I nternat. J. Math. & Math. Si.
Vol. 4 No. 4 (1981) 775-794
775
FAST METHODS FOR THE SOLUTION OF SINGULAR
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
L.F. ABD-ELAL
Department of Mathematics
Faculty of Science
Cairo University
Egypt
Giza
(Received July 3, 1979)
ABSTRACT.
Uniform methods based on the use of the Galerkin method and different
Chebyshev expansion sets are developed for the numerical solution of linear integrodifferential equations of the first order.
time
O(N21n
These methods take a total solution
N) using N expansion functions, and
are cheap to compute.
ential equations.
a_]so provide error extimates which
These methods solve both singular and regular integro-differ-
The methods are also used in solving differential equations.
KEY WORDS AND PHRASES. Integro-drfferential equations, numerical solution,
Galerkrn method, Cebyshev expansion.
1980 ATEoIATICS SUBJECT CLASSIFICATION CODES.
i.
45J05.
INTRODUCTION.
consider the Galerkin solution for those integro-differential equations of
the first order having the form
Lf(x)
g(x)
subject to the boundary condition f(a)
x E [-i,i]
,
I
a
E [-i,i].
dy
(x,y
(i.I)
1
L
Z {Pk(X)-dxk
k=O
1
where f and g are elements of a Hilbert space H
I
I
and
L:
H
-
H is linear.
L. F. ABD-ELAL
776
We assume that
(x,y)
and
Pk(X)
are either regular or have singularities
provided that the singularities are of known and standard form
For a given Chebyshev expansion set
weak or logarithmic singularities.
{ho(x)}
1
like for example,
H, the solution f(x) defines approximations
c
N
Z
i=0
N (x)
a
(N)
where L
is the
.
L(N)
h.l (x)
[I],
Using Galerkin technique given in Mikhlin
equations
(N)
i
we modify the linear system of
_a(N)
g(N)
(1.3)
(N + 1 x N + 1) leading minor of the matrix L with elements
1
L..13
dx T
i(x)
Lhj(x)//l-x p
N
0,i,
i, j
(1.4)
-i
and
g(N)
is the leading (N
+ l)-vector of the
vector g with elements
i
I
gi
dx
ri(x) gx)/l-x 2
i
0,i
,N
(1.5)
-i
T. is the
i
th
Now to determine the vector a (N)
Chebyshev polynomial, as usual.
we replace the first equation of the system,
(1.3), by the boundary
condition
equation
N
(N)f.3
where
a’
fj
hj(a)
(1.6)
Thus the linear system of equations (1.3) is replaced by
L,(N) a_(N)
g,(N)
where for i >_ i the i-chequation is the (i
(N)
(1.7)
i)- equation in (1.3) and the vector
According to [2], provided the set {h.}
J
is suitably complete, the exact solution has the expansion
can be determined by solving (1.7).
f(x)
Z
i=0
b
i
h
i
(i.8)
and b satisfies the infinite matrix equation
L b
g
(1.9)
777
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
Further,
fN
f"
In this paper, we consider two different Chebyshev expansion sets:
T. (x)}
{h. (x)
(i.i0}
and
{h 0(x)
I,
h
I(x)
hi(x)
x,
x
(i
2) Tj_ 2 (x),
j -> 2}
(i.ii)
leading to three different methods (I), (II), (III).
Methods based on different techniques have been described before for solving
integro-differential equations of the first order; Linz [2], Ei-Gendi [3], Abd-elal
[4]
in all these papers integro-differential equations of the type (i.i) with
Kl(X,y)
0 are reduced to integral equations and a quadrature rule is used to
All of these methods are limited to integro-diff-
establish numerical procedures.
erential equations with no f’ under the integral sign, also they do not treat
boundary conditions in a very uniform way.
Ei-Gendi’s method [3] used Chebyshev
expansion (1.2, i. I0) in approximating the solution of the equation and produce
the solution in time
0(N3).
The methods we describe in this paper not only over-
come these limitations, but also (the last two methods) produce the solution at a
cost of total solution time
cheap to compute.
0(N21n
N) and give reliable error estimates which are
Method (I) is a straightforward method in which a Chebyshev
expansion set (i.i0) is used to approximate the solution f(x), and then we solve
the linear system of equations (1.7) to get the coefficient vector
we consider it a standard method.
a(N);
hence,
Method (II) uses a modified Chebyshev expansion
set (I.Ii) to approximate f(x) and so we consider it a modified method.
Method
(III) uses Chebyshev expansion set (I.i0) to approximate not only f(x) by expansion
(1.2), but also
f (x)
w
i=O
i
Ti(x)
(i.i)
by
d’(N)l
fN (x)
T.I (x).
(1.13)
We solve the corresponding linear system
e*(N)
d
(N)
(1.14)
L.F. ABD-ELAL
778
for the vector d
tems
(N).
An iterative procedure [5] is used to solve the linear sys-
(1.7) and (1.14).
The three methods effectively handle singularities in any or all of
k
(x,y),
0, i, g(x), the solution f(x), and its derivative f (x), provided that the
singularities are of known form and have a known Chebyshev expansion (see [b]).
These requirements limit the applicability of the method to those cases where the
singularitie-s which appear are of "standard" form- for example, weak singularities
or logarithmic singularities.
The methods can also treat some other types of
singularities modifying the integro-differential equation; for example, a simple
pole can be changed to a logarithmic singularity using integration by parts.
give in section 2 the analysis which leads to the structure of the matrix L
We
(N)
for
the three methods, while a comparison between the convergence rate attained by the
Section 4 shows, by example, that in the three cases
methods is given in section 3.
rapid convergence is obtained.
2.
THE MATRIX
L(N).
We wish to investigate the construction of the matrix L
Using the expansion (].2), the matrix
different methods considered in this paper.
L(N)
(N) for the three
reduces to
LIN)
A
(2.1)
B
with
A
where A
j
0,i
(k)
and B
,N; k
A
(0)
+ A (I)
B
and
B
(0)
+ B (I)
(k)
(k) are
(N + I x N + i) matrices with elements A.
B
(k)
i
0,I defined by
1
Aij(k)
dx
Pk(X) T i(x)
-i
(2.2)
1
1
Bi j(k)
2
--dxk {hj(x)} / -x
dx T
’2
i(x)//l-’x
-I
dy
(x,y)
-I
--dyk {hj(y)}
(2.3)
The integrals appearing in Equations(2.2), (2.3) and (1.5) must be approximated
numerically.
We do this by relating
Aij(k), Bij(k),
and
gi’ i, j
O,1,...,N to
779
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
Chebyshev coefficients
n
Pr(X), Kr(x,y)/l’y7
the expansions of
and g(x) respect-
These later coefficients are evaluated numerically using the fast
ively.
alorithm
given by Delves, Abd-Elal, and Hendry [6] in which Fourier transform technique is
used
Thls algorithm leads to small quadrature errors whether K r (x,y)
g(x) are singular or regular functions; also, it takes
0(N21n
evaluating the coefficients of the expansion for K r (x,y)
Aj
(r)
BI]
and
N) operations for
l-y2
and 0(N in N) for
evaluating the coefficients of the expansions for P (x) and g(x).
uate
P r (x)
Indeed
to eval-
(r) and
gl of Equations (2.2), (2.3), and (1.5) numerically, let
Kr(x,y)/1-y2 have Chebyshev
(r)
r--0,1
p1
r](x)
us assume that the functions P r (x) and
P (x)-r
2’
j=0
(r)
r
T i(x)
i,j=0
g(x)
gj
j--O
Tj(y)
r
expansions
(2.4)
(2.5)
0,i
(2.6)
Tj(x)
where the expansion coefficients
1
I
pj(r) 2
dx
rj(x) Pr(X)//-
r
(2.7)
0,I
-i
satisfy the inequality
-
Ipj(r)
-< C r
r
(2.8)
0,i
and the expansion coefficients
Kij
i
i
dx T
2
i(x)/l-x
dy
-i
Kr(x,y) Tj (y)
r
0,i
(2.9)
-i
satisfy the inequality
[Kij (r)
-<
-Yr
Dr
-Br
i,j > 0
which we can replace by the weaker bounds
IKij
(r)
IKij
(r)
<-
Dr
.-Yr
i
i> j
(2.10)
< D
r
J
j > i
780
L. F. ABD-ELAL
Also,
of Equation (1.5) has the bound
gi
Igil
Cr,
and
Dr’
-< G
i _> 0
1
i
when
j
0
j
when
j
>- 1
denotes a sum with first term halved.
G are constants,
algorithm [6], we can easily calculate the elements A.
lJ
,N of the matrices A
0,1
j
(2.4) in (2.2) we get for i
Ai-’3
P0(x)
1, then
Aij(0)
Also
Ai0
and B
lJ
(k) for
methods I, II, and III as follows"
In this method we choose the expansion set (i.i0) and by substituting
Method (I).
when
(k)
(k)
using the fast
K..
(k) and B (k) i
p.(k),
Knowing the numerical values for the coefficients
()
Aij
N
0,i
_(0) + (0)
P li-j ’)
(0)
J
(pi+j
(o) reduces
> 0
(2.ii)
to
0 for all i,j except
A00
(0)
T[,
A i
i
(0)
--for i
2
>_i
0
[J-o
Al3
()
’-,’ [p(1)
i+2 r+2 j_+_12
r
3
J+12 ]) + p(1)]i-2r-2( +2
i
Pl(X)
i, then
Aij
(i)
reduces to A.
lj
(i)
1
N
1,2
j
When
+2
-[
0 for all i,j except for j > 1
i
and j of different parity where by different parity, we mean one even and one odd.
Aoo
(i)
lj
[s]
2r
is the integer part of s, and
+ 2(
j
+
j
i
+
i
2
2
1)
means halving the term with
O.
Substituting (2.5) in (2.3), we get for i
B
(0)
4
J
Bi@
Bij
?r
(i)
(i)
2
K..
0,1,...,N
(0)
j
0,i
N
(2.13)
j
1,2,...,N
(2.14)
0
=2J
r=O
Ki,2r+2
J+i2
--J+i ]),
781
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
Notice that we take
0(N21n
N) operations to get the matrices A
(2.12) and (2.14) it is clear that we take 0(N
A
(I) and (I)
B
the matrix L
Bo
(i)
*(N"
(i)
unless explicit forms for A
0(N21n
3)
operations to set up
N) operations to get the matrix L
In this method we choose the expansion set (i.ii).
For i
(0")
+ P li-j+2[
(0)
,,,qen
Po(x)
Ai j(0)
A02
A
1,
A22
(0)
Also
Aio
Ail
(0) +
1
(0)
(Pi+j
Pli-jl
reduces to
-
AI3
(0)
for i >_ 3, A
i,i
-7
8
ii
(o)
A00
0 for all i,j except,
(0)
(0)
Ai_.j
i) and
*(N)"
,N
0,i
j =0 1
(0)
[(Pi+j-2
Pl(X)
(for example, case
_(o)
(o)
Pi+j +p li- jl
A..
but from
operations to obtain the matrices
and in general, tliis makes method I take 0(N
are achieved; then, it takes
Method II.
3)
(0), B (0),
(0)
(0)
7, A
A(0)
i,i
-3
8
-__
+ 4
8
t0)
Ii
_
1
(0)
(Pi+lj-4
(0)
+ Pli-lj-4ll
)1
j >_ 2
(2.15)
2
for i >_ 2,
+ 2
for i >_ 0
(I) =0
(1)
Pi
(1)
(2.16)
Ai2
Aij
(1)
2
(1)
A
4)7
(j
8
When
Aij
(p (1)
Pl(X)
(1)
i,
+ 3
(1)
p
(1)
(1)
(Pi
+
j
4
j8
+ P i-j+31
3
(i) reduces to
0 for all i,j except
(I)
i,i
Aij
+
A01
(1)
I)
(i- i) for i > i, A
+ 2,
i
AI2
7,
+
i
(1)
3
Corollary (II.i).
IAij (0)] _< A0
(i
j)
A0
(j
i
_<
-0
-0
(1)
(1)
Pi+j-i + P li-j+ll
-, A03 (1)
4
(i + 3)
i > j
j > i
for i >_ 0
j > 3
L. F. ABD-ELAL
782
oro llar (II.2.
IAij (1)
---
<
A
<
A i j(j
i > j
(i- j)
I
i)
-El
j > i
Both Corollaries (11.1,2) follow directly using inequality (2.8) and Equations
(2.15,16) respectively.
Bi0
Bil
Bij
Bi0
Bil
Bi2
mz]
(0)
(o)
(0)
(I)
(1)
(1)
(i)
2
Kio
2
=-
Kil
A0 and A I are constants.
(0)
(o)
(2.17)
2
1
Ki
7 (Kij
21
J
(0)
+
Ki,
(0)
J
41 )l
j >_ 2
0
2
Ki0
(1)
(2.18)
2
Kil
2
T
1)
z) K
[(
2
3
i,j
j >- 3
i
x,J
Corollary (11.3).
IBij (0)
B0
x
-YO
i > j
-<B 0
3
.-B 0
j > i
<
-71
x
i
<
C.orollary (II.4.).
IBij(1)
B
<B
I
^-(l-
1
1
> j
j > i
j
Corollary (11.3,4) follows directly from inequalities (2.10) and Equations
(2.17,18) respectively.
Method
(III)..
B
0
and B
I
are constants.
In this method we use expansion set (i.i0), and hence approximate
f(x) by (1.2) and f’(x) by (1.13).
ai
(di
i
Using the relation connecting a
di
+ 1 )/ 2i,
i
1,2
,N
i
and d
i
[7]:
(2.19)
783
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
hence
.
N
fN(x)
where
a
0
+
dj [Tj
j=0
means that the term
the unknown vector
+ 1 (x)/ 2(j + i)
Tj
[c 0, c
1
_.c
l(X)/ 2(j
CN+1]
l(X)/
Tj
i)
0 for j
where c
ao,
0
i)]
2(j
0,i.
cj +
(2 20)
Now consider
dj,
i
j
0
then the integral equation (i.i) is now reduced to
L(N) c(N)
where L
(N)
+
N
is the
x N
i
+ 2)
g(N)
(2.217
matrix defined by
(2.1); hence we need one
,
equation more, and this comes from the boundary condition f(a)
which we write
in the form
N+I
Z c’(N)
]
j=0
e.
(2.22)
]
where
e
I
0
ej +
Tj
+
l(a)/
2(j + i)
+
l(a)/
2(j
leading to the (N
+ 2 X
with elements, j
L*..
l]
N
+ 2)
(0)
A.
1J
g i
2,3
linear system of equations
(N)
,(N)
(2.24)
g
+ A
(i)
B
1J
(0)
B
mJ
(i)
i
3
N
0 1
(2.25a)
j
i
gi’
P0(x)
c
e.
j
where the elements A..
13
i
dx
l(a)/
-Tj
i)
2(j
N+I
0,i
L*N+I,
I
j
(Z. 23)
+ I)
L,(N)
(0)
0,I
1
Tj
Ai0
j
(k)
0,1,...,N
Bij
(k)
i
+
a
i
N; k
0,I
(2.25b)
0,I defined by
Ti(x)/i/- x 2
-i
.(0)
i, j+l
dx
2(j + i)
i
-I
P0(x) Ti(x) Tj+l(X)
/
i
x2
j
0,i
(2.26)
-i
dx
P0(x)Ti(x) [Tj+l(x)/
2(j + 17
Tj_ l(x)/
2(j
j= 2
i)]/
N
784
Aio
L. F. ABD-ELAL
(i)
0
1
I
I
(1)
i,j+l
Tj(x)//
Pl(X) T i(x)
dx
(2.27)
x2
j
,N
0,i
-i
i
Bio
(o)
i
Ti(x)//l
dx
I K0(x’Y)
x2
dy
i
dx
2(j + i)
B
Ti(x)/- x
l(y)
+
j
0,i
(0)
I
i,j+l
i
I
BiO
(I)
I Ko(x,y)[Tj+l(Y)/2(
Ti(x)//l-x2
dx
-I
j
+ i)-
Tj_l (Y) /2 (j
l)]dy
2,3,...,N
j
0
;
(I)
i,j+l
1
Ti(x)/l
dx
I
x2
-i
(2.29)
Kl(X
y)
Tj(y)
dy
(o)
j
0,i
N
j
0,I
-i
Substituting_ (2.4) in (2.26, 27) we get for i
Ai0
(2.28)
-I
1
B
K0(x,y Tj
dy
Pi
N
0,i
(o)
+ P (0)
]i-j-1]
(0)
r
8(j + i)
(Pi+j+l
.(o)
(2.30)
Ai’j+l
(0)
1
’(j+l)’
(Pi+j+l
(0)
i
+ P i-j-ll
(j -I)
o)
pi0)
i+j -I
P i_j+l
When
Po(x)
Aij(O)
Aij(O)
reduces to
0 for all i,j except
(0)
A
i,
/(4j)
jj
j
N
2,3
j
3
N+I
A00
(0)
7r,
(0)
Aj -i ,j+l
All
(0)
=-l(4(j
r14,
A22
i)), j
(0)
r18
2
N
1 2
,N
and
Ai0
A
(i)
o
(I)
_(i)
(pi+j
i,j+l
When
Aij
Pi(x)
(i)
i,
Aij
(i)
+ P li-j
j
0,1,2
N
(i) reduces
to
0 for all i,j except
A0i
(i)
’ Aj(1)
,j+l
z12
j
785
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
Corollary (III.i.
(0)
<
A0(i-
j)
<
AO( j
i)
-<
Al(i
j)
_<
AI( j
^i)
Corollary (111.2).
IAij(1)
_
i
j
-0
-
-i
> j
j > i
j
i >
J
j > i
Both corollaries (111.1,2) follow directly using inequality (2.8) and equations
(2.30,31) respectively.
Now substituting (2.5) in equations (2.28,29) we get for i
Bio
(0)
2
K+/-0
4
-
8(j
O)
.
i,j+l
and
Bi0
B
(l)
o
2
(i)
i,j+l
4
2
0,i
N
(0)
2
+ 1)
K(0)
i,j+l
j
0,i
(2.32)
[
(j+l)
i,j-I
(j-l)
m ,j+l
]
j
2,3,...,N
(2.33)
K..(1)
j
0,1,...,N
Corollary (III. 3).
IBij (0)
<_
B
c-Y0
j
^-(So
+ 1)
0
-i
i > j
j > i
j
-<B0
Corollary (III.4).
IBij (i)] _< BI
_<
B
i
i
.-I
j
1
> j
j > i
Bo’th corollaries (111.3,4) follow directly from inequality (2.10) and equations
(2.32,33).
To evaluate
algorit.hm [6].
(r)
Pi
Kij
(r)
r
0,i numerically, we refer to the fast
L. F. ABD-ELAL
786
ASYMPTOTICALLY DIAGONAL MATRICES.
3.
Definition:
L, let the matrix F have elements
Given an infiniet matrix
ILijl
(ILiil ILjj I) 1/2
Fij
(3.1)
The matrix L is said to be "Asymptotically lower diagonal (A.L.D.) of type B(p,r;c)"
if constants p,r e 0, c > 0 exist such that
Fo.
-P
_< c
(i
j)-r
-
i > j
(3.2)
and is said to by "Asymptotically upper diagonal" (A.U.D.) of the same type if
-P
F.. -< c
(j
i)
-r
j > i
(3.3)
In an obvious notation we shall then refer to systems of type B as:
B(PL, PU’ rL’ ru; CL’ Cu)
Type
(3.4)
For systems (A.D.) of type B, Freeman and Delves [8] provide estimates of the
convergence rate.
In order to compare the convergence rate attained by the three
methods of this paper we need to study the matrices given by each methe@.
We construct the elements of the matrix L *(N)
An( k
B n(k) are (N+I
An(0)
B*
B *(0) + B *(I)
+
N+I)matrices with elements
,* (k)
i
i-l,j
* (0)
* (0)
* (i)
* (k)
fj A0j
B0j
Bi-’j
where
f0
*mi-l,j
(k)
i,
fl
i
a,
L*.
lJ
fj
(N)
g(N)
a
+
’
0
(3.5)
0,i
,N.. k
0,i
N
1,2
(i
* (i)
B0j
N
1,2
j
n (k)
B* where
A*(1)
A*
A0j
Aij
An
2) Tj_2(a),
2 _< j <
N(Ifjl
<
i, j
i,
N
{A*. (r)- B*. (r)}
j
J
n(N)
gi
,
gi-i
(N)
i
0,i
N)
787
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
As is clear from equation (2.12) the matrix A (I) is not (A.D.); indeed,
Method (I):
for j >
Aij(1)
the elements
So L *(N) is not (A.U.D.) and hence
increase with j.
the analysis of Freeman and Delves [8] is not applicable to method I, so as we will
see later we do not suggest a value for the truncation error.
Also we can not use
the iterative method given by Delves [5] to solve the linear system (1.7) and hence
any standard method for solving linear equations (Gauss elimination method) can be
used.
Method (II)
-
The matrix L *(N) of equation (1.7) is (A.D.) of type
Lemma i.
B(0’O’min(0’ i’ Y0’ YI )’ min(0’ i -I’
BO’ 81-1); LL’
LU)
From corollaries (II.i, 2, 3, 4) we get
Proof:
ILj] -< Al(i-j)
Hence
,
ILij
-< L 1 j(i
ILij*
-<
j)
Also
AIJ^(J
i)
+
A0(i-j)
-min(0’
-o + B0(i- -o + Bl(ij)
j)
-YI
i > j
i’ Y0’ Yl
i > j
-E + A0(j-l) -o + B 0(j-l) -o + Bl(J-i -81+I j
(3.6)
> i
Hence
e*..l
-< L 2
ILil
> L
(J
2 i) -min(l’
0+
I,
80 +
I,
81
j > i
3.7)
Also
where
LI, L2,
3
"
and L
3
are constants.
From (3.6, 3.8, for i
j,
L ^-2
I
---i 2j
L
_<
^ (I
j)
-min(o’ $i’ Y0’
#i
3
LL(i
j)
-mini0’ i’ Y0’ YI
(3.9)
Also from (3.7, 3.8), for i > j,
-< L2
L
3
_<
-1/2
Lu(J-I)
^-1/2 j (j-l) -min( I,
j
-min($1-1,
0+I, B0+I, 81
0’ B0’ 81-1
j > i
(3.10)
788
L. F. ABD-ELAL
since j(j
i)
-i
< 2i for j > i.
From (3.9) and (3.10), the lemma follows.
Method (III):
The matrix L *(N) of equations (2.24) is (A.D.)of type
Lemma 2.
B(0,0,min(0,Ei,Y0, yl),
min(E 0
+
i,
EI,80
+
1,81);
LL, L
U)
From corollary (III.i,2,3,4) we get:
Proof:
,
-0
ILijl < Al(i
J)
-El
-< L (i
4
j)
-min(E0’ El’ Y0’ YI
-< A l(j
i)
<-
i)
-E I
+
+ AO(j
-min(
L5( j
J)
A0(i
0
i)
+ I,
^-i
3
-E 0-i
El,
+
+ B I (i- j)
i > j
j)
B0(i
j > i
(3.11)
+ B l(j
8 0 + i, B
i)
-81
+
B O(j
I)
i)
-(80
+ i)
j > i
j > i
(3.12)
Also
IL+/-+/-*
>L
where
L4,
From
(3.13)
6
L 5 and L 6 are constants.
(3.11,313)
,
lJ
(ILii
1/2
Ljj I)
_<
LL(i
j)
-<
LU(
i)
i > j
-min(E0, El, 0’ I
(3.14)
From (3.12,13)
-min(E
j
0
+ i, E l, B 0 + i,
81
j > i
(3.15)
From (3.14,15), the lemma follows.
From Lemma 1,2 the matrix L
*(N)
is U.A.D. and hence the analysis of Freeman and
Delves [8] is applicable to methods II, III and a value of the truncation error is
suggested as given later.
(1.7) and (2.24) in 0(N
2)
Also the iterative Method (III) may be used to solve
operations.
Now by virtue of Theorems 6 and 7 of Freeman and Delves [8] with normalization we have the following theorem.
789
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
THEOREM i.
rL, r
U
If L is L.A.D. or U.A.D. of type
> i,
>- 0
PL’ PU
2
then if, for some constant
>- ^i,
Liil
and
B(PL, PU’ rL’ ru; CL’ Cu)
e i, there exist some positive constants
with
Ib i
a
(N)
< M
i
N
-(Pu
+ t)
-r
(N+
M
M1,
i)
g
c
il < i-
such that
2
+
U
with
A
-i
i
i _< N
-I
t
min(6,
PL
Theorem
ru)"
leads us to our two main theorems"
In Method (II)
THEOREM 2.
l’b i
+
(N)
a
< M
i
Ibil
where t
_<
min(,
M
-t
N
i
2
-min(E
(N +
i)
-(t + 1/2)
-2
O-
B 0-
B -2) ^-1/2
i
iNN
i > N
min(Eo, El, YO’ YI ))"
This follows directly from Lemma
and
Theorem 1.
In Method (III)
THEOREM 3.
lwi
d
i(N)
_<
lwil
M
<M
N
2
-t
i
(N + 1- i)
-t
min(,
4.
ERROR ESTIMATES.
E
i,
80,
B
i)
i
-< N
i>N
min(E0, $i’ Y0’ i ))"
t
-min(E0,
This follows directly from Lemma 2 and Theorem i.
The error in any of the three methods contains three distinct components:
(a)
The truncation error due to cutting off the expansion (1.2) at the
N
(b)
th
term.
The discretization error stemming from the quadrature errors A
the matrix A-B, and
(c)
g
B
in
in the vector g.
There are also in principle errors arising from the numerical solution
of the linear equations.
Now according to Delves [9] we measure the error
eN(x)
fN(x)
f(x), with
L. F. ABD-ELAL
790
]eN(x)
IS I
$31
+ S2 +
where the first two summands represent the truncation
error (a), and are defined by
N
Sl
S
i
2
Ii
0
(N)
b
Method
(I) and (II)
Method
(III)
Ibil
i--N+l
while
N
S
S
3
li (N)
E
i=O
S
2
lwil
i=N+l
is the quadrature error
.
.
N
S
3
ai
i=0
N
di
i=O
(N)
In the above,
(a)
i(N)
(b), defined by
(N)
hi(N)
for Method (I) and (II)
(N)
i(N)
for Method (III).
are the computed coefficients
(S + S 2)
Truncation error estimates:
From Theorems2 and 3 and for Methods (II) and (III), S dominates S
2
S
+ S2
and so
S2
Hence for Method (II)
S
2
M
-t
2
N
+ 1/2 /(t
N
]aNl
(4.1)
/(t- i)- N
IdNl
(4.2)
i)
while for Method (III)
$2
M2 N-t
+
For Method (I), since we are unable to apply Theorem I, we do not suggest a value
for S
(b)
+ S 2.
quadrature error estimates:
S
3
As given in Delves [9]
S
3
lle-lll (lel]
W
+
II gll)/( I -lle-lll
llell)
(4.3)
791
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
lall
lldll
Here W
for Method (I) and (II),
IIL---III
for Method (III) and a rough estimate for
lie-ill II gll / W
II LII, since we refer
We require only an estimate for
is taken to be
to Delves et al
[6]
to
II gll
estimate
LII"
Evaluation of
Method (I)"
from Equations (2.11,12), we get
iI A(0)II
-<
N
’T
p(O)II
II A(1)II
/2,
N3 II p(1)II
-<
/3
Hence
lldp (0) [1 /2 +
IIAIL -< Tr(N
N
II p(,1)II
3
/3)
Also, from Equations (2.13,14)we get
I1 B(0)]L -< T2
(K(0) []
6B(1)II
/4
-<
T2 N2 It 6K(I-) Ii
/4
Hence
N2 I 6K(1)II
(I] 6K(0) I +
Then
I LII
)/4
(4.6)
(4.5) + (4.6).
Method (II)"
from Equations (2.1,16)
II 6A(0) II
-<
II AI L -<
6p(0)11
N
}6p (’*)
(N
]1 6A(1) It
/2
IL/2
+. 2
.
-<
(])11/4
2
11 6p(1) I/4
(4.7)
From Equations (2.17,18),
[I
6B(0)[
_<
72 ]1 K(0)]I
B (1)
/4
-< N
T
2
6K(1) I
/4
Hence
II dBI1
I LII
Method (III)"
(A(0) ]1
-<
T2
II 6K(0)
+ N
I (K(1)II)
/4
(4.8)
(4.7) + (4.8)
from Equations (2.30,31),
< 2rr(2
+
in N)
II (p(0)II
II 6A(I)IL -<
N
I16p(1)II
/2
Hence
I {SAil
-< T(2(2 + in N))
II Sp()ll
+ N
II 6P (1) II
/2
(4.9)
L. F. ABD-ELAL
792
from Equations (2.32,33),
IIdB (i) I}
/4;
I 6B I
<
’v2(ll 6K(0) II
2 I @K(1)II
-<
@K(1) I
+
/4
/4
(4.10)
(4.9) + (4.10)
We refer to Delves et al [6] for the numerical estimations of
@P(r)II
and
II +K(r) l!
(c)
Error stemming from solving the linear system of equations.
To solve the linear system of equations (1.7) or (2.24), for Method II or
we use an iterative scheme given in
[5] with 0(N
2)
operations.
II<
The error due to
this iterative solution is small and so we neglect it, but for Method I and accord-
ing to Delves [5], we cannot use this iterative method and hence the error could
We will now see that this error
be relatively large due to error cancellation.
cancellation has no serious effect in a numerical example.
5.
NUMERICAL EXAMPLE.
We give in this section the numerical results for a singular integro-differential equation of the first order.
i
I
I K0(x’Y)
g(x) +
f’ (x) + f(x)
f(Y) dy +
-I
K0(x,y)
2 in(x- y) (x- 2y
K i(x,y)
(x
g(x)
2
It has the exact solution f(x)
ON
where
x.]
+ 2xy 2
2y
3)
x
2e(x + 1) ln(x + 1) + 2xe
with boundary condition f(a)
N
-i
y2)
e-
The computed errors o
] Kl(x,y)
,
e
a E [-i,i].
2
x
are defined to be
j Nj0 eN2 (xj)/N
cos (jw/N), j
0,1,...,N and e N
i
e
N
(x) dx
fN- fexact
f’(y) dy
793
INTEGRO-DIFFERENTIAL AND DIFFERENTIAL EQUATIONS
Computed results for the numerical example.
Method (I)
6.
3
3.8x i0
5
4.9x I0
7
4.6x i0
9
2.9x i0
ii
1.4 x i0
13
6.0 x I0
15
3.3 x i0
17
2.8 x i0
19
2.8 x I0
(II)
Method
f(-l)=e
oN Method (II)
f(-l)=e
-i
3.7 x i0
-2
7.1 x i0
-3
6.5x i0
-4
4.1 x I0
-5
2.1 x i0
-7
8.6 x i0
-8
3.2 x 10
-8
3.3x i0
-8
2.1 x i0
f(0)
-1
3.6 x I0
-2
8.5 x 10
-3
5.0 x 10
-4
2.2 x i0
-5
8.6 x 10
-7
2.9 x i0
-8
8.6x i0
-9
3.8x i0
-9
1.3x i0
I
(III)
oN Method
f(-l)=e
-I
1.2 x I0
-2
1.3 x 10
-3
9.1 x 10
-4
4.9x i0
-6
2.0 x 10
-7
6.8 x i0
-9
2.1 x i0
-i0
2.0x I0
-I0
i.i x 10
-I
-2
-4
-5
-6
-8
-9
-I0
-10
COMMENTS ON THE METHODS.
(i) As is clear from the analysis, although Method (I) is a standard method,
in fact Methods
(II) and (III) are preferable because they provide easy error
estimates.
(2) All the three methods work very well when applied
to the numerical example,
and this suggests that Method (I) is probably a stable method.
(3) The three methods can be applied to ordinary differential equations of the
first order which have the form (i.I) but with
/-
h
k=0
Pk (x)
d
k
dx
Hence the same analysis holds with the matrix B
k
0.
(4) The three methods represent a uniform way of treating boundary conditions,
so we recommend methods
(II) and (III) be extended
equations of the second order.
to include integro-differential
We are now working on this.
(5) A standard Galerkin calculation has an operations count of 0(N
3)
for both
setting up and solving the linear equations defining the coefficient vector"
however, Methods (II) and (III) of this paper need"
L. F. ABD-ELAL
794
Setting up equations:
Iterative solution"
ACKNOWLEDGMENT.
O(N21n N) operations
O(N 2) operations.
The author wishes to thank Professor L. M. Delves for useful
discussions on this work.
REFERENCES
i.
MIKHLIN, S.G. "Variational methods in Mathematical Physics", New York:
Pergamon Press (1964).
2.
LINZ, P.
3.
EL-GENDI, S.E.
4.
ABD-ELAL, L.F.
"A Method for the Approximate Solution of Linear Integro-Differential
Equations", Siam J. Numer. Anal. 11 (1974), no. I.
"Chebyshev Solution of Differential, Integral and IntegroDifferential Equations", Comput. J. I__2 (1969) pp. 282-287.
"An Asymptotic Expansion for a Regularization technique for
Numerical Singular Integrals and its Application to Volterra Integral
Equations", J. Inst. Maths. Applics. 2__3 (1979) pp. 291-309.
"The Numerical Solution of sets of Linear Equations Arising from
Ritz-Galerkin Methods", J. Inst. Maths. Applics, 2__0 (1977) pp. 163-171.
5.
DELVES, L.M.
6.
DELVES, L.M., L.F. ABD-ELAL, and J. HENDRY. "A Fast Galerkin Algorithm for
Singular Integral Equations", J. Inst. Maths. Applics. 23 (1979)
pp. 139-166.
7.
FOX, L. and I.B. PARKER.
"Chebyshev Polynomials in Numerical Analysis",
Oxford University Press (1968).
8.
FREEMAN, T.L. and L.M. DELVES.
methods III.
pp. 311-323.
9.
Unsymmetric
"On the Convergence Rates of Variational
Systems" J. Inst. Maths. Applics. 14 (1974)
DELVES, L.M. "A Fast Method for the Solution of Fredholm Integral Equations"
J. Inst. Maths. Applic_s. 2__0 (1977) pp. 173-182.