st1+s 0 at t>1+s, , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGebGaaiikaiaadwhacaGGSaGaamiEaiaacMcacqGH9aqp daGabaqaauaabaqadiaaaeaacaaIXaGaai4laiaadwhaaeaacaqGHb GaaeiDaiaabccacaWG0bGaeyipaWJaaGymaiabgkHiTiaadohacaGG SaaabaGaaiikaiaaigdacqGHRaWkcaWGZbGaeyOeI0IaamiDaiaacM cacaGGVaGaaiikaiaaikdacaWG1bGaam4CaiaacMcaaeaacaqGHbGa aeiDaiaabccacaaIXaGaeyOeI0Iaam4CaiabgsMiJkaadshacqGHKj YOcaaIXaGaey4kaSIaam4CaaqaaiaaicdaaeaacaqGHbGaaeiDaiaa bccacaWG0bGaeyOpa4JaaGymaiabgUcaRiaadohacaGGSaaaaaGaay 5EaaGaaiilaaaa@67A4@

Where  t=2|x|/u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0JaaGOmaiaacYhacaWG4bGaaiiFaiaac+ca caWG1baaaa@3E0A@ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Caa aa@377D@  is any real parameter from the interval  (0,1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aaicdacaGGSaGaaGymaiaacMcaaaa@3A03@ . The plot of this function together with the axis of abscissas forms an isosceles trapezium, the top and bottom bases of which have the lengths equal to u(1s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bWdaiaayEW7peGaaiikaiaaigdacqGHsislcaWGZbGa aiykaaaa@3D47@  and  u(1+s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bWdaiaayEW7peGaaiikaiaaigdacqGHRaWkcaWGZbGa aiykaaaa@3D3C@ , respectively, and the height of which is equal to  1/u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aac+cacaWG1baaaa@38ED@ . The less is value s, the less this trapezium differs from a rectangle.

Besides the trapezoid dependence, in the developed software package the following dependences are offered to the user’s choice:

  1. Gaussian

D(u,x)= 1 u 2π e 1 2 (x/u) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGebGaaiikaiaadwhacaGGSaGaamiEaiaacMcacqGH9aqp daWcaaWdaeaapeGaaGymaaWdaeaapeGaamyDa8aacaaMh8+dbmaaka aapaqaa8qacaaIYaGaeqiWdahabeaaaaWdaiaayEW7peGaamyza8aa daahaaqabeaapeGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbi aaikdaaaGaaiikaiaadIhacaGGVaGaamyDaiaacMcapaWaaWbaaKqb GeqabaWdbiaaikdaaaaaaaaa@4D88@  ;

  1. Lorentzian

D(u,x)= 2 2πu(1+ (x/u) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadseacaGGOaGaamyDaiaacYcacaWG4bGaaiykaiabg2da 9KqbaoaalaaapaqaaKqzGeWdbiaaikdaaKqba+aabaqcLbsapeGaaG Omaiabec8aWjaadwhapaGaaG5bV=qacaGGOaGaaGymaiabgUcaRiaa cIcacaWG4bGaai4laiaadwhacaGGPaqcfa4damaaCaaajuaibeqaaK qzadWdbiaaikdaaaqcLbsacaGGPaaaaaaa@4F4A@  .

In fact, the function  D(u,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWG1bGaaiilaiaadIhacaGGPaaaaa@3B4E@  defined by one of the two latter relations plays the part of a smoothing function, which allows us to transform any function from  L 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aaCaaajuaibeqaaiaaikdaaaaaaa@3862@  into an infinitely differentiable function by means of the operation of convolution [10,11].

If the pollution sources do not operate continuously, but they operate for a limited interval of time  [0,T] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaai4wai aaicdacaGGSaGaamivaiaac2faaaa@3A88@ , then the capacity of each source should be smoothed not only by spatial coordinates, but also by time. It means that this function should be proportional to A(v,tT) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiDaiabgkHiTiaadsfa caGGPaaaaa@3D2E@ , where  v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2baaaa@37A0@  is the additional parameter, and  A(v,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiDaiaacMcaaaa@3B68@  is the function satisfying the following conditions: at  t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyOKH4QaeyOeI0IaeyOhIukaaa@3BE9@  it tends to unit, and at  t+ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyOKH4Qaey4kaSIaeyOhIukaaa@3BDE@  to zero. Its plot represents a quasi-stepped curve, which steepness is maximum at  t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9iaaicdaaaa@393E@ , and this maximum value of the steepness is proportional to  1/v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aac+cacaWG2baaaa@38EE@ . The parameter  v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2baaaa@37A0@  characterizes the time of diminution of the function of discharge. At  v0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeyOKH4QaaGimaaaa@3A47@ , the function  A(v,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiEaiaacMcaaaa@3B6C@  tends to  1ϑ(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaeyOeI0Iaeqy0dOKaaiikaiaadIhacaGGPaaaaa@3C4B@ , where  ϑ(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHrpGscaGGOaGaamiEaiaacMcaaaa@3AA3@  is Heaviside’s stepped function. In the developed software package, it is possible to choose an explicit form of the function  A(v,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiEaiaacMcaaaa@3B6C@  from the following types:

  1.  

A(v,t)= t/v 1 2π e τ 2 /2 dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiDaiaacMcacqGH9aqp daWdXaWdaeaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbmaakaaapa qaa8qacaaIYaGaeqiWdahabeaaaaaapaqaa8qacaWG0bGaai4laiaa dAhaa8aabaWdbiabg6HiLcGaey4kIipapaGaaG5bV=qacaWGLbWdam aaCaaabeqaa8qacqGHsislcqaHepaDpaWaaWbaaKqbGeqabaWdbiaa ikdaaaqcfaOaai4laiaaikdaaaWdaiaayEW7peGaamizaiabes8a0b aa@53B6@ ;

  1.  

A(v,t)= 1 2 1 π arctan(t/v) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiDaiaacMcacqGH9aqp daWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaaaacqGHsisldaWcaa WdaeaapeGaaGymaaWdaeaapeGaeqiWdahaa8aacaaMh8+dbiGacgga caGGYbGaai4yaiaacshacaGGHbGaaiOBaiaacIcacaWG0bGaai4lai aadAhacaGGPaaaaa@4D28@ .

As the computation results show, the smoother are the functions of discharge, the more accurate are the results of solution of the equations in the vicinities of the points of discharge. For given types of functions  D(u,x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWG1bGaaiilaiaadIhacaGGPaaaaa@3B4E@  and  A(v,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaaiikaiaadAhacaGGSaGaamiDaiaacMcaaaa@3B68@ , the accuracy of the results increases as the values of parameters  u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDaa aa@377F@  and  v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2baaaa@37A0@  increase.

Capabilities of the package

Input and editing of the initial data

The input of information necessary for operation of the package is performed in the form of separate constants, tables or the choice of the appropriate line from the pull-down. Data for editing are chosen by means of teams of the main menu (Figure 1).

Figure 1: Data for editing are chosen by means of teams of the main menu.

The entered data which form sequences of elements of the same type are represented in the form of tables (Figure 2). When there are several editing windows with tables on the screen, different windows correspond to different sections of the river.

Figure 2: The entered data which form sequences of elements of the same type are represented.

Realization of computation and representation of the results

The results of calculations are displayed in the form of text messages, tables, graphs, etc. For example, the formats of display of computation results as diagrams and tables are shown in Figure 3 & 4, respectively.

Figure 3: The results of calculations are displayed in the form of text messages.

Figure 4: the formats of display of computation results.

The represented text and diagrams may be printed or written in a text- or graphic files, the name of which is determined by default or indicated by the user.

Software for Identification of Excessive Pollution Sources

 The software for identification of river water excessive pollution sources located between two controlled cross-sections of the river has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers (described above) and statistical hypotheses testing methods. Depending on the available a priori information, different methods of statistical hypotheses testing can be chosen in the software for identification. With increasing of a priori information, application of more compli­ca­ted methods becomes possible, thus providing their higher assurance. The fol­lo­wing statistical tests are realized in this package:

  1. The Euclidean distance method;
  2. The Makhalanobis distance method;
  3. The unconstrained Bayesian method with the stepwise and arbitrary loss functions;
  4. The Constrained Bayesian Method (CBM);
  5. The quasi-optimal test based on CBM.

 The peculiarities of CBM methods are described in [13-18]. Therefore we will not spend space here for their description. Other methods are widely known and described in many scientific works. Let’s show only some characteristics of the package. In particular, in Figure 5 are shown the modes realized in the package, the input of the initial data and the calcu­la­tion regimes. In Figure 6 is shown the graphical view of package output information where the 3-rd point in the bed of the river corresponds to the pollution source guilty of the excessive pollution (Figure 6).


Figure 5: some characteristics of the package.

Figure 6: the graphical view of package output information where the 3rd point in the bed of the river corresponds to the pollution source guilty of the excessive pollution.

Discussion

The results of detailed experimental research of algorithms and appropriate programs included in computer packages described ­above are given in [5]. There are considered the results of research of algorithms of calculation of concentration of polluting substances in the rivers with the help of diffusion equations and also sensitivity of models of different dimensions to the geometrical sizes of a section of the river and dependence of quality of identification ­of emergency pollution sources on the level of the noise, deforming results of measurement. They show truth and reliability both the created algorithms and programs of their realization [19-23].

From the results of calculations, the following practical recommendation concerning the use of the de­ve­­lo­ped models may be given. If geometric sizes, locations of pollution sources, hydrologic characteristics and pollution conditions of the river are such that full mixing of water takes place upstream of the lower controlled section, then it is enough to use the one-dimen­si­onal river models, which are considerably faster and require less computer me­mo­ry than the models of greater dimensionality. Otherwise it is necessary to use the models of greater dimensionality. When choosing a working model for a concrete section of a certain river, it is necessary to per­form preliminary studies with due regard for the above factors. The proper choice of the model and its parameters ensures the qualitative modeling and identification of excessive discharge sources.

Conclusion

The computer packages of realization of mathematical models of propagation of polluting substances in rivers and for identification of exce­ssi­ve pollution sources of the river have been created. The packages comprise original models, methods and algorithms developed by the authors. The software packages are realized for IBM-compatible personal computers according to the generally accepted standard for similar production all over the world. The consumer can use them as a modern, convenient, simple and reliable tool for resolution the problems he deals with in the considered area. Versatile experimental investigation of the developed software packages and the algorithms realized in them, have confirmed their high computing, operational and service qualities. They are good tools for the experts of different professions for qualified solving many practical problems.

References

  1. Primak AV, Kafarov VV, Kachiashvili KJ (1991) System analysis of air and water quality control. Naukova Dumka: Kiev, Ukraine.
  2. Karaushev AV (1969) River hydraulic. Leningrad: Hydrometeoizdat, Canada.
  3. Nikanorov AM, Trunov NM (1999) Intrareservoir processes and quality control of natural water. Hydrometeoizdat: St.-Petersburg, Canada.
  4. Gallakher L, Khobbs DD (1981) Dissemination of pollutions in estuary: mathematical models of the control of water pollution. Moscow: Mir p. 228-243.
  5. Kachiashvili KJ, Melikdzhanian DI (2012) Advanced Modeling and Computer Technologies for Fluvial Water Quali­ty Research and Control. Nova Science Publishers, New York.
  6. Gordeziani DG, Gordeziani EG, Kachiashvili KJ (1999) About some mathematical models of propagation of solutions in rivers. IX intergovernmental conference Problems of an ecology and exploitation of objects of power engineering. Sevastopol, p. 77-79.
  7. Kachiashvili KJ, Gordeziani, DG, and Melikdzhanian DI (2001) Mathematical models of Pollutants Transport with Allowance for Many Affecting Pollution Sources. Urban Drainage Modeling Symposium, Florida, pp. 692-702.
  8. Gordeziani DG, Samarskii AA, (1978) Some problems of the thermo-elasticity of plates and shells, and the method of summary approximation. Complex analysis and its applications, Moscow, pp. 173-186.
  9. Kachiashvili KJ, Melikdzhanian DI (2002) Analytical description of the bank line of the river for simplification and improvement of the process of calculation of polluting substances concentration. In: Reports of enlarged session of the workshop of I. Vekua Institute of Applied Mathematics 17(3): 101-109.
  10. Kachiashvili KJ, Gordeziani DG, Lazarov RG, Melikdzhanian DI (2007) Modeling and simulation of pollutants transport in rivers. Applied Mathematical Modelling 31(7): 1371-1396.
  11. Kachiashvili KJ, Melikdzhanian DI (2006) Parameter optimization algorithms of difference calculation schemes for improving the solution accuracy of diffusion equations describing the pollutants transport in rivers. Applied Mathematics and Computation 183(2): 787-803.
  12. Kachiashvili KJ, Melikdzhanian DI, Prangishvili, AI (2015) Computing Algorithms for Solutions of Problems in Applied Mathematics and Their Standard Program Realization: Part 1-Deterministic Mathematics. Nova Science Publishers, New York, USA.
  13. Kachiashvili KJ, Mueed A (2013) Conditional Bayesian Task of Testing Many Hypotheses. Statistics 47(2): 274-293.
  14. Kachiashvili KJ (2011) Investigation and Computation of Unconditional and Conditional Bayesian Problems of Hypothesis Testing. ARPN Journal of Systems and Software 1(2): 47-59.
  15. Kachiashvili GK, Kachiashvili KJ, Mueed A (2012) Specific Features of Regions of Acceptance of Hypotheses in Conditional Bayesian Problems of Statistical Hypotheses Testing. Sankhya A: The Indian Journal of Statistics 74(1): 112-125.
  16. Kachiashvili KJ, Hashmi MA, Mueed A (2012) Sensitivity Analysis of Classical and Conditional Bayesian Problems of Many Hypotheses Testing. Communications in Statistics -Theory and Methods 41(4): 591-605.
  17. Kachiashvili KJ, Hashmi MA (2010) About Using Sequential Analysis Approach for Testing Many Hypotheses. Bulletin of the Georgian Academy of Sciences 4(2): 20-25.
  18. Kachiashvili KJ, Hashmi MA, Mueed A (2013) Quasi-optimal Bayesian procedures of many hypotheses testing. Journal of Applied Statistics 40(1): 103-122.
  19. Kachiashvili KJ (2014) Comparison of Some Methods of Testing Statistical Hypotheses: Part I. Parallel Methods. International Journal of Statistics in Medical Research 3(2): 174-189.
  20. Kachiashvili KJ (2014) Comparison of Some Methods of Testing Statistical Hypotheses: Part II. Sequential Methods. International Journal of Statistics in Medical Research 3: 189-197.
  21. Kachiashvili KJ (2014) The Methods of Sequential Analysis of Bayesian Type for the Multiple Testing Problem. Sequential Analysis 33(1): 23-38.
  22. Kachiashvili KJ (2014) Investigation of the method of sequential analysis of Bayesian type. Journal of Advances in Mathematics 7(4): 1367-1380.
  23. Kachiashvili KJ (2015) Constrained Bayesian Method for Testing Multiple Hypotheses in Sequential Experiments. Sequential Analysis: Design Methods and Applications 34(2): 171-186.
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