In this paper we investigate two problems arise in pollutant transpor t in [626148]

Abstract
In this paper we investigate two problems arise in pollutant transpor t in
rivers, and we give some numerical results to approximate this solut ions.
We determined the approximate solutions using a two numerical meth ods:
1. B-splines combined with Runge-Kutta methods ,
2. BVP4C solver of MATLAB
and then we compare the run-times.
General Mathematics Vol. xx, No. x (201x), xx–xx
Some Numerical Results of Multipoint
Boundary Value Problems arise in
Environmental Protection1
Daniel N. Pop
2010 Mathematics Subject Classification: 65D07,34B10,65L06
Key words and phrases: B-spline, Multipoints boundary value problem,
Multistep, Runge-Kutta and extrapolation method.
1 The statement of the problems
The aim of this work is to compare the run-times of two numeric al methods used
to determine the approximate solutions of multipoints valu e problems with bound-
ary conditions at infinity, they appears in pollutants trans port in rivers. Ames and
Lohner(1981) [1], study models for a transport, reaction and dissi pation of pollu-
tants in rivers. One model gives rise to a system of three first -order PDE in one
space variable xand time t. They suggest to use the substitution:
y=x−t
then the wave solutions depend only the variable y.Also they reduce PDE, to ODE:
f′′=βgf (1)
g′′=−βgh
h′′=αβgh
1Received dd month, yyyy
Accepted for publication (in revised form) dd month, yyyy
1

2 Daniel N.Pop
Herehrepresents carbon, fa pollutant and gbacteria; and the physical
parameters α,βare constants. After showing that the equations for gandhimply
that:
(2) g(y) =U−h(y)
α
whereUis a given value for g(∞). We rewrite the system (1) thus:
h′′=αβ(U−h
α)h, (3)
f′′=β(U−h
α)f. (4)
We solved this equations with boundary conditions:
(5) h(0) = 1, h(∞) = 0, f(0) = 1, f(∞) = 0
Sinceh(∞) = 0 the equation (3) can be approximated for large yby:
(6) h′′=Uαβh
Forβ=α= 10 and U= 1,the equation (6) becomes:
(7) h′′= 100h
with solution:
h(y) =Ae10y+Be−10y
Because h(0) = 1, h(∞) = 0 implies A= 0 and B= 1,then the approximate
solution h(y) of the problem: (7) with conditions (5) is asymptotically m ultiple of
e−10y.
Withh(y) =e−10y, β=α= 10, U= 1 the equation (4) becomes:
(8) f′′= 10 (1−e−10y
10)f,
subject to boundary conditions:
(9) f(0) = 1, f(∞) = 0.
Also for large ythe equation (8) can be approximate by:
f′′= 10f.
Solving this approximating ODE we find that:
f(y) =Ae2√
10y+Be−2√
10y
for constants A,B. The solution is to have f(∞) = 0,soA= 0.From this we see
that the solution of the given equation is approximately a mu ltiplye−√
10y.
Using the idea gives by [6] it is interesting solution behavi or intransient area
(an interval bounded by that starts from 0) and not the behavi or of the asymptot-
ically area whereh(y)≃0, f(y)≃0.

Some Numerical Results arise in Environmental Protection 3
2 Numerical results
We present two approximation methods for exact solutions of the problems (7) and
(8) with conditions (5),(9). We divide the interval [0 ,∞) in two subintervals like: a
transient area [0,10] andthe asymptotically area (10,∞),wherey= 10 is the
positive solution of equation:
z2−100 = 0.
2.1 B-splines and Runge-Kutta methods
A natural choice of the start solution for this problems is:
h(t) =1
(t+1)3, f(t) =1
(t+1)3
since:
h(0) = 1, h(∞) = 0,
f(0) = 1, f(∞) = 0.
We obtained the approximation solutions of the problem (7) w ith conditions (5),
like:
1. On the interval [0 ,10] with B-splines functions of order ( k+1),
2. And on the interval [10 , δ) withRunge-Kutta methods ( k−stages), solver
ode113orode45.
The convergence of this method was proof in: ([3], [4]). The r esults are depicted
in fig 1(a),1(b),and the errors in figures: 2(a) and 2(b).
2.2 The solver BVP4C
Following the idea giving in ([6], pp:146-152), we solve the problems (7) + (5) and
(8)+(9) using the solver BVP4c of MATLAB suggest by ([6], pp: 153) and obtained
the following results, depicted in figures: 3(a),3(b),the e rrors are depicted in figures
4(a),4(b).
Remark 1 TheMATLAB code can be found in the work [5, pp. 47-60.].

4 Daniel N.Pop
24681012141618200.10.20.30.40.50.60.70.80.91
xB−splines+Runge−Kutta
←(a,alpha)
←(g,delta)
→(e,gama) →(b,beta)Start solution
Aprox solution
Inner points
(a) Approximate solution for the
problem: (7)+ (5)24681012141618200.10.20.30.40.50.60.70.80.91
xB−splines+Runge−Kutta
←(a,alpha)
←(g,delta)
→(e,gama) →(b,beta)Start solution
Aprox solution
Inner points
(b) Approximate solution for the
problem: (8)+ (9)
Figure 1: B-spline + Runge-Kutta
0 5 10 15 2010−5010−4010−3010−2010−10100
xabs(fe(x)−y)Error
(a) Errors problem:(7)+ (5)0 5 10 15 2010−2010−1510−1010−5100105
xabs(fe(x)−y)Error
(b) Errors problem: (8)+ (9)
Figure 2: Errors B-spline + Runge-Kutta

Some Numerical Results arise in Environmental Protection 5
24681012141618200.10.20.30.40.50.60.70.80.91Shampine−bvp4c−2(iterations)

Start solution
Aprox solution
(a) Approximate solution for the
problem: (7)+ (5)246810121416182000.10.20.30.40.50.60.70.80.91Bvp4c−3(iterations)

Start solution
Aprox solution
(b) Approximate solution for the
problem: (8)+ (9)
Figure 3: BVP4C
0 5 10 15 2010−1410−1210−1010−810−610−410−2
xabs(fe(x)−y)Error
(a) Errors problem: (7)+ (5)0 10 20 30 40 50 6010−1210−1010−810−610−410−2
xabs(fe(x)−y)Error
(b) Errors problem: (8)+ (9)
Figure 4: Errors for BVP4C

6 Daniel N.Pop
2.3 Conclusions
Using the functions tic-toc of Malab we obtained the following results:
Problems Method Tolerance Runtimes
(7)+(5) Bsplines+RungeKutta 10−150.990373seconds,
(7)+(5) BVP4C 10−151.9354seconds,
(8)+(9) Bsplines+RungeKutta 10−151.067721seconds,
(8)+(9) BVP4C 10−151.9752seconds.
For more details concerning the run-times you can be used the function profile
viewerof Matlab. This problems is set on an infinite interval, so som e experimen-
tation is necessary to verify that a sufficiently large δhas been specified.
3 Acknowledgements
The writing of this work benefited enormously from a lot of dis cussion with prof.
dr. Damian Trif, assoc. prof. dr. Radu T.Trˆ ımbita¸ s from Ba be¸ s-Bolyai University
of Cluj-Napoca and prof dr.Eugen Draghici from Lucian Blaga University of Sibiu.
4 Bibliography
References
[1] W.Ames, E.Lohner, Nonlinear models of reaction-diffusssion in rivers, New
Brunswick, NJ:IMACS, 1981.
[2] U.M. Ascher, R.M. Mattheij, R.D Russel, Numerical Solution of boundary Value
Problems for Ordinary Differential Equations Philadelphia S.I.A.M, 1995.
[3] G. Goldner, Radu T.Trˆ ımbita¸ s, A combined method for two point boundary value
problem, P.U.M.A, Vol 11 pag 255-264, 2000.
[4] Daniel N. Pop, Radu T.Trˆ ımbita¸ s, An approximation methods for second order
nonlinear value polylocal problems using B-splines and Run ge-Kutta methods ,
N.A.T 2010 , Studia Univ ”Babes-Bolyai” Cluj-Napoca, Vol. L VI, Number 2 ,
pag 515-526, June 2011.
[5] Daniel N.Pop, Aspects concerning some numerical method for approximate s o-
lution of two point boundary value problems, Ed. Presa Universitar˘ a Clujean˘ a,
2011.
[6] L.F. Shampine, I.Gladwell, S.Thomson, Solving ODEs with Matlab, Cambridge
University Press,2003.

Some Numerical Results arise in Environmental Protection 7
Author : Daniel N.Pop
University Lucian Blaga
Faculty Inginery
Department Computers and Electric Inginery
Address Emil Cioran 4Sibiu550024Romania
e-mail: popdaniel 31@yahoo.com