Mathematics - 404 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

Design Feed Forward Neural Network to Determine Doses 
of the Decongestant for Cold Pills 

 
L. N. M. Tawfiq    and   Y. R. B. Al-Mayouf*  
Department of Mathematics, College of Education Ibn-Al-Haitham, 
University of Baghdad 
*Department of computer science, College of Education Ibn-Al-Haitham, 
University of Baghdad 
 
Received in: 30  May  2012  Accepted in: 24   September   2012 
 
Abstract 
       The aim of this paper is to design feed forward neural network to determine the effects of 
cold pills and cascades from simulation the problem to system of first order initial value 
problem. This problem is typical of the many models of the passage of medication throughout 
the body. Designer model is an important part of the process by which dosage levels are set. 
A critical factor is the need to keep the levels of medication high enough to be effective, but 
not so high that they are dangerous. 
Keyword: Ann, feed forward neural network, propagation training algorithm. 
Introduction 
       The study of cold pills based on the work of the contemporary applied mathematician, 
Edward Spitznagel Professor of Mathematics at Washington University, St. Louis, Missouri 
[1]. In this paper we design feed forward neural network(FFNN) to determine the doses of 
decongestant since FFNNs are parallel computational models comprised of densely 
interconnected, simple, adaptive processing units, characterized by an inherent propensity for 
storing experiential knowledge and rendering it available for use, FFNNs resemble the human 
brain in two fundamental respects; Firstly, knowledge is acquired by the network from its 
environment through a learning process, and secondly, interneuron connection strengths, 
known as synaptic weights are employed to store the acquired knowledge[2]. In the proposed 
approach the model function is expressed as the sum of two terms: the first term satisfies the 
initial conditions (IC)  and contains no adjustable parameters. The second term can be found 
by using feed forward neural network(FFNN) which is trained so as to satisfy the differential 
equation and such technique is called collocation neural network. Since it is known that a 
multilayer FFNN with one hidden layer can approximate any function to arbitrary accuracy 
[3],[4] thus our FFNN contains one hidden layer. Now, we illustrate how our approach can be 
used to find the approximate solution of the system of first order differential equation: 
yi׳(x) = Fi(x, y1, y2)   ,    i =1,2         ,        (1) 
where a subject to certain IC’s and x ∈ R, D ⊂ Rn denotes the domain and y(x) is the solution 
to be computed. 
           If yt(x, p) denotes a trial solution with adjustable parameters p, the problem is 
transformed to a discretize form: 
Minp 

i
ˆx D∈

∑


 F(xi ,yt1(xi ,p), yt2(xi ,p) )              ,         (2) 

subject to the constraints imposed by the IC’s. 
In the proposed approach, the trial solution yt employs a FFNN and the parameters p 
correspond to the weights and biases of the neural architecture. We choose a form for the trial 



 
 
 

Mathematics - 405 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

function yt(x) such that it satisfies the IC’s. This is achieved by writing it as a sum of two 
terms:  
      yt(xi , p) = A(x)  + G( x, N(x, p) )            ,      (3) 
where N(x, p) is a single output FFNN with parameters p and n input units fed with the input 
vector x. The term A(x) contains no adjustable parameters and satisfies the IC’s. The second 
term G is constructed so as not to contribute to the IC’s, since yt(x) satisfy them. This term 
can be formed by using a FFNN whose weights and biases are to be adjusted in order to deal 
with the minimization problem. 
Computation of the Gradient 
An efficient minimization of (2) can be considered as a procedure of training the FFNN, 
where the error corresponding to each input vector xi is the value E (xi) which has to force 
near zero. Computation of this error value involves not only the FFNN output but also the 
derivatives of the output with respect to any of its inputs. Therefore, in computing the 
gradient of the error with respect to the network weights consider a multi layer FFNN with n 
input units (where n is the dimensions of the domain) one hidden layer with H sigmoid units 
and a linear output unit . 
  For a given input vector x = ( x1, x2, …, xn ) the output of the FFNN is :     

    N = 
H

i i
i 1

(z )
=

ν σ∑ ,  where  zi = 
n

ij j i
j 1

w x b
=

+∑  
wij denotes the weight connecting the input unit j to the hidden unit i  
vi denotes the weight connecting the hidden unit i to the out put unit , 
bi denotes the bias of hidden unit i , and  
σ (z) is the sigmoid transfer function ( tanhsig. ). 
The gradient of FFNN, with respect to the parameters of the FFNN can be easily obtained as: 
                   

i

N∂
∂ν

 = σ (zi)           ,                          (4) 

                   
i

N
b
∂
∂

 = viσ′(zi)         ,                         (5) 

                  
ij

N
w
∂
∂

 = viσ′(zi) xj     ,                         (6) 

Once the derivative of the error with respect to the network parameters has been defined, then 
it is a straight forward to employ any minimization technique. It must also be noted, the batch 
mode of weight updates may be employed.  
Cold Pills and Cascades 
           At the first sign of a cold many of us start to take cold pills. These pills usually contain 
a decongestant to relieve stuffiness. The pill dissolves in the gastrointestinal tract, and the two 
medications diffuse into the bloodstream. The bloodstream takes the medications to the sites 
where they have therapeutic effect. Both medications are eventually removed from the blood 
by the kidneys and the liver. The dynamics of the rising and falling levels of the medications 
in the GI tract and in the bloodstream may be modeled by system of first-order linear 
differential equations. Pharmaceutical companies do extensive testing to determine the 
movement medications through the body. This process is modeled by treating various 
components of the human body as compartments and following the medication as it enters 
and leaves these compartments. Examples of body compartments are the GI tract, the 
bloodstream, the tissues and the excretory system. Testing has shown that a typical cold 
medication leaves one compartment (e.g., the GI tract) and moves into another (e.g., the 
bloodstream) at a rate proportional to the amount present in the first compartment. The 



 
 
 

Mathematics - 406 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

coefficient of proportionality depends upon the specific medication, the compartments 
involved, and the age and general health of the individual. In this section we look at three 
models based on different dosage strategies suppose that a single dose of a fast-dissolving 
cold pill is taken. The pill dissolves "instantaneously" in the GI tract and releases A 
milligrams of decongestant. Each medication independently diffuses into the bloodstream.  
 
One Inst ant l y Di ssol vi ng  Dose:  T he M od el  IVP  
         L et  y 1 ( t ) b e t h e a mo u n t  o f med i ci n e o f d eco n g est a n t  i n  t h e G I  t r a ct  a t  
t i me t  a ft er  t h e p i l l  h a s d i sso l v ed .  W e sh a l l  u se t h e B a l a n ce L a w t o  co mp u t e 
t h e r a t e o f ch a n g e o f t h e a mo u n t  i n  ea ch  co mp a r t men t .  B eca u se t h e 
med i ca t i o n  mo v es o u t  o f t h e G I  t r a ct  a n d  i n t o  t h e b l o o d -st r ea m a t  a  r a t e 
p r o p o r t i o n a l  t o  t h e a mo u n t  i n  t h e G I  t r a ct ,  a n d  b eca u se n o t h i n g  i s co mi n g  i n ,  
we h a v e t h a t :   

        ),(
)(

11
1 tyk
dt

tdy
−=  Ay =)0(1  

wh er e k 1  >  0  i s t h e co effi ci en t  o f p r o p o r t i o n a l i t y ,  a n d  A i s t h e i n i t i a l  a mo u n t  
t h e u n i t s fo r  t h e r a t e co n st a n t  k 1  a r e r eci p r o ca l s o f t h e u n i t s ch o sen  t o  
mea su r e t h e t i me t .  F o r  ex a mp l e t h e u n i t s o f k 1  a r e ( h o u r s) 1−  i f t i me i s g i v en  
i n  h o u r s.  
       T h e l ev el  o f med i ca t i o n  i n  t h e b l o o d st r ea m wi l l  b u i l d  u p  fr o m zer o  
( i n i t i a l l y ,  t h er e a r e n o  co l d  med i ca t i o n s i n  t h e b l o o d ) ,  b u t  t h en  wi l l  fa l l  a s t h e 
k i d n ey s a n d  l i v er  d o  t h ei r  j o b  o f r emo v i n g  " fo r ei g n "  su b st a n ces fr o m t h e 
b l o o d .  I f y 2 ( t )  d en o t es t h e a mo u n t  o f med i ca t i o n  i n  t h e b l o o d st r ea m a t  t i me t ,  
t h en  t h e B a l a n ce L a w i mp l i es t h a t  :   

                 ),()(
)(

2211
2 tyktyk
dt

tdy
−=      0)0(2 =y  

      T h e fi r st  t er m o n  t h e r i g h t  si d e o f t h e r a t e eq u a t i o n  mo d el s t h e o b ser v ed  
fa ct  t h a t  t h e med i ca t i o n  l ea v i n g  t h e G I  t r a ct  g o es d i r ect l y  i n t o  t h e 
b l o o d st r ea m ;  t h e seco n d  r a t e t er m mo d el s t h e cl ea r a n ce o f med i ca t i o n  fr o m 
t h e b l o o d  a n d  i n t o  t h e ex cr et o r y  sy st em.  
C o n seq u en t l y ,  t h e sy st em o f fi r st -o r d er  O D E s a n d  fl o w o f med i ca t i o n  i s g i v en  
b y  t h e I VP  :  

              11
1 yk

dt
dy

−=    ,   Ay =)0(1  

              ,2211
2 ykyk

dt
dy

−=     0)0(2 =y            … … . …  ( 7 )  

wh er e t h e ( p o si t i v e)  co effi ci en t  k 2  i s t y p i ca l l y  sma l l er  t h a n  k 1 .  
S y st em ( 7 )  mo d el s t h e fl o w o f d eco n g est a n t .  H o wev er ,  t h e fl o w r a t es d i ffer  
b eca u se t h e ex p er i men t a l l y  d et er mi n ed  v a l u es o f t h e r a t e co n st a n t s k 1  a n d  k 2  
a r e q u i t e d i ffer en t  fo r  t h e t wo  med i ca t i o n s ( see T a b l e 1 ) .  T h ese d i ffer en ces 
l ea d  t o  v er y  d i ffer en t  l ev el s o f t h e t wo  med i ca t i o n s i n  t h e b l o o d st r ea m.   
Observe from table (1) how much smaller k2 is than k1. Consequently, the medication will 
stay in the bloodstream longer than in the GI tract. 
 
 Solving The Model IVP  
         Professor Edward Spitznagel in [1] solved this problem analytically and obtain the 
following solution for 21 kk ≠  : 



 
 
 

Mathematics - 407 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

    tkAety 1)(1
−=     ,      )()( 12

21

1
2

tktk ee
kk

Ak
ty −− −

−
=      …..     (8) 

where time t is measured forward from the instant the initial dose of A units of medication is 
released into the GI tract. 
From the solution set (8) see that, as expected, the levels of the medication in the GI tract and 
the bloodstream tend to zero as time increases. Since, the medication is initially at the zero 
level in the bloodstream, these levels reach a maximum value at some later time. 
      Now, we solve system(7) for A=1 where k1 = 1.386, k2 = 0.1386 using three layers feed 
forward neural network with back propagation training algorithm  having one input unit, five 
hidden units with tanhsig. transfer function and one linear output unit. Table (2) gives the 
analytic and neural solutions with different training algorithm such as : Levenberg – 
Marquardt ( trainlm ) [5], and quasi – Newton (trainbfg)[6] and its errors given in table (3), 
table(4) gives the weight and bias of the designer network and table(5) gives the performance 
of the train with epoch and time  
 Figures (1) show the levels of decongestant in the GI tract and the blood over a six-hour 
period if the values of k1 and k2 given in Table (1) are used and A = 1. 
T h e cl ea r a n ce co e f fi ci en t  k 2  o f  m ed i ca t i o n  fr o m t h e b l o o d -s t r ea m i s o ft e n  
mu ch  l o w er  f o r  o l d  a n d  si ck  p e o p l e t h a n  i t  i s fo r  t h e y o u n g  a n d  h ea l t h y .  T h i s  
mea n s t h a t  fo r  s o me p eo p l e t h e me d i ca t i o n  l e v el s i n  t h e b l o o d  ma y  b ec o m e  
d a n g er o u sl y  h i g h  ev en  wi t h  a  " st a n d a r d "  d o sa g e.   
F e w o f  u s st o p  wi t h  t a k i n g  o n e co l d  p i l l .  T h e  m o d el s t h a t  f o l l o w h ea t  t h e  
mo r e r ea l i st i c s et t i n g  o f t a k i n g  me d i ca t i o n  o v er  a  p er i o d  o f t i m e.  
Cont i n u ous D oses  
         I f a  med i ca t i o n  i s e m b e d d ed  i n  r e si n s t h a t  d i s so l v e a t  v a r y i n g  r a t es,  a  
fi x ed  fl o w o f  m ed i ca t i o n  ca n  b e a s su r ed .  T i n y  b ea d s o f t h e mi x t u r e a r e  
p a ck ed  i n t o  a  ca p su l e,  a n d  t h e m ed i ca t i o n s a r e r el ea s ed  a t  a  co n st a n t  r a t e i n t o  
t h e C I  I  t r a ct  o v er  a  p er i o d  o f h o u r s.  O n e ca p su l e o f c o n t i n u o u s -a ct i n g  
me d i ca t i o n  su ch  a s C O N T AC  ( S mi t h - Kl i n e B ee ch a m)  ma y  n e ed  t o  b e  t a k en  
o n l y  t wi ce a  d a y .  T h e ma t h e ma t i ca l  m o d el  fo r  t h i s si t u a t i o n  i s sl i g h t l y  
d i f f er e n t  fr o m I VP  ( 7 ) .   T h er e i s n o  me d i ca t i o n  i n  t h e G I  t r a ct  a t  t h e st a r t  o f  
t h e co n t i n u o u s d o sa g e  r eg i m en ,  b u t  i n st ea d  t h er e i s a  co n st a n t  i n fl o w  r a t e I  
d u e t o  t h e g r a d u a l l y  d i s s o l v i n g  b ea d s.  
Cont i n u ous- Act i ng  Cap sul es:  M od el  IVP  and i ts S ol u t i on  
       T h e m o d el  I VP  fo r  t h e a m o u n t s y 1 ( t ) a n d  y 2 ( t )  o f c o n t i n u o u s -a ct i n g  
me d i ca t i o n  i n  t h e GI tract and bloodstream, respectively, is : 

11
1 ykI

dt
dy

−=        ,   0)0(1 =y  

2211
2 ykyk

dt
dy

−=   ,    0)0(2 =y                       ….…    (9) 

Where I is a positive constant modeling the rate of release of each medication into the GI tract 
. 
        The system of IVP (9) solved analytically by Professor Edward  Spitznagel in [1] and he 
found the following solution :  

)1()( 1
1

1
tke

k
I

ty −−=  

])(11)( 21 12
212

2
tktk ekek

kkk
I

ty −− −



−

+=      ………   (10) 



 
 
 

Mathematics - 408 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

Observe that as time increases, the level of medication in the GI tract tends to I / k1 and that in 
the bloodstream to I / k2 . The solution formulas (10) are valid for any positive values of the 
coefficients k1 and k2 , k1 ≠ k2.  
Now, we solve system (9) where k1 = 1.386 , k2 = 0.1386 and I = 12 , using FFNN . 
Table (6) gives the analytic and neural solutions with different training algorithm such as: 
Levenberg–Marquardt (trainlm), and quasi–Newton (trainbfg) and its errors given in table (7), 
table(8) gives the weight and bias of the designer network and table(9) gives the performance 
of the train with epoch and time . 
 
          Suppose that the input rate I is taken to be one unit of each medication per hour. Thus, a 
total dose of twenty-four units is administered continuously over a period of one day. Figures 
(2) show that, if the values of k1 and k2 from Table (1) are used, the decongestant builds up 
quite rapidly to its equilibrium level in the blood.  
Repeated Doses of Fast-Dissolving Pills 
        Many cold pills dissolve fairly quickly in the GI tract, releasing their medications over a 
period of no more than half an hour. Repeated doses must then be taken every four to six 
hours in order to keep up the medication levels in the blood. In this case, the input rate I into 
the GI tract is not constant, but operates at a high level for a short time, then cuts off entirely 
until the next dose is taken four to six hours later. 
One Dose livery Six Hours: The Model IVP 
        In contrast to a continuous acting medication, other forms of medications dissolve very 
rapidly in the GI tract. This means that the medication is delivered at a high constant rate but 
only over a short time. Suppose that the rate is 12 units / hour, but delivered in half an hour 
and then repeated at six-hour intervals. The model IVP is given below: 

,)( 11
1 yktI

dt
dy

−=         0)0(1 =y   

,2211
2 ykyk

dt
dy

−=      0)0(2 =y  

I(t) = 12 sqw ( t , 12 / 100 , 6 ) 
where sqw ( t , 12/100, 6 ) denotes the pulse function with period 6 hours, which is "on" for 
half an hour at the start of each period and " off " otherwise ( 1/2 hour = 12/100 % of the 6-
hour period ). See [7] for a description of sqw and other piecewise continuous functions.       
 
         The total dosage of 24 units in one day is the same as in the continuous- acting model 
.Figures 2 display the medication levels corresponding to this repeated on-off dosage pattern 
over a forty – eight hour period if values of k1 and k2 are taken from Table 1, using FFNN. 
Medication levels rise during the half hour the cold pill is dissolving in the GI tract, but then 
fall until the next dose. The amount of decongestant in the blood quickly reaches and then 
oscillates around an equilibrium level.  
 
 
 
References 
1. Robert L. B. and Courtney, S. C. (1996) Differential Equations A Modeling perspective; 
United States of America. 
2.  Galushkin I. A.( 2007) "Neural Networks Theory", Berlin Heidelberg. 



 
 
 

Mathematics - 409 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

3. Tawfiq, L.N.M. and Naoum, R.S. (2007) " Density and approximation by  using feed 
forward Artificial neural networks ", Ibn Al-Haitham Journal for Pure & Applied Sciences, 
Vol. 20 (1).  
4. Jabber A. K. (2009)," On Training Feed Forward Neural Networks for Approximation 
Problem ", MSc Thesis, Baghdad University, College of Education (Ibn Al-Haitham). 
5. Tawfiq, L. N. M. and Oraibi, Y. A.( 2012) Improve Levenberg-Marquardt Training 
Algorithm For Feed Forward Neural Networks, Second Conference on Computer and 
Information Technology/College of Computer – University of Anbar.   
6. Stanevski, N. and Tsvetkov, D. (2004) "On the Quasi-Newton Training Method for Feed-    
Forward Neural Networks", International Conference on Computer System and Technologies. 
7. Abd-Al-Razak H. A. (2010) Solution of a System of Ordinary Initial Value Problems using 
Semi-Analytic Method, MSc Thesis, Baghdad University, College of Education ( Ibn Al - 
Haitham ). 
 
 

Tabel (1) :  Rate Coefficients of D e c o n g e s t a n t  for Cold Medications 
k 1        1 . 3 8 6  h o u r 1−  

k 2        0 . 1 3 8 6  h o u r 1−  
 

Table (2):The analytic & neural result of the problem (7) 
Neural solution 
(Trainbfg )(y1t2) 

 

Neural solution 
(Trainlm) (y1t1) 

 

)y1a ( Analytic 
solution x 

0.999837751002654 1 1 0 
0.870576189297933 0.870645179547517 0.870576189297161 0.1 
0.757902901371998 0.757902901371166 0.757902901371166 0.2 
0.659810076684370 0.659812219732971 0.659812219732971 0.3 
0.574416254623237 0.574424576246922 0.574416807906831 0.4 
0.500073595698388 0.500073595695768 0.500073595695768 0.5 
0.435352165307395 0.435344727835639 0.435352165308950 0.6 
0.379007285987416 0.379002100324034 0.379007229076934 0.7 
0.329954669210264 0.329954669205873 0.329954669205873 0.8 
0.287250678555912 0.287250678558054 0.287250678558054 0.9 
0.250073601113749 0.250073601112094 0.250073601112094 1 

 
Neural solution 
(Trainbfg) (y2t2) 

Neural  solution 
(Trainlm) (y2t1) 

)y2a( Analytic solution x 

-3.059617923823967e-12 1.530880578316805e-
16 

0 0 

0.128510464758401 0.128419804302230 0.128510464762339 0.1 
0.238627462671385 0.238604066374236 0.238619746993297 0.2 
0.332737486328395 0.332733745257234 0.332733745257234 0.3 
0.412946642432535 0.412945914088550 0.412946642430978 0.4 
0.481081253754288 0.481081253748791 0.481081253748791 0.5 
0.538724152783951 0.538724152786095 0.538724152786095 0.6 
0.587254349184960 0.587256251241918 0.587256251241918 0.7 
0.627876612156372 0.627880263180820 0.627879420907607 0.8 
0.661639670057443 0.661639670057993 0.661639670057994 0.9 
0.689447320203182 0.689447320205629 0.689447320205629 1 



 
 
 

Mathematics - 410 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

        Table (3) : A comparison between analytic and neural solutions 
| y1a – y1t1  | | y2a – y2t1  | | y1a – y1t2 | | y2a – y2t2 | 

0 1.530880578316805e-16 1.622489973456265e-04 3.059617923823967e-12 
6.899025035600470e-05 9.066046010927642e-05 7.728262474415715e-13 3.937739023740505e-12 

0 1.568061906176421e-05 8.324452238639424e-13 7.715678088132671e-06 
0 5.551115123125783e-17 2.143048601244146e-06 3.741071161966758e-06 

7.768340090974490e-06 7.283424278692330e-07 5.532835937183123e-07 1.557254325490476e-12 
1.110223024625157e-16 0 2.620459405022757e-12 5.497102772977769e-12 
7.437473311000531e-06 1.110223024625157e-16 1.555866546709694e-12 2.144506794365952e-12 
5.128752899485889e-06 2.220446049250313e-16 5.691048232669971e-08 1.902056958269505e-06 

0 8.422732126200927e-07 4.391098595846188e-12 2.808751235261475e-06 
0 1.110223024625157e-16 2.141509192199465e-12 5.505595979116151e-13 

5.551115123125783e-17 1.110223024625157e-16 1.654731907052565e-12 2.447597680088620e-12 
 
Table (4): Weight and bias of the network for different training algorithm   

Weight for trainlm of (y1) Weight for trainlm 
of (y1) 

Bias for trainlm of 
(y1) 

Net.IW{1,1} Net. LW{2,1} Net.B{1} 
0.241569248284151 0.017611616453924 0.252859672695871 
0.351195035228651 0.681708279346507 0.024385651630710 
0.126738675949935 0.880833084532577 0.530723581584766 
0.739408639260099 0.139821021807629 0.934854674058909 
0.445679823858157 0.810853784598876 0.769328503579249 

 

Weight for trainbfg of (y1) Weight for trainbfg of (y1) bias for trainbfg of (y1) 
Net.IW{1,1} Net. LW{2,1} Net.B{1} 

0.715931252353248 0.670860771332138 0.413824842108441 
0.248146352936429 0.982859184594668 0.180845024472233 
0.531900325168972 0.936821328628590 0.995631187232624 
0.382209682936579 0.576272354062037 0.520386194349656 
0.801761242973036 0.080186187030087 0.885282695498373 

 

Weight for trainbfg of (y2) Weight for trainbfg of (y2) Bias for trainbfg of (y2) 
Net.IW{1,1} Net. LW{2,1} Net.B{1} 

0.869492229395656 0.688725578355128 0.431049305282942 
0.947906559807061 0.596021891944303 0.190738233247638 
0.254304842122024 0.214739790766330 0.327775459738820 
0.048491632422091 0.535169282982425 0.937657078181142 
0.831838695092381 0.055755557082901 0.847367010662909 

 

bias for trainlm of (y2) Weight for trainlm of 
(y2) 

Weight for trainlm of (y2) 

Net.B{1} Net. LW{2,1} Net.IW{1,1} 
0.241305720471971 0.618786526718974 0.101305687973029 
0.594729968864710 0.699150113317535 0.468935089858187 
0.868232914442518 0.314828670017582 0.710646362418379 
0.818668373595483 0.804788645898101 0.601346407503426 
0.590084132290161 0.178022430412338 0.204916976156494 

 
 
 

 
 



 
 
 

Mathematics - 411 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

         Table(5) : The performance of the train with epoch and time  
Performance Epoch Time Performance for MSE Train code 

2.20e-33 295 0:00:04 4.010417891598336e-10 Trainlm of y1 
3.96e-33 468 0:00:06 6.927087884925925e-10 Trainlm of y2 
5.07e-28 1268 0:00:28 2.154243203877806e-09 Trainbfg of y1 
2.26e-25 543 0:00:12 6.957344125042582e-12 Trainbfg of y2 

 
Table (6) : The analytic & neural result of the problem (9) 

Trainbfg )(y1t2)( Neural 
solution 

Trainlm) (y1t1)(Neural 
solution 

)ya1 (  Analytic solution        x 

0.008076966501195 0 0 0 
1.121576309950494 1.119074768489855 1.120552473617658 0.1 
2.096078776010794 2.096078776007224 2.096078776007225 0.2 
2.945329004027621 2.945519665978265 2.945348746900685 0.3 
3.684702961851884 3.684702961845623 3.684702961845622 0.4 
4.328367136844937 4.328367136833180 4.328367136833180 0.5 
4.888725841491276 4.888755428434839 4.888725841480949 0.6 
5.376560787208372 5.376560787212696 5.376560787212696 0.7 
5.801258275281827 5.801258275273830 5.801258275273828 0.8 
6.170989796037555 6.171052575520006 6.170989796034165 0.9 
6.492853256454569 6.492869254440744 6.492869254440744 1 

 

Trainbfg) (y2t2)( Neural 
solution 

Trainlm) (y2t1)( Neural 
solution 

 )2ya( Analytic solution x 

6.970449786967482e-08 0 0 0 
0.079077614840971 0.083579667139451 0.079077570606103 0.1 
0.296617057945247 0.301069405674204 0.301069405674204 0.2 
0.643383165800754 0.645370996519114 0.645370996519114 0.3 
1.094073602030272 1.094073563826114 1.094073563826115 0.4 
1.631757503744028 1.629619175798611 1.631614766704876 0.5 
2.244474633309778 2.242721178439358 2.244474623805575 0.6 
2.920910793699243 2.920910794904590 2.920910794904590 0.7 
3.650728106463634 3.650322736313918 3.650728128703034 0.8 
4.425158477200711 4.425078041900662 4.425078041900662 0.9 
5.236283818105677 5.236283868595361 5.236283868595361 1 

 
Table (7) : A comparison between analytic and neural solutions 

| y1a – y1t1  | | y2a – y2t1  | | y1a – y1t2 | | y2a – y2t2 | 
0 0 0.008076966501195 6.970449786967482e-08 

0.001477705127803 0.004502096533347 0.001023836332836 4.423486724136616e-08 
4.440892098500626e-16 6.106226635438361e-16 3.569144979564953e-12 0.004452347728957 
1.709190775800451e-04 0 1.974287306349964e-05 0.001987830718360 
4.440892098500626e-16 6.661338147750939e-16 6.261213769676033e-12 3.820415694910651e-08 

0 0.001995590906265 1.175681774157056e-11 1.427370391517968e-04 
2.958695388954880e-05 0.001753445366216 1.032685048585336e-11 9.504203646315545e-09 
8.881784197001252e-16 0 4.324540725519910e-12 1.205346933375040e-09 
1.776356839400251e-15 4.053923891160771e-04 7.998046669399628e-12 2.223940009926650e-08 
6.277948584099136e-05 0 3.389288849575678e-12 8.043530004897548e-05 

0 0 1.599798617490933e-05 5.048968443333024e-08 
 

 



 
 
 

Mathematics - 412 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

 
Table( 8): Weight and bias of the network for different training algorithm 

Weight for trainlm of 
(y1) 

Weight for trainlm of 
(y1) 

Bias for trainlm of 
(y1) 

Net.IW{1,1} Net. LW{2,1} Net.B{1} 
0.505692448683919 0.792456860497627 0.134776995255468 
0.004625760582352 0.837900449095441 0.569914349014358 
0.188468748383506 0.687358399403697 0.574691506274377 
0.779639443833603 0.081434100056796 0.016102750791896 
0.610825949687488 0.806140569013718 0.934828315776462 

 

bias for trainlm of 
(y2) 

Weight for trainlm of 
(y2) 

Weight for trainlm of 
(y2) 

Net.B{1{ Net. LW{2,1} Net.IW{1,1} 
0.912782147048597 0.922429224841606 0.478376283649921 
0.534170292387813 0.934308461414860 0.124994240255344 
0.451461598784967 0.824811938202454 0.064155985584096 
0.285596533472468 0.183582125058882 0.386664880532908 
0.622774097441083 0.393108093448545 0.558245691279696 

        

Weight for trainbfg of (y2) Weight for trainbfg of (y2) Bias for trainbfg of (y2) 
Net.IW{1,1} Net. LW{2,1} Net.B{1} 

0.018010467945131 0.002291815245818 0.612551084945355 
0.503631782070622 0.772712775811915 0.133072433875949 
0.126991991291519 0.661798871106240 0.283353859003173 
0.305174566418702 0.397966331038198 0.837400333016416 
0.256058502513567 0.501051178487892 0.865931228918039 

 

Weight for trainbfg of (y1) Weight for trainbfg of (y1) Bias for trainbfg of (y1) 
Net.IW{1,1} Net. LW{2,1} Net.B{1} 

0.735194614493405 0.100480242633400 0.828924078623420 
0.879697065220770 0.522668514129408 0.724963349991644 
0.851423388162803 0.058688252904172 0.337293776914633 
0.672029350928386 0.813646068464573 0.211463983876867 
0.798496214678467 0.345685331495202 0.339840122932540 

 
 
Table (9) : The performance of the train with epoch and time  

Train code Performance Epoch Time Performance for MSE  
Trainlm of y1 2.82e-32 302 0:00:05 1.814434713408252e-07 
Trainlm of y2 4.83e-32 102 0:00:02  2.249195719271235e-06 
Trainbfg of y1 2.49e-26 1461 0:00:35 5.423422452091749e-06 
Trainbfg of y2 2.20e-23 243 0:00:05 1.947413043462885e-06 

 
 
 
 
 
 
 
 
 



 
 
 

Mathematics - 413 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

0 2 4 6
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:ex
y2:ex

0 2 4 6
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:approx
y2:approx

 
Fig. (1): decongestant in GI tract(solid)and bloodstream(dashed):single unit dose 
( k1 = 1.386, k2 = 0.1386 ) 

0 20 40 60
0

10

20

30

40

50

60

70

80

90

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:ex
y2:ex

0 20 40 60
0

10

20

30

40

50

60

70

80

90

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:approx
y2:approx

 
Fig. (2): Decongestant in GI tract (solid) and bloodstream (dashed) : continuous 
acting capsules ( k1 = 1.386, k2 = 0.1386 ) , I = 12 . 

0 20 40 60
0

5

10

15

20

25

30

35

40

45

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:ex
y2:ex

0 20 40 60
0

5

10

15

20

25

30

35

40

45

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:approx
y2:approx

 
 

 
 
 
 
 
 



 
 
 

Mathematics - 414 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

0 20 40 60
0

5

10

15

20

25

30

35

40

45

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:ex
y2:ex

0 20 40 60
0

5

10

15

20

25

30

35

40

45

t(hours)

y1,
y2(

dec
ong

est
ant

)

 

 
y1:approx
y2:approx

 
Fig. (3): Decongestant in GI tract (solid) and bloodstream (dashed): continuous  
acting capsules ( k1 = 1.386 , k2 = 0.1386 ) , I = 6  
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



 
 
 

Mathematics - 415 
 

 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية

 2012 السنة 25 المجلد 3 العدد

Ibn Al-Haitham Journal for Pure and Applied Science  

 No. 3 Vol. 25 Year 2012 

تصميم شبكة عصبية صناعية ذو تغذية تقدمية لتحديد جرعات دواء 
 الديكونجيستد

 
 

 لمى ناجي محمد توفيق  و  يسر رافد بحر المعيوف
 ، جامعة بغدادابن الهيثم -كلية التربية قسم الرياضيات،
 2012ايلول   24البحث في : بل ق   2012 ايار  30استلم البحث في:

 
 الخالصة

الهدف من هذا البحث هو تصميم شبكة عصبية صناعية ذو تغذية تقدمية لتحديد تأثير حبوب البرد و تعاقبها من خالل      
استخدام النمذجة و تحويل المشكلة إلى منظومة معادالت  مسائل القيم االبتدائية من الرتبة األولى .هذه المسألة مثال لعدد من 

سم. االنموذج المصمم هو جزء مهم في العالج والذي يحدد مستوى الجرعات و االج النماذج التي تمثل مرور العالج خالل
 خطر       معدلها حيث ان التحليل الحرج هو الحاجة لبقاء مستوى العالج عال و فعال لكن ليس العلو الذي يجعله

 الشبكة العصبية، الشبكة العصبية التقدمية،استخدام النمذجة .الكلمات المفتاحية:. 
 
 
 
 

 
 
 


	Received in: 30  May  2012  Accepted in: 24   September   2012