Assistant professor dr.mat.Daniel N. Pop Faculty of engineering, Department of Computers and Electrical Engineering Lucian Blaga University of Sibiu… [626146]

SOME NUMERICAL RESULTS OF
MULTIPOINTS BOMNDARY VALUE
PROBLEMS ARISE IN
ENVIRONMENTAL PROTECTION
Assistant professor dr.mat.Daniel N. Pop
Faculty of engineering,
Department of Computers and Electrical Engineering
Lucian Blaga University of Sibiu
August 3, 2016
Abstract
In this paper we investigate two problems arise in pollutant transport
inrivers, andwegivesome numericalresultstoapproximate thissolutions.
We determined the approximate solutions using a two numeric al methods:
1. B-splines combined with Runge-Kutta methods , 2. BVP4C so lver of
MATLAB
and then we compare the run-times.
Keywords :
B-spline, Nonlocal and multipoint boundary value problem, Multistep,
Runge-Kutta and extrapolation method.
AMS Classifications (2010) :
65D07,34B10,65L06
1 The statement of the problems
The aim of this work is to compare the run-times of two numerical met hods
used to determine the approximate solutions of multipoints value pro blems with
boundary conditions at infinity, they appears in pollutants transpo rt in rivers .
Ames and Lohner (1981) [1], study models for a transport, reaction and
dissipation of pollutants in rivers. One model gives rise to a system of three
first-order PDE in one space variable xand time t. By looking for traveling
wave solutions, that depend only the variable:
z=x−t
they reduce PDE, to ODE:
f′′=βgf (1)
g′′=−βgh
h′′=λβgh
1

Herefrepresents a pollutant, gbacteria, and hcarbon; the physical param-
etersλandβare constants. After showing that the equations for gandhimply
that:
(2) g(z) =E−h(z)
λ
whereEis a given value for g(∞). They reduce the system of ODE so:
h′′=λβ(E−h
λ)h, (3)
f′′=β(E−h
λ)f. (4)
This equations are be solved subject to boundary conditions:
(5) h(0) = 1, h(∞) = 0, f(0) = 1, f(∞) = 0
Sinceh(∞) = 0 the equation (3) can be approximated for large zby:
h′′=Eλβh
Forβ=λ= 10 and E= 1,this equation becomes:
(6) h′′= 100h
with solution:
h(z) =Ae10z+Be−10z
Becauseh(0) = 1,h(∞) = 0 implies A= 0andB= 1,then theapproximate
solutionh(z) of the problem: (6) with conditions (5) is asymptotically multiple
ofe−10z.
Withh(z) =e−10z, β=λ= 10, E= 1 the equation (4) becomes:
(7) f′′= 10 (1−e−10z
10)f,
subject to boundary conditions:
(8) f(0) = 1, f(∞) = 0.
Also for large zthe equation (7) can be approximate by:
f′′= 10f.
Solving this approximating ODE we find that:
f(z) =Ae2√
10z+Be−2√
10z
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 multiply e−√
10z.
For practical purposes it is interesting solution behavior in transien t area
(an interval bounded by that starts from 0) and not the behavior of the asymp-
totically area where h(z)≃0, f(z)≃0.
2

2 Numerical results
We present two approximation methods for exact solutions of the p roblems (6)
and (7) with conditions (5),(8). We divide the interval[0 ,∞) in twosubintervals
like: a transient area [0 ,10] and the asymptotically area (10 ,∞),wherez= 10
is the positive solution of equation:
r2−100 = 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 (6) with cond itions
(5), like:
1. On the interval [0 ,10] with B-splines functions of order ( k+1),
2. And on [10 , δ) withRunge-Kutta methods ( k−stages), solver ode113or
ode45.
The convergence given in: ([3], [5]).
After 5 iterations obtained the following results fig :1(a), fig:1(b):
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: (6)+ (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: (7)+ (8)
Figure 1: B-spline + Runge-Kutta
2.2 The solver BVP4C
Following the idea giving in ([6], pp:146-152), we solve the problems (6) + (5)
and (7)+(8) using the solver BVP4c of MATLAB suggest by ([6], pp: 153)and
obtained the following results fig:2(a) and fig :2(b):
3

24681012141618200.10.20.30.40.50.60.70.80.91Shampine−bvp4c−2(iterations)

Start solution
Aprox solution
(a) Approximate solution for the
problem: (6)+ (5)246810121416182000.10.20.30.40.50.60.70.80.91Bvp4c−3(iterations)

Start solution
Aprox solution
(b) Approximate solution for the
problem: (7)+ (8)
Figure 2: BVP4C
2.3 Conclusions
Using the functions tic-toc of Malab we obtained the following results:
Problems Method Tolerance Runtimes
(6)+(5) Bsplines+RungeKutta 10−150.990373seconds,
(6)+(5) BVP4C 10−151.9354seconds,
(7)+(8) Bsplines+RungeKutta 10−151.067721seconds,
(7)+(8) BVP4C 10−151.9752seconds.
For more details concerning the run-times you can be used the func tion
profile viewer of Matlab. This problems is set on an infinite interval, so
some experimentation is necessary to verify that a sufficiently large δhas been
specified.
3 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]Matlab2011, The MathWorks. Inc.,Natick MA, USA.
[5] Daniel N. Pop, Radu T.Trˆ ımbita¸ s, An approximation methods for second
order nonlinear value polylocal problems using B-splines a nd Runge-Kutta
methods , N.A.T 2010 , Studia Univ ”Babes-Bolyai” Cluj-Napoca, Vol. LVI,
Number 2 , pag 515-526, June 2011.
[6] L.F. Shampine, I.Gladwell, S.Thomson, Solving ODEs with Matlab, Cam-
bridge University Press,2003.
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