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Journal of Computational and Applied Mathematics 239 (2013) 333–345

Contents lists available at SciVerse ScienceDirect

Journal of Computational and Applied
Mathematics
journal homepage: www
...
com/locate/cam

New algorithms for the numerical solution of nonlinear Fredholm and
Volterra integral equations using Haar wavelets
Imran Aziz a,b,∗ , Siraj-ul-Islam a
a

Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan

b

Department of Mathematics, University of Peshawar, Pakistan

article

info

Article history:
Received 20 March 2012
Received in revised form 6 June 2012
Keywords:
Haar wavelets
Fredholm integral equations
Volterra integral equations

abstract
Two new algorithms based on Haar wavelets are proposed
...
These methods are designed to exploit the special characteristics of Haar wavelets in
both one and two dimensions
...
These formulae are then used in the proposed
numerical methods
...
The methods are validated on test problems, and numerical
results are compared with those from existing methods in the literature
...
V
...


1
...
Analytical solutions of integral equations,
however, either do not exist or are hard to find
...
Recent contributions in this regard include Chebyshev polynomials [1],
the modified homotopy perturbation method [2,3], wavelet methods [4–7], radial basis functions (RBFs) [8,9], Bernstein’s
approximation [10], the Toeplitz matrix method [11], the linear multistep method [12], and the triangular function
method [13]
...
The need to develop a generic algorithm which can be applied to a general type of
nonlinear integral equations is felt necessary in order to have a single platform to be used for the numerical solution of
these types of problem
...

The use of wavelets has come to prominence during the last two decades
...
This is largely due to the fact that wavelets provide a natural mechanism for decomposing the solution
into a set of coefficients, which depend on scale and location
...
Researchers have employed various methods in applying wavelets to numerical
approximations
...
A survey of some of the early works can be found in [18]
...

E-mail address: imran_aziz@upesh
...
pk (I
...


0377-0427/$ – see front matter © 2012 Elsevier B
...
All rights reserved
...
1016/j
...
2012
...
031

334

I
...
These methods involve different types of wavelet
...
On account of their simplicity, Haar wavelets have received the attention of many researchers
...

Various types of wavelet have been applied for numerical solution of different kinds of integral equation
...
Many
researchers have applied Haar wavelets for the numerical solution of integral equations
...
Lepik and Tamme [46] have applied
Haar wavelets to nonlinear Fredholm integral equations, but their method involves approximation of certain integrals
...
The main advantage
of our method is that it can be applied to general types of nonlinear Fredholm and Volterra integral equations
...
The first type is nonlinear Fredholm
integral equations of the second kind, given as follows:
u(x) = f (x) +

1



K (x, t , u(t )) dt ,

(1)

0

and the second type is nonlinear Volterra integral equations of the second kind, given as follows:
u(x) = f (x) +

x



K (x, t , u(t )) dt ,

(2)

0

where K (x, t , u(t )) is a nonlinear function defined on [0, 1] × [0, 1]
...

The organization of the rest of the paper is as follows
...

In Section 3, formulation of the method based on Haar wavelets is defined for nonlinear Fredholm and Volterra integral
equations
...

2
...


(3)

All other functions in the Haar wavelet family are defined on subintervals of [0, 1), and are given as follows:


haari (x) =

1

−1
0

for x ∈ [α, β)
for x ∈ [β, γ )
elsewhere,

(4)

where

α=

k
m

,

β=

k + 0
...
, 2M
...
, J , M = 2J ,and integer k = 0, 1,
...
The integer j indicates the level of the
wavelet and k is the translation parameter
...
The relation between i, m and k
is given by i = m + k + 1
...

The Haar wavelet functions are orthogonal to each other because
1



haarj (x)haark (x)dx = 0,

whenever j ̸= k
...


(7)

i =1

The above series terminates at finite terms if f (x) is piecewise constant or can be approximated as piecewise constant during
each subinterval
...
Aziz, Siraj-ul-Islam / Journal of Computational and Applied Mathematics 239 (2013) 333–345

335

We introduce the following notation:
x



ihaari,1 (x) =

haari (x′ ) dx′
...
(4), and is given as follows:


x − α
ihaari,1 (x) = γ − x


for x ∈ [α, β),
for x ∈ [β, γ ),

(9)

elsewhere
...
Numerical methods
In this section, we will discuss the proposed numerical methods for two different types of integral equation
...

In the third and fourth subsections we apply these results for finding numerical solutions of Fredholm and Volterra integral
equations
...
5

,

p = 1, 2,
...
, 2N
...
5
2N

3
...
One-dimensional Haar wavelet system
Any square integrable function f (x) can be approximated using Haar wavelets as follows:
f ( x) =

2M


ai haari (x)
...
(10), we obtain the following linear system of equations:
f ( xp ) =

2M


ai haari (xp ),

p = 1, 2,
...


(13)

i=1

This is a 2M × 2M linear system of equations whose solution for the unknown coefficients ai can be calculated using the
following theorem
...
The solution of system (13) is given as follows:
a1 =

ai =

2M
1 

2M j=1
1



ρ

β


f (xj ),

f (xp ) −

p=α

α = ρ(σ − 1) + 1,
ρ
β = ρ(σ − 1) + ,
ρ=

2M

,
τ
σ = i − τ,
τ = 2⌊log2 (i−1)⌋
...
, 2M ,

(15)

336

I
...
For proof of Eq
...
For the second part of the proof we use induction on J
...
(15) for
i = 2, 3,
...
For J = 0, we have M = 1, and the linear system in this case is given as follows:
f (x1 ) = a1 + a2

(16)

f (x2 ) = a1 − a2
...


(17)

Now let us substitute i = 2 and M = 1 in Eq
...
(17), and hence the formula is true for J = 0
...
, and consider the linear system with J = n
...
From this system we obtain a new system by adding equations
corresponding to p = 2k − 1 and p = 2k for k = 1, 2,
...
This new system is an M × M linear system, and it can be
expressed as follows:
M
 
1
ai haari (xp ),
f (x2p−1 ) + f (x2p ) =
2
i=1

p = 1, 2,
...


(18)

Using the induction hypothesis, the solution of this system is given as follows:

ai =

1



ρ′

β

1
p=α ′

2

γ

1
′

′

(f (x2p−1 ) + f (x2p )) −

p=β ′ +1

2


(f (x2p−1 ) + f (x2p )) ,

n

i = 2, 3,
...
(19) can be written in a more compact form as follows:

ai =



1
2ρ ′

2β


2γ

′

′

f (xp ) −

p=2α ′ −1

ρ′
2

(19)

and γ ′ = ρ ′ σ ′
...
, M
...
(20) becomes exactly the same as Eq
...
, M
...
, M
...
, 2M
...
, M
...
, M in the system given in Eq
...
, M
...
(15)
matches Eq
...
, M
...

3
...
Two-dimensional Haar wavelet system
A real-valued function F (x, t ) of two real variables x and t can be approximated using a two-dimensional Haar wavelet
basis as
F (x, t ) ≈

2M 
2N

i=1 j=1

bi,j haari (x)haarj (t )
...
Aziz, Siraj-ul-Islam / Journal of Computational and Applied Mathematics 239 (2013) 333–345

337

In order to calculate the unknown coefficients bi,j , the collocation points defined in Eqs
...
(22)
...
, 2M , q = 1, 2,
...
The solution of this system can be
calculated using the following theorem
...
The solution of system (23) is given below:
b1,1 =

2M 
2N


1

2M × 2N p=1 q=1



1

b i ,1 =

ρ1 × 2N


β2
2M 


2M × ρ2



1

bi,j =

ρ 1ρ 2

Fp,q −

β1 
β2


Fp,q −


Fp,q

,

i = 2, 3,
...
, 2N ,

(26)


Fp,q

p=1 q=β2 +1

Fp,q −

p=α1 q=α2

γ2


γ1
2N


p=β1 +1 q=1

p=1 q=α2

γ1


+

β1 
2N


(24)

p=α1 q=1

1

b1,j =

Fp,q ,

γ2
β1


p=α1 q=β2 +1

Fp,q −

γ1
β2



Fp,q

p=β1 +1 q=α2


Fp,q

,

i = 2, 3,
...
, 2N ,

(27)

p=β1 +1 q=β2 +1

where

α1 = ρ1 (σ1 − 1) + 1,
ρ1
β1 = ρ1 (σ1 − 1) + ,
2

γ1 = ρ1 σ1 ,
ρ1 =

2M

(28)

,

τ1
σ1 = i − τ1 ,
τ1 = 2⌊log2 (i−1)⌋ ,

and similarly,

α2 = ρ2 (σ2 − 1) + 1,
ρ2
β2 = ρ2 (σ2 − 1) + ,
2

γ2 = ρ2 σ2 ,
ρ2 =

2N

,

(29)

τ2
σ2 = j − τ2 ,
τ2 = 2⌊log2 (j−1)⌋
...
We will prove the theorem by using induction on J = J1 + J2 , where M = 2J1 and N = 2J2
...
Aziz, Siraj-ul-Islam / Journal of Computational and Applied Mathematics 239 (2013) 333–345

b2,1 =
b2,2 =

1
4
1
4

F1,1 + F1,2 − F2,1 − F2,2



F1,1 − F1,2 − F2,1 + F2,2 ,



which agree with Eqs
...
Thus the theorem is true for J = 0
...
, m + n − 1, and suppose that J = m + n
...

First, assume that m = 0 and n > 0
...
, 2n , we obtain the following system:
Gp,q =

2 
2n


bi,j haari (xp )haarj (tq ),

p = 1, 2, q = 1, 2,
...




(31)

Applying the induction hypothesis to system (30) we obtain the following values of bi,j :
b1,1 =

b2,1 =

b1,j =

2 
2n


1

2 × 2n p=1 q=1



1
2 × 2n
1
2 × ρ2′


,

G2,q

(33)

q =1

β2


β2


′

G1,q −

(34)

G1,q −

2

, σ2 = j − τ2 , ρ2 =
′

′

′

′

G2,q +

2

2

G2,q  ,

j = 2, 3,
...
, 2n ,

p=1 q=β ′ +1
2

p=1 q=α ′
2

where τ2 = 2
′

2n




γ2′
β2′
2
2 




Gp,q  ,
Gp,q −

1
2 × ρ2′

G1,q −

(32)

q =1


b2,j =

2n


Gp,q ,

, α2 = ρ2′ (σ2′ − 1) + 1, β2′ = ρ2′ (σ2′ − 1) +
′

ρ2′
2

and γ2′ = ρ2′ σ2′
...
, 2n ,

(38)

p=1 q=2β ′ +1
2

Now let us assume that

ρ2 = 2ρ2′ =

F2,q  ,

p=1 q=2α ′ −1



2 × 2ρ2′





2β2′
2γ2′
2
2





Fp,q −
Gp,q  ,


b2,j =

(36)

,
τ2
α2 = 2α2′ − 1 = ρ2 (σ2 − 1) + 1,

2 β2

′

F1,q −

q =2 α ′ −1
2

2γ2

′

F2,q +

q=2β2′ +1


F2,q  ,

j = 2, 3,
...


(39)

I
...

Then Eqs
...
(24)–(27) for i = 1, 2 and j = 1, 2,
...
Next, subtracting equations corresponding
to p = k, q = 2l from equations corresponding to p = k, q = 2l − 1, respectively, for k = 1, 2 and l = 1, 2,
...
, 2n

F2,2l−1 − F2,2l = 2b1,2n +l − 2b2,2n +l ,

l = 1, 2,
...


(40)

This implies the following:


1
F1,2l−1 + F2,2l−1 − F1,2l − F1,2l ,
4

1
b2,2n +l =
F1,2l−1 − F1,2l − F2,2l−1 + F2,2l ,
4

l = 1, 2,
...
, 2n
...
(41) agree with the values obtained from Eqs
...
This proves the theorem when m = 0
and n > 0
...

Finally, we assume that m > 0 and n > 0
...
, 2n and l = 1, 2,
...
, 2m , q = 1, 2,
...

2

(43)

Applying the induction hypothesis to system (42), we obtain the following values of bi,j :
b 1 ,1 =

n+1
2m 2



1

2m × 2n+1 p=1 q=1


bi,1 =

b1,j =

1

β2
2m 


β1
β2





ρ1′ ρ2

′


Gp,q  ,

i = 2, 3,
...
, 2n ,

(46)

p=1 q=β2 +1

β1
γ2



γ1
β2



′

Gp,q −

p=α1′ q=α2

γ1


+

Gp,q −

′

1

n+1
γ1
2



′

′

p=1 q=α2


bi,j =

(44)

p=α1′ q=1



2m × ρ2

β1 2n+1





ρ1′ × 2n+1
1

Gp,q ,

′

Gp,q −

p=α1′ q=β2 +1

Gp,q

p=β1′ +1 q=α2



γ2


Gp,q  ,

i = 2, 3,
...
, 2n ,

(47)

p=β1′ +1 q=β2 +1
m
ρ
where τ1′ = 2⌊log2 (i−1)⌋ , σ1′ = i − τ1′ , ρ1′ = 2τ ′ , α1′ = ρ1′ (σ1′ − 1) + 1, β1′ = ρ1′ (σ1′ − 1) + 21 , γ1′ = ρ1′ σ1′ , τ2′ = 2⌊log2 (j−1)⌋ , σ2′ =
1

′

n+1

j − τ2′ , ρ2′ = 2 τ ′ , α2′ = ρ2′ (σ2′ − 1) + 1, β2′ = ρ2′ (σ2′ − 1) +
2
written as follows:
b1,1 =

×


2n+1


b i ,1 =

2

and γ2′ = ρ2′ σ2′
...
(44)–(47) can be

2m+1 2n+1

1
2m+1

ρ2′

1
2ρ1′ × 2n+1

Fp,q
...
, 2m ,

(49)

340

I
...
, 2n+1 ,

(50)

p=1 q=β2 +1
2β1

′

Fp,q −

p=2α1′ −1 q=α2

2γ1


+

Fp,q −

p=1 q=α2


bi,j =

γ2
 

2m+1

2m+1

γ2


2γ1

′

Fp,q −

p=2α1′ −1 q=β2 +1

β2


Fp,q

p=2β1′ +1 q=α2



γ2


Fp,q  ,

i = 2, 3,
...
, 2n+1
...
(48)–(51) yield (24)–(27) for i = 1, 2,
...
, 2n+1
...
, 2m+1 and l = 1, 2,
...
, 2m+1 and j = 1, 2,
...
It only remains to obtain Eq
...
, 2m+1 and
j = 2n + 1, 2n + 2,
...
For this, we first add equations corresponding to p = 2k − 1, q = 2l − 1 and p = 2k, q = 2l, and
then from the resultant equation obtained we subtract equations corresponding to p = 2k, q = 2l − 1 and p = 2k − 1, q = 2l
for k = 1, 2,
...
, 2n
...

4

(52)

These values agree with the values obtained from Eq
...
, 2m+1 and l = 2n + 1, 2n + 2,
...

Hence the theorem is proved
...
3
...
The kernel function K (x, t , u(t )) is approximated using the two-dimensional
Haar wavelets given in Eq
...
Substituting this approximation of the kernel function in the Fredholm integral equation
(1), we obtain the following:
u(x) = f (x) +

2M 
2M


bi,j haari (x)

1



haarj (t ) dt
...
,

(54)

Eq
...


(55)

i=1

Substituting the collocation points defined in Eq
...
, 2M
...
Aziz, Siraj-ul-Islam / Journal of Computational and Applied Mathematics 239 (2013) 333–345

341

Now the coefficients bi,1 can be replaced with their expressions given in (24)–(25), and we obtain the following system of
nonlinear equations:
u(xr ) = f (xr ) +

+

2M 
2M


1

2M × 2M p=1 q=1



2M


1

i =2

ρ1 × 2M

β1 
2M


K (xp , tq , u(tq ))haar1 (xr )

K (xp , tq , u(tq )) −

p=α1 q=1

γ1
2M




K (xp , tq , u(tq )) haari (xr ),

p=β1 +1 q=1

r = 1, 2,
...


(57)

Let us introduce the following notation:
u(xr ) = u(tr ) = ur ,

r = 1, 2,
...


(58)

With this notation, the above system can be written as
ur = f (xr ) +


×

2M 
2M


1

2M × 2M p=1 q=1

β1 
2M


K (xp , tq , uq )haar1 (xr ) +
γ1
2M



K (xp , tq , uq ) −

p=α1 q=1

2M


1

i=1

ρ1 × 2M


K (xp , tq , uq ) haari (xr ),

r = 1, 2,
...


(59)

p=β1 +1 q=1

Now Eq
...
, 2M
...
The algorithm is terminated when the required tolerance
has been achieved; that is, when

∥u(k+1) − u(k) ∥ < ϵ,
where ϵ is the required tolerance
...
Once the solution at collocation points is known, we can find solution at any point with the help of
Theorem 1
...
4
...
Substituting the approximation of the kernel function given in Eq
...


(60)

0

i=1 j=1

Using the property of Haar functions, this equation reduces to the following:
u(x) = f (x) +

2M 
2M


bi,j haari (x)ihaarj (x)
...
, 2M
...
(24)–(27):
ur = f (xr ) +

+

+

2M 
2M


1

2M × 2M p=1 q=1

2M


1

i=2

ρ1 × 2M

2M


1

j =2

2M × ρ2





β1 
2M


K (xp , tq , uq )haar1 (xr )ihaar1 (xr )

K (xp , tq , uq ) −

γ1
2M



p=α1 q=1

p=β1 +1 q=1

β2
2M 


γ2
2M



p=1 q=α2

K ( x p , t q , uq ) −

p=1 q=β2 +1


K (xp , tq , u(tq )) haari (xr )ihaar1 (xr )


K (xp , tq , uq ) haar1 (xr )ihaarj (xr )

342

I
...

2M

Haar wavelet method [46]

Present method

4
8
16
32
64
128

3
...
7E−3
1
...
7E−4
1
...
1E−5

3
...
0E−3
2
...
6E−5
1
...
2E−6

Fig
...
Comparison of exact and approximate solutions for Example 1
...
, 2M
...

4
...
Consider the following nonlinear Fredholm integral equation [46]:
u(x) = −x2 −

x
3


√
(2 2 − 1) + 2 +

1

xt



u(t )dt
...
In Table 1 we have compared the maximum absolute errors of the proposed method with those from the Haar wavelet method [46]
...
In Fig
...
It is evident from the
figure that the maximum absolute error decreases with the increase in number of collocation points
...
In our
case we approximate the integrand with the Haar wavelet basis and perform exact integration of Haar functions
...
Consider the following nonlinear Fredholm integral equation [13]:
u(x) = sin(π x) +

1
5

1



cos(π x) sin(π t )(u(t ))3 dt
...


(66)

I
...
2
...


Table 2
Comparison of errors with the triangular function method [13] for Example 2
...
9E−2
9
...
5E−3
7
...
8E−16
2
...
8E−16
3
...
3
...


In Fig
...
Table 2 shows the comparison of
errors of the present method with those of the triangular function method [13]
...

Example 3
...


(67)

0

The exact solution of this problem is exp(−x)
...
3, we have compared the maximum absolute errors of the present
method with those of the triangular function method [13]
...
This shows the stability of our method while the errors in the
triangular function method [13] show an oscillatory behavior
...
Aziz, Siraj-ul-Islam / Journal of Computational and Applied Mathematics 239 (2013) 333–345

Fig
...
Comparison of exact and approximate solutions for Example 4
...
Consider the following nonlinear Volterra integral equation [49]:
u(x) = f (x) +

x



xt 2 (u(t ))2 dt ,

(68)

0

where
f (x) =


1+

11
9

x+

2
3

x2 −

1
3

x3 +

2
9

x4



ln(x + 1) −

1
3

(x + x4 )(ln(x + 1))2 −

11
9

x2 +

5
18

x3 −

2
27

x4
...
In Fig
...

5
...
A two-dimensional Haar wavelet basis is used for
this purpose
...
The new algorithms do not need
any linear system solution for evaluation of the wavelet coefficients and are more efficient than conventional Haar wavelet
based methods
...

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