Errors in the Discretized Solution of a Differential Equation

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 3 Issue 5, August 2019 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
@ IJTSRD | Unique Paper ID – IJTSRD27937 | Volume – 3 | Issue – 5 | July - August 2019 Page 2266
Errors in the Discretized Solution of a Differential Equation
Wai Mar Lwin1, Khaing Khaing Wai2
1Faculty of Computing, 2Department of Information Technology Support and Maintenance
1,2University of Computer Studies, Mandalay, Myanmar
How to cite this paper: Wai Mar Lwin |
Khaing Khaing Wai "Errors in the
Discretized Solution of a Differential
Equation" Published
in International
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Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-3 |
Issue-5, August
2019, pp.2266-2272,
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ABSTRACT
We study the error in the derivatives of an unknown function. We construct
the discretized problem. The local truncation and global errors are discussed.
The solution of discretized problem is constructed. The analytical and
discretized solutions are compared. The two solution graphs aredescribed by
using MATLAB software.
KEYWORDS: Differential Equations, MATLAB, Heat Equation
1. INTRODUCTION
1.1. Measuring Errors
In order to discuss the accuracy of a numerical solution, it is necessary to
choose a manner of measuring that error.Itmayseemobviouswhatis meant by
the error, but as we will see there are often many different ways tomeasurethe
error which can sometimes gives quite different impressions as totheaccuracy
of an approximate solution.
1.1.1. Errors in a Scalar Value
First we consider a problem in which the answer is a single value Rz .
Consider, for example, the scalar ODE
 )0()),(()( utuftu (1.1)
and suppose we aretrying to compute the solution at someparticular time T, so
z=u( T). Denote the computed solution by
zˆ . Then the error in this computed solution is
zzE  ˆ (1.2)
1.1.2. Absolute Error
A natural measure of this error would be the absolute value
of E,
zzE  ˆ (1.3)
This is called the absolute error in the approximation.
1.1.3. Relative Error
The error defined by
z
zz ˆ is called relative error.
1.1.4. “Big-oh” and “little-oh” notation
In discussing the rate of convergence of a numerical method
we use the notation )( p
O  , the so-called “big-oh”notation.If
)(f and )(g are two functions of  then we say that
))(()(  gOf  as 0 .
If there is some constant C such that C
g
f

)(
)(

 for all 
sufficiently small, or equivalently, if we can bound
)()(  gCf  for all  sufficiently small.
It is also sometimes convenient to use the “little-oh”
notation ))(()(  gOf  as 0 . This means that 0
)(
)(



g
f as
0 . This is slightly stronger than the previous statement,
and means that )(f decays to zero faster than )(g . If
))(()(  gof  then ))(()(  gOf  though theconversemay
not be true. Saying that )1()( of  simply means that the
0)( f as 0 .
Examples 1.1.1
)(2 23
 O as 0 , since 12
2
2
3
 

 for all
2
1
 .
)(2 23
 o as 0 , since 02  for all
2
1
 .
)()sin(  O as 0 ,since 

  ...
53
)sin(
53
for all
0 .
)()sin(  o as 0 , since   )(
)sin( 2



O
 .
1.1.5. Taylor Expansion
Each of the function values of u can be expanded in a Taylor
series about the point x, as e.g.,
)()(
6
1
)(
2
1
)()()( 432
 Oxuxuxuxuxu  (1.4)
)()(
6
1
)(
2
1
)()()( 432
 Oxuxuxuxuxu  (1.5)
1.2. Finite Difference Approximations
Our goal is to approximate solutions to differential equation,
i.e. to find a function (or some discrete approximation to this
function) which satisfies a given relationship between
IJTSRD27937

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
various of its derivatives on some given region of space and/
or time along with someboundaryconditionsalongtheedges
of this domain. A finite difference method proceeds by
replacing thederivativesinthedifferentialequationsbyfinite
difference approximations. This gives a large algebraic
system of equations to be solve in place of the differential
equation, something that is easily solved on a computer.
We first consider the more basic question of how we can
approximate the derivatives of a known function by finite
differenceformulas based only onvaluesofthefunctionitself
at discrete points. Besides providing a basis for the later
development of finite difference methods for solving
differential equations, this allows us to investigate several
key concepts such as the order of accuracy of an
approximation in the simplest possible setting.
Let u(x) represent a function of one variable that will always
be assumed to be smooth, defined bounded function over an
interval containing a particular point of interest x,
1.2.1. First Derivatives
Suppose wewant to approximate )(xu by a finitedifference
approximation based only onvalues of u at a finitenumberof
points near x. One choice would be to use



)()(
)(
xuxu
xu


)()(
2
1
)( 2
 Oxuxu  (1.6)
for some small values of . It is known as forward
difference approximation.
Another one-sided approximation would be



)()(
)(


xuxu
xu
)()(
2
1
)( 2
 Oxuxu  (1.7)
for two different points.It is known as backward difference
approximation.
Another possibility is to use the centered difference
approximation



2
)()(
)(0


xuxu
xu
)]()([
2
1
xuxu   
)(
6
1
)( 42
 Ouxu  (1.8)
1.2.2. Second Order Derivatives
The standard second order centered approximation is
given by
2
2 )()(2)(
)(





xuxuxu
xu (1.9)
)()(
12
1
)( 42
 Oxuxu 
)()( 2
Oxu 
1.2.3. Higher Order Derivatives
Finite difference approximations to higher order derivatives
can be obtained.
)]()(3)(3)2([
1
)( 3
2


  xuxuxuxuxu (1.10)
)()(
2
1
)( 2
 Oxuxu 
)()( Oxu 
The first one equation (1.10) is un-centered and first order
accurate:
)]2()(2)(2)2([
2
1
)( 30 

  xuxuxuxuxu (1.11)
)()(
4
1
)( 42
 Oxuxu 
)()( 2
Oxu 
This second equation (1.12) is second order accurate.
2. Comparison Of Analytical and Discretized Solution
Of Heat Equation
2.1 Solutions for the Heat Equation
2.1.1. Finite Difference Method
We will derive a finite difference approximation of the
following initial boundary value problem:
xxt uu  for 0),1,0(  tx
0),1(),0(  tutu for 0t (2.1)
)()0,( xfxu  for )1,0(x
Let 0n be a given integer, and define the grid spacinginthe
x-direction by
)1(
1


n
x 
The grid points in the x-direction are given by
j=xj for j=0,1,…,n+1. Similarly, we define tm=mh for
integers 0m , where th  denotes the time step. Then, we
let
m
jv denote an approximation of u(xj,tm). We have the
following
approximations
)(
),(),(
),( hO
h
txuhtxu
txut 

 (2.2)
and
)(
),(),(2),(
),( 2
2



O
txutxutxu
txuxx 

 (2.3)
These approximations motivate the following scheme:

2
11
1
2

m
j
m
j
m
j
m
j
m
j vvv
h
vv 




for j=1,…,n, 0m (2.4)
By using the boundary conditions of (2.1), we have
00 m
v and 01 
m
nv , for all 0m .
The scheme is initialized by
)(0
jj xfv 
, for j=1,…,n.
Let
2

h
r 
. Then the scheme can be rewritten in a more
convenient form
m
j
m
j
m
j
m
j rvvrrvv 11
1
)21( 


, j=1,…,n, 0m (2.5)
When the scheme is written in this form, we observe that the
values on the time level tm+1 are computed using only the
values on the previous time level tm and we have to solve a
tridiagonal system of linear equations.
2.1.2 Approximate Solution
The first step in our discretized problem is to derive a family
of particular solutions of the following problem:
2
11
1
2

m
j
m
j
m
j
m
j
m
j vvv
h
vv 




for j=1,…,n, 0m (2.6)
with the boundary conditions
00 m
v
and
01 
m
nv
, for all 0m . (2.7)
The initial data will be taken into account later. We seek
particular solutions of the form
mj
m
j TXv 
for j=1,…,n, 0m (2.8)
Here X is a vector of n components, independent of m, while
0}{ mmT is a sequence of real numbers. By inserting (2.8)
into (2.6), we get
2
111 2

mjmjmjmjmj TXTXTX
h
TXTX  


Since we are looking only for nonzero solutions, we assume
that
0mjTX
, and thus we obtain
j
jjj
m
mm
X
XXX
hT
TT
2
111
2





.
The left-hand side only depends on mandtheright-handside
only depends on j. Consequently, both expressions must be
equal to a common constant, say )(  , and we get the
following two difference equations:
j
jjj
X
XXX



 
2
11 2
, for j=1,…,n, (2.9)
m
mm
T
h
TT

1
for 0m . (2.10)
We also derive from the boundary condition (2.7) that
010  nXX (2.11)
We first consider the equation (2.10). We define T0=1 and
consider the difference equation
mm ThT )1(1  for 0m . (2.12)
Some iterations of (2.12)
mm ThT )1(1 
1
2
)1(  mTh
.
.
.
0
1
)1( Th m
 
1
)1( 
 m
h
Clearly indicate that the solution is
m
m hT )1(  for 0m . (2.13)
This fact is easily verified by induction on m. Next we turn
our attention to the problem (2.9) with boundary condition
(2.11). In fact this is equivalent to the eigenvalue problem.
Hence, weobtain thattheneigenvalues n ,...,, 21 aregiven
by
)cos(
22
22


 k
))cos(1(
2
2



)
2
(sin2
2 2
2



)
2
(sin2
4 2
2


 k
for n,...,1 (2.14)
and the corresponding eigenvectors
,),...,,( ,2,1,
n
nk RXXXX   n,...,1
have components given by
)sin(,  jX j 
)sin(, jj jxX  
, j=1,…,n.
It can be easily verified that
1,2,21,2,
121
 

 jjjj XXXAX 

))1(sin(
1
)sin(
2
))1(sin(
1
222






 jjj
)sin(
1
)sin(
2
))sin(
1
222






 jjj









)sin()cos()cos()sin(
)sin(2)sin()cos()cos()sin(1
2


 jj
jjj
))cos(1)(sin(
2
2


 j
))
2
(sin2)(sin(
2 2
2



j

)sin()
2
(sin
4 2
2



j
jX ,
Hence, we obtain particular solutions
m
jv ,
of the form
)sin()1(, j
mm
j xhv  
(2.15)
Wehavederivedafamilyofparticularsolutions
n
v 1}{  with
values
m
jv ,
at the grid point (xj,tm). Next, we observe that
any linear combination of particular solutions


n
vv
1

, (  is scalar)
is also a solution of (2.6) and (2.7). Finally, we determine the
coefficients {  } by using the initial condition
)(0
jj xfv 
, for j=1,…,n.
Since  Xv  at t=0 , we want to determine {  } such that
)(
1
, j
n
j xfX 

, for j=1,…,n. (2.16)
Hence, it follows from


n
jj xxf
1
)sin()(

 
h
n
jjhjsj xsxxfX 

1
sin,sin)(,

 
that


n
j
jj Xxf
1
,)(2  
for n,...,1
2.1.3. Exact Solution
To find a solution of the initial-boundaryvalueproblem(2.1),
assume that
)()(),( tTxXtxu  (2.17)
Using boundary conditions we get
X(0) = X(1) = 0
If we insert the (2.17) in the equation (2.1), we have
)(
)(
)(
)(
tT
tT
xX
xX 


(2.18)
Now we observe that the left hand side is a function of x ,
while the right hand side just depends on t. Hence, both have
to be equal to the same constant R  . This yields the
eigenvalue problem for x
XX  , X(0) = X(1) = 0 (2.19)
Nontrivial solutions only exist for special values of 
They are so-called the eigenfunctions. In this special case we
have the eigenvalues
2
)(  for ,.....2,1 (2.20)
with the eigenfunctions
)sin()( xxX   for ,.....2,1 (2.21)
Further,the solutionof TT 
isgivenby
t
etT
2
)(
)( 


 for
,.....2,1 (2.22)
Finally, weuse the superpositionprincipletoasolutiontothe
initial-boundary value problem (2.1), and we get




1
)()(),(

 xXtTctxu
)sin(),(
1
)( 2
xectxu t








The unknown coefficients can be determined from the initial
condition such that
)sin()(
1
xcxf 





(2.23)
The solution of the continuous problem is given by
)sin(),(
1
xectxu t










(2.24)
where
2
)(  fourier coefficient

1
0
)sin()(2 dxxxfc 
(2.25)
2.2. Comparison of Analytical and DiscretizedSolution
2.2.1. We want to comparethisanalyticalsolutionwith
the discretized solution given by


n
j
mm
j xhv
1
)sin()1(

 
(2.26)
where







2
sin
4 2
2



(2.27)
and


n
j
jj xxf
1
)sin()(2  
for n,...,1 (2.28)
In order to comparethe analytical and discretizedsolutionat
a grid point (xj,tm), we define
),( mj
m
j txuu 
, i.e,
)sin(
1
j
tm
j xecu m










(2.29)
Our aim is to prove
m
j
m
j uv 
under appropriateconditionsonthemeshparameters  and
h. To avoid technicalities, we consider a fixed grid point
(xj,tm) where
ttm  for t >0 independent of the mesh

parameters. Furthermore, we assumethattheinitialfunction
f is smooth and satisfies the boundary conditions, i.e. f(0)
=f(1) =0. Finally we assume that the mesh parameters h and
 are sufficiently small.
In order to compare
m
ju
and
m
jv
, we note that
)sin(),(
1
j
t
mj xectxu m










)sin()sin(
11
j
n
t
j
n
t
xecxec mm















Here we want to show that
0)sin(
1




j
n
t
xec m





(2.30)
Since f is smooth, it is also bounded and then the Fourier
coefficients c are bounded for all  .
Obviously, we have
1)sin( jx
and
)sin()sin(
11
j
n
t
j
n
t
xecxec mm
















1





1
)( 2
max
n
tm
ec




 




1
,
2
n
m
t
tteC



    


 




...
21 22 n
t
n
t
eeC 
   ...1
222
2
1
 

 tt
n
t
eeeC 
  t
n
t
e
eC 2
2
1
11







0 , for large value of n.
Since we have verified (2.30) it follow that
)sin(
1
j
n
tm
j xecu m








(2.31)
Now we want to compare the finite sums (2.26) and (2.31):


n
j
mm
j xhv
1
)sin()1(

 
Motivated by the derivation of the solutions, we try to
compare the two sums termwise. Thus wekeep  fixed,and
we want to compare
)sin( j
t
xec m



and
)sin()1( j
m
xh   
.
Sincethe sine part here is identical, itremainstocomparethe
Fourier coefficients c and  , andthetime-dependentterms
mt
e 
and
m
h )1(  .
2.2.2. Comparison of Fourier coefficient c and
coefficient 
Westart by consideringtheFouriercoefficients,andnotethat
 is a good approximation of c because

n
j
jj xxf
1
)sin()(2 
is the trapezoidal-rule approximation of

1
0
)sin()(2 dxxxf 
In fact we have
 

1
0 1
)sin()(2)sin()(2
n
j
jj xxfdxxxfc 
   
 
n
j
n
j
jjjjj xxfxxfxf
1 1
1 )sin()(2)sin()()(
2
2 

 

n
j
jjj xxfxf
1
)sin()()( 








n
j
jjjjj xxfxfxfxf
1
2
)sin()(...)(
2
)()( 


)sin(...)(
2
)(
1
3
2
j
n
j
jj xxfxf 

















n
j
jj xfxf
1
3
2
...)(
2
)(


)( 2
O , for f sufficiently smooth.
2.2.3 Comparison of the Terms
m
h )1(  and
mt
e 
We will compare the term
m
h )1(  approximates the term
mt
e 
, we simplify the problem a bit by choosing a fixed time
tm , say tm=1 and we assume that 2
2

h
. As a consequence,
we want to compare the terms

 
 e (2.32)
and
h
h
1
)1(    (2.33)
Since both  and  are very small for large values of  , it
is sufficient to compare them for small  . In order to
compare  and  for small values of  , we start by
recalling that
)()sin( 3
yOyy 
Thus, we get
2
2
5
5
2
3
3
2
...
!5
2
)(
!3
2
)(
2
2
2
sin2




































hh
hh



 
22
42
2
...
!5
2
)(
!3
2
)(
1
2
2



























hh
h


 
2
2
2 h

, for h sufficiently small
h22

By using these facts, we derive
h
h
1
)1(   
h
h
1
2
2
)
2
(sin
4
1 








h
h
1
2
2
sin21
















 
h
h
1
22
)1(  
h
hh
hh
h
h
1
2222222
)(...)(
!2
1
11
)(
1
1  








h
h
h
1
2222222
)(...)(
2
1
1  


22

 e , for h sufficiently small

 e
  
This shows that also the time dependent term
mt
e 
is well
approximated by its discrete
m
h )1(  .
2.3 Consistency
Lemma 2.3.1.
The finite difference scheme (2.6) is consistentoforder(2,1).
Proof:
Local truncation error of the finite difference scheme (2.6) is
given by
 m
j
m
j
m
j
m
j
m
jm
j uuu
h
uu
l 112
1
2
1






 m
j
m
j
m
j
m
j
m
j
m
j uuu
h
uuhl 112
1
2 


























...),(
24
),(
6
),(
2
),(),(),(2...),(
24
),(
6
),(
2
),(),(
),(
...),(
6
),(
2
),(),(
432
4
32
2
32
mjxxxxmjxxxmjxx
mjxmjmjmjxxxx
mjxxxmjxxmjxmj
mj
mjtttmjttmjtmj
txutxutxu
txutxutxutxu
txutxutxutxu
h
txu
txu
h
txu
h
txhutxu














...),(
12
),(
...),(
2
),(
4
2
2
2
mjxxxxmjxx
mjttmjt
txutxu
h
txu
h
txhu



...),(
12
),(...),(
2
),(
2
 mjxxxxmjxxmjttmjt
m
j txutxutxu
h
txul

...),(
12
),(
2
),(),(
2
 mjxxxxmjttmjxxmjt txutxu
h
txutxu

...),(
12
),(
2
2
 mjxxxxmjtt txutxu
h 
),( 2
hO 
The finite difference scheme (2.6) is consistent of order
(2,1).
Example 2.3.1
Solve the IVP
xxt uu  for )1,0(x , t > 0
0),1(),0(  tutu for 0t (2.34)
)()0,( xfxu  for )1,0(x .
The exact solution is given by
xeu t

sin
2


Approximate solution and exact solution are
illustrated by 2.1.
Figure2.1: Comparison of Exact and Approximate
Solutions
MATLAB codes for Figure 2.1
N=100 ; x=zeros (N,1) ;
g= zeros (N,1) ;
v=zeros (N,N) ;
%delta x=h, delta t=p
%f (x)=cos (2*pi*x) ;
H=1/(N+1) ;
P=1/(N+1) ;
for j = 1:N
x ( j )=j*h ;
end
%for m=1:N
% t ( m )=m*p ;

%end
%gamma k
for k=1:N
sum=0 ;
for j=1:N
term=2*h*sin ( 2*pi*x ( j ))*sin (k*pi*x ( j ));
sum=sum+term ;
end
g ( k )=sum ;
end
%vk
for k=1 : N
for j = 1 : N
v ( k , j )=p*(4/h^2)*(sin(k*pi*h/2))^2)*sin(k*pi*x(j)) ;
end
v ( k , : ) ;
end
sum 1 =0 ;
for k= 1 : N
term1=g ( k ) . *v ( k , :) ;
sum1=sum1+term1 ;
s =sum1 ;
end
%%%Approximate solution
plot ( x , s , ‘ – ‘) ; hold on
%%%%%%Exact solution
Plot ( x ,-exp (-*pi^2*0.01 )*sin(pi*x) ) ;
3. CONCLUSION
The aim of this research paper describe the Errors between
Analytical solution and Discretized solution of Differential
equations.
4. REFERENCES
[1] R. J. LeVeque, Finite difference methods for differential
equations, Washington, 2006.
[2] J. W. Thomas, Numerical partial differential equation:
Finite difference methods, Springer-Verlag, New York,
1995.

Errors in the Discretized Solution of a Differential Equation

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