ةكلمملا ةيبرعلا
ةيدوعسلا
ةرازو ميلعتلا
يلاعلا
ةيلك مولعلا تانبلل
مسق تايضايرلا
A New Modification of the Method of Lines for a First Order Hyperbolic
Differential Equation
F.M.Alabdli
Supervisor
Dr.Huda Omar Bakodah
1435/1434
ليدعت ديدج
ةقيرطل طوطخلا
يف لح
ةلداعم ةيدئاز
ةيلضافت نم
ةبترلا ىلولأا
The general formula for first order linear equation of two independent variables, is as follows :
is constant, represent time and is distance .
it is appear in many physical and chemical applications, as well as in the biological processes .
t and x
.1 First Order Hyperbolic Differential Equation
1) 0 (
t
x vu
u
v t x
2
. Method of Lines approximations
Suppose we have the following equation
the conditions are
when using the method of lines to solve this problem, we discretize one of the two independent variables and the other remains continuous .
(discretization of the -variable)
2) 0 (
x u t
u
0 ) 0, ( , ) ( 0)
,
(x f x u t u
x
ih n x
a
h b , i
(I) If we approximate the spatial derivative x by the central differences as follows:
substituting 3 into 2 , we get the following problem:
putting we get in the following system
4) ( ,....,
2 1, 2 ,
1
1 i n
h u
ui ui i
r h 2
1
2 1
1 3
2
0 2
1
. .
5) ( .
n n
n ru ru
u
ru ru
u
ru ru
u
3) 2 (
1 1
h u u
x
ui i i
equation (5) are a system of first order ordinary differential equations that can be solved by using numerical
methods such as Euler or Runge-Kutta.
For studying stability by using matrix method ,expressed in a form of matrix as follows:
6)
1u ( A u
(
II) when using non-central differences formula to approximate the spatial derivative
putting we get in the following system
for studying stability by using matrix method ,expressed in
a form of matrix as follows:
7) ( 4 ]
3 2 [
1
1
1
i i
i u u
h u x
u
9)
2u ( A u
4 ] 3
[ .
.
8) ( .
4 ] 3
[
4 ] 3
[
1 1
1
1 2
3 2
0 1
2 1
i i n
n r u u u
u
u u
u r u
u u
u r u
r h 2
1
Example (1): linear case
Consider the advection equation with the conditions
if we calculate the error by using and error and it is defined
as follows :
4
. Numerical examples
0
x
t u
u
, 0 sin
) 0, (
1 , 0
sin 0)
, (
t t t
u
x x
x u
a i e
i i
a i n
i
e i
u u
L
u u
h L
max
]
[ 0.5
0 2
L2 L
Table 1: and error for example 1
0.00101545 0.000579369 0.01 0.00233093 0.00118751 0.02 0.00397634 0.00184354 0.03
0.00598185 0.0025641 0.04
0.00837779 0.00337482 0.05
0.0111947 0.00428697 0.06
0.0144633 0.0053202049 0.07
0.0182144 0.00649227 0.08
0.0224789 0.00781859 0.09
0.0272878 0.00931508 0.10
L
L2
t
01 0. 1 ,
0.
x t
L2 L
Example (2): nonlinear case
Consider the advection equation with the condition
and the exact solution
0
x
t uu
u
x x
u( ,0)
t t x
x
u
) 1 , (
2.56842
1.52351 0.1
3.65622
2.16876 0.2
4.02893
2.38985 0.3
4.05249
2.40383 0.4
3.90807
2.31851 0.5
3.6900
2.1888 0.6
3.44423
2.04302 0.7
3.19589
1.89572 0.8
2.95723
1.77415 0.9
2.73389
1.62166 1.0
L
L2
t
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
01 0. 1 ,
0.
x t
L2 L
In this paper we use modified method of the lines to approximate
the first order hyperbolic differential equation . These equations
are one of the most difficult class of PDEs to integrate
numerically. To overcome this, we will suggest a modified MOL
scheme. the results are in good agreement with the exact solution
as shown in tables(1,2,3). The presented method is a
5
. Conclusion
G.D.Smith, Numerical Solution of Partial Differential Equations(Finite Difference Methods),Third Edition, Oxford University Press, 1985
.
[ 1
]
W.E.Schiesser, "The Numerical Method of Lines,Integration of partial differential , , Ed ,Clarendon press Oxford , 1978
.
[ 2
]
A.A.Sharaf and H.O.Bakodah, "A Good Spatial Discretization in the Method of , Applied Mathematics and Computation,vol.
171-2 , PP.
1253-
(1263 2005
) .
[ 3
]
M.B.Carver and H.W.Hinds, The Method of lines and Advective Equation ,Simulation,PP.
59-69 ,August,(
1978 )
.
[ 4
]
References
equationn 2nd
linesn
