# Haar Wavelet Method for the System of Integral Equations

Authors: Hassan A. Zedan; Eman Alaidarous
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We employed  the Haar wavelet method  to find  numerical solution of the system of Fredholm integral equations (SFIEs) and  the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of  the results is obtained by the Haar wavelet method  with the exact solution.


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## Body

## 1. Introduction

The Haar wavelet was initiated and independently developed by Haar [1] in the early nineteen tens. In recent years, many different methods and different basis functions have been used to estimate the solution of the system of integral equations, such as Adomian decomposition method [1, 2], Taylor’s expansion method [3, 4], homotopy perturbation method [5, 6], projection method and Nystrom method [7], Spline collocation method [8], Runge-Kutta method [9], sinc method [10], Tau method [11], block-pulse functions, hat basis functions method [12], and operational matrices [13, 14].In the present paper, we use Haar wavelet method to solve the system of linear Fredholm integral equations (SLFIEs) of the second kind:(1)U(x)-∫01K(x,t)U(t)dt=F(x),0<x<1,
and the system of linear Volterra integral equations (SLVIEs) of the second kind:
(2)U(x)-∫0xK(x,t)U(t)dt=F(x),0<x<1,
where
(3)U(x)=[u1(x),u2(x),…,un(x)]T,F(x)=[f1(x),f2(x),…,fn(x)]T,K(x,t)=kij(x,t),i,j=1,2…,n.In (1) and (2), the functions K and F are given, and U is the vector function of the solution of systems (1) and (2) that will be determined. Also, we assume that (1) and (2) have a unique solution.


## 2. Haar Wavelet Method

Let us confine to the time intervalt∈[0,1]. The Haar wavelet family is
(4)hi(x)={1fort∈[τ1,τ2]-1fort∈[τ2,τ3]0elsewhere.
Here the notations are as follows:
(5)τ1=km,τ2=k+(1/2)m,τ3=k+1m.The integer=2j,   j=0,1,…,J, indicates the level of the wavelet; k=0,1,…,m-1 is the translation parameter. The integer J determines the maximal level of resolution. The index i is calculated from the formula i=m+k+1; the minimal value for which (4) holds is i=2 (then m=1,k=0); the maximal value is i=2M, where M=2J. The index i=1 corresponds to the scaling function of the Haar wavelet h1(t)≡1.Simple calculations show the following.We have that(6)∫01hi(t)hl(t)dt={1mfori=l0fori≠l;
consequently, the functions hi(t) are orthogonal.Next we discretize the functionshi(x) by dividing the interval x∈[0,1] into 2M parts of equal length Δx=1/(2M) and introduce the collocation points:
(7)xl=l-(1/2)2M,l=1,2,…,2M.Following Chen and Hsiao [15, 16] the coefficients matrix Hil=hi(tl) is introduced (this is a 2M×2M matrix). A function u(t) which is defined in the interval x∈[0,1] can be expanded into the Haar wavelet series:
(8)u(x)=∑i=12Maihi(x),
where a is the wavelet coefficients. The discrete form of this equation is
(9)u(xl)=∑i=12Maihi(xl)=∑i=12MaiHil
or in a matrix presentation u=aH, where u and a are 2M dimensional row vectors.


## 3. System of Fredholm Integral Equation

Let us consider SLFIEs (1) at (i,j,n=1,2):
(10)u(x)-∫01k11(x,t)u(t)dt-∫01k12(x,t)v(t)dt=f1(x),v(x)-∫01k21(x,t)u(t)dt-∫01k22(x,t)v(t)dt=f2(x),aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaax∈[0,1].Let(11)u(x)=∑i=12Maihi(x),v(x)=∑i=12Mbihi(x).Replacing (12) into (10) and (11) we find
(12)∑i=12Maihi(x)-∑i=12MaiG1i(x)-∑i=12MbiG2i(x)=f1(x),∑i=12Mbihi(x)-∑i=12MaiG3i(x)-∑i=12MbiG4i(x)=f2(x),
where
(13)G1i(x)=∫01k11(x,t)hi(t)dt,G2i(x)=∫01k12(x,t)hi(t)dt,G3i(x)=∫01k21(x,t)hi(t)dt,G4i(x)=∫01k22(x,t)hi(t)dt.Next we will evaluate the wavelet coefficientsai and bi in the following way.Collocation Method. Satisfying (12) only at the collocation points (7) we get a system of linear equations:
(14)∑i=12M[ai(hi(tl)-G1i(xl))-biG2i(xl)]=f1(xl),∑i=12M[bi(hi(tl)-G4i(xl))-aiG3i(xl)]=f2(xl),aaaaaaaaaaaaaaaaaaaaaaal=1,2,…,2M.
Therefore the matrix form of this system is given by
(15)a(H-G1)-bG2=F1,b(H-G4)-aG3=F2,whereGil=Gi(xl),Fl=f(xl).Now, we can present the following problems.Example 1.
Consider the following SLFIEs [3, 16]:
(16)u(x)-∫01(x-t)3u(t)dt-∫01(x-t)2v(t)dt=120-1130x+53x2-13x3,v(x)-∫01(x-t)4u(t)dt-∫01(x-t)3v(t)dt=-130-4160x+320x2+2312x3-13x4.
Carrying out the integration in (13) we obtain
(17)G1(x)m={x(1+x2)-1+6x24fori=17-24mx+24(k+k2-2kmx+m2x2)32m4fori>1,G2(x)m={13+(x-1)xfori=1-1+2k-2mx4m3fori>1,G3(x)m={15+x(x-1)(1+x(x-1))fori=1-((1+2k-2mx)(3+8k2+k(8-16mx)mmmmmmmmmm+8mx(-1+mx)3+8k2)(1+2k-2mx))×(16m5)-1mmmmmmmmmmmmmimfori>1,G4(x)m={x(1+x2)-1+6x24fori=17-24mx+24(k+k2-2kmx+m2x2)32m4fori>1.
We apply collocation method, and vectorsai and bi can be calculated from (15); the functions u(x) and v(x) are evaluated from (11).
Computations were carried out for different values ofJ. These results were compared with the exact solution u(x)=x2 and v(x)=-x+x2+x3.
The accuracy of the results (see Table1 and Figure 1) was estimated by the error function for u(x) and v(x):
(18)eJ=max⁡1≤l≤2M⁡|u(xl)-uex(xl)|,eJ=max⁡1≤l≤2M⁡|v(xl)-vex(xl)|,
where xl is defined by (7).Table 1
Error ofu(x) and  v(x) of Example 1.
J
2
M
Error of  functionu(x)
Error of  functionv(x)
2
8
1.45
E
-
3
1.57
E
-
3
3
16
3.65
E
-
4
4.16
E
-
4
4
32
9.19
E
-
5
1.06
E
-
4
5
64
2.29
E
-
5
2.68
E
-
5(a) Exact and Haar wavelet solution ofu(x) at J=5. (b) Exact and Haar wavelet solution of v(x) at J=5.
(a)
(b)Example 2.
Consider the following SLFIEs given by [10, 12]:
(19)u(x)+∫01ex-tu(x)dt+∫01e(x+2)tv(x)dt=2ex+ex+1-1x+1,v(x)+∫01extu(x)dt+∫01ex+tv(x)dt=ex+e-x+ex+1-1x+1.
Carrying out the integration in (13) we obtain
(20)G1(x)={(-1+ⅇ)ⅇ-1+xfori=1ⅇ-((1+k)/m)+x(-1+ⅇ1/2m)2fori>1,G2(x)={-1+ⅇ2+x2+xfori=1-ⅇk(2+x)/m(-1+ⅇ(2+x)/2m)22+xfori>1,G3(x)={-1+ⅇxxfori=1-x-ⅇkx/m(-1+ⅇx/2m)2/x4m2fori>1,G4(x)={(-1+ⅇ)ⅇxfori=1-ⅇ(k/m)+x(-1+ⅇ1/2m)2fori>1.
We apply collocation method, and vectorsai and bi can be calculated from (15); the functions u(x) and v(x) are evaluated from (11).
Computations were carried out for different values ofJ. These results were compared with the exact solution (see Table 2 and Figure 2) u(x)=ex and v(x)=e-x.Table 2
Error ofu(x) and  v(x) of Example 2.
J
2
M
Error of  functionu(x)
Error of  functionv(x)
2
8
2.28
E
-
2
7.30
E
-
3
3
16
5.79
E
-
3
1.82
E
-
3
4
32
1.45
E
-
3
4.56
E
-
4
5
64
2.95
E
-
4
4.62
E
-
5(a) Exact and Haar wavelet solution ofu(x) at J=5. (b) Exact and Haar wavelet solution of v(x) at J=5.
(a)
(b)Example 3.
Consider the following SLFIEs given by [10]:
(21)u(x)-∫01x+t3u(t)dt-∫01x+t3v(t)dt=x18+1736,v(x)-∫01xtu(t)dt-∫01xtv(t)dt=x2-1912+1.
Carrying out the integration in (13) we obtain
(22)Gj(x)={16+x3fori=1j=1,2,3,4-112m2fori>1.
We apply collocation method, and vectors ai and bi can be calculated from (15); the functions u(x) and v(x) are evaluated from (11).
Computations were carried out for different values ofJ (see Table 3 and Figure 3). These results were compared with the exact solution u(x)=x+1 and v(x)=x2+1.
By comparing the results obtained with the results found in [3, 10, 12], we find that the results we have obtained are accurate and error rate, which is much less but almost nonexistent compared to other methods, is also shown in tables of error attached and this shows the importance of the method used.Table 3
Error ofu(x) and  v(x) of Example 3.
J
2
M
Error of  functionu(x)
Error of  functionv(x)
2
8
6.15
E
-
3
8.22
E
-
3
3
16
1.57
E
-
3
2.13
E
-
3
4
32
3.96
E
-
4
5.42
E
-
4
5
64
9.95
E
-
5
1.36
E
-
4(a) Exact and Haar wavelet solution ofu(x) at J=5. (b) Exact and Haar wavelet solution of v(x) at J=5.
(a)
(b)


## 4. System of Volterra Integral Equations

Let us consider system of linear Volterra integral equations (SLVIEs) (2) at (i,j=1,2,n=2):
(23)u(x)-∫0xk11(x,t)u(t)dt-∫0xk12(x,t)v(t)dt=f1(x),v(x)-∫0xk21(x,t)u(t)dt-∫0xk22(x,t)v(t)dt=f2(x),aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiix∈[0,1].Its discrete form is(24)u(xl)-∫0xlk11(xl,t)u(t)dtihh-∫0xlk12(xl,t)v(t)dt=f1(xl),v(xl)-∫0xlk21(xl,t)u(t)dtihh-∫0xlk22(xl,t)v(t)dt=f2(xl),mmmmmmmmmmmmx∈[0,1],
where xl defined in (7) are the collocation points.Let(25)u(x)=∑i=12Maihi(x),v(x)=∑i=12Mbihi(x).
Replacing (25) with (24) we find
(26)∑i=12Maihi(xl)-∑i=12Mai(G11)i(xl)-∑i=12Mbi(G12)i(xl)=f1(xl),∑i=12Mbihi(xl)-∑i=12Mai(G21)i(xl)-∑i=12Mbi(G22)i(xl)=f2(xl).The matrices(Gsr)il=(Gs)i(xl),   s,r=1,2, are now defined as
(27)(G11)il=∫0xlk11(xl,t)hi(t)dt,(G12)il=∫0xlk12(xl,t)hi(t)dt,(G21)il=∫0xlk21(xl,t)hi(t)dt,(G22)il=∫0xlk22(xl,t)hi(t)dt.By computing these integrals the following cases should be distinguished:(28)(Gsr)ilm={0ifxl<τ1∫τ1xlksr(xl,t)dtifτ1≤xl<τ2∫τ1τ2ksr(xl,t)dt-∫τ2xlksr(xl,t)dtifτ2≤xl<τ3∫τ1τ2ksr(xl,t)dt-∫τ2τ3ksr(xl,t)dtifxl≥τ3.Next we will evaluate the wavelet coefficientsai and bi in the following way.Collocation Method. Satisfying (27) and (28) only at the collocation points (7) we get a system of linear equations
(29)∑i=12M[ai(hi(tl)-(G11)i(xl))-bi(G12)i(xl)]=f1(xl),∑i=12M[bi(hi(tl)-(G22)i(xl))-ai(G21)i(xl)]=f2(xl),l=1,2,…,2M.
The matrix form of this system is
(30)a(H-G11)-bG12=F1,b(H-G22)-aG21=F2.Example 4.
Consider the following (SLVIEs) [17]:
(31)u(x)-∫0xxtu(t)dt-∫0x(x+t)v(t)dt=e2x(-12x2+14x+1)+e-2x(x+14)-34x-14,v(x)-∫0x(x-t)u(t)dt-∫0x(x+t)2v(t)dt=e-2x(2x2+x+54)-14e2x-12x2.
Carrying out the integration in (28) we obtain
(32)(G11)ilm={0forxl<τ112(xl3-xlτ12)iforτ1≤xl<τ2-12xl(xl2+τ12-2τ22)iforτ2≤xl<τ3-12xl(τ12-2τ22+τ32)iforxl≥τ3,(G12)ilm={0forxl<τ112(xl-τ1)(3xl+τ1)iiforτ1≤xl<τ2-3xl22-τ122-xl(τ1-2τ2)+τ22iiforτ2≤xl<τ3-τ122+τ22-τ322-xl(τ1-2τ2+τ3)fforxl≥τ3,(G21)ilm={0forxl<τ112(xl-τ1)2forτ1≤xl<τ212(-xl2+τ12-2xlmi×(τ1-2τ2)-2τ22-xl2+τ12-2xl)forτ2≤xl<τ312(τ12-2τ22+τ32nn-2xl(τ1-2τ2+τ3)τ12-2τ22+τ32)forxl≥τ3,(G22)ilm={0forxl<τ18xl33-13(xl+τ1)3iiforτ1≤xl<τ213(-8xl3-(xl+τ1)3+2(xl+τ2)3)iiforτ2≤xl<τ313(-(xl+τ1)3+2(xl+τ2)3-(xl+τ3)3)iiforxl≥τ3.
We apply collocation method, and vectorsai and bi can be calculated from (30); the functions u(x) and v(x) are evaluated from (25).
Computations were carried out for different values ofJ (see Table 4 and Figure 4). These results were compared with the exact solution u(x)=e2x and v(x)=e-2x.
The accuracy of the results was estimated by the error function foru(x) and v(x):
(33)eJ=max⁡1≤l≤2M⁡|u(xl)-uex(xl)|,eJ=max⁡1≤l≤2M⁡|v(xl)-vex(xl)|,
where xl is defined by (7).Table 4
Error ofu(x) and  v(x) of Example 4.
J
2
M
Error of  functionu(x)
Error of  functionv(x)
2
8
1.51
E
-
2
9.43
E
-
3
3
16
4
32
5
64
3.14
E
-
4
1.91
E
-
4(a) Exact and Haar wavelet solution ofu(x) at J=5. (b) Exact and Haar wavelet solution of v(x) at J=5.
(a)
(b)Example 5.
Consider the following (SLVIEs) [4, 18]:
(34)u(x)=-1-xcos⁡(x)2+2cos⁡(x)+sin(x)(1-12x+cos⁡(x))+∫0x((sin(x-t)-1)u(t)+(1-tcos⁡(x))v(t))ⅆt,v(x)=-x+sin⁡(x)+∫0x(u(t)+(x-t)v(t))ⅆt.
Carrying out the integration in (28) we obtain
(35)(G11)il={0forxl<τ11-cos⁡[xl-τ1]-xl+τ1forτ1≤xl<τ2-1-cos⁡[xl-τ1]+2cos⁡[xl-τ2]+xl+τ1-2τ2forτ2≤xl<τ3-cos⁡[xl-τ1]+2cos⁡[xl-τ2]-cos⁡[xl-τ3]+τ1-2τ2+τ3forxl≥τ3,(G12)il={0forxl<τ1xl-τ1+12cos⁡[xl](-xl2+τ12)forτ1≤xl<τ2-xl-τ1+2τ2+12cos⁡[xl](xl2+τ12-2τ22)forτ2≤xl<τ312(12-2(τ1-2τ2+τ3)mm+cos⁡[xl](τ12-2τ22+τ32)12)forxl≥τ3,(G21)il={0forxl<τ1xl-τ1forτ1≤xl<τ2-xl-τ1+2τ2forτ2≤xl<τ3-τ1+2τ2-τ3forxl≥τ3,(G22)il={0forxl<τ112(xl-τ1)2iiforτ1≤xl<τ212(-xl2+τ12-2xl(τ1-2τ2)-2τ22)iiforτ2≤xl<τ312(τ12-2τ22+τ32-2xl(τ1-2τ2+τ3))iiforxl≥τ3.
We apply collocation method, and vectorsai and bi can be calculated from (29); the functions u(x) and v(x) are evaluated from (25).
Computations were carried out for different values ofJ (see Table 5 and Figure 5). These results were compared with the exact solution u(x)=cos⁡(x) and v(x)=sin(x).Table 5
Error ofu(x) and  v(x) of Example 5.
J
2
M
Error of  functionu(x)
Error of  functionv(x)
2
8
1.74
E
-
3
1.94
E
-
3
3
16
4.78
E
-
4
5.07
E
-
4
4
32
1.20
E
-
4
1.29
E
-
4
5
64
30.3
E
-
5
3.27
E
-
5(a) Exact and Haar wavelet solution ofu(x) at J=5. (b) Exact and Haar wavelet solution of v(x) at J=5.
(a)
(b)Example 6.
Consider the following (SLVIEs):(36)u(x)=1-x2+∫0xv(t)ⅆt,v(x)=x+∫0xu(t)ⅆt.
Carrying out the integration in (28) we obtain
(37)(G12)il={0forxl<τ1xl-τ1forτ1≤xl<τ2-xl-τ1+2τ2forτ2≤xl<τ3-τ1+2τ2-τ3forxl≥τ3,(G21)il={0forxl<τ1xl-τ1forτ1≤xl<τ2-xl-τ1+2τ2forτ2≤xl<τ3-τ1+2τ2-τ3forxl≥τ3.
We apply collocation method, and vectorsai and bi can be calculated from (29); the functions u(x) and v(x) are evaluated from (25).
Computations were carried out for different values ofJ (see Table 6 and Figure 6). These results were compared with the exact solution u(x)=1 and v(x)=2x.Table 6
Error ofu(x) and  v(x) of Example 6.
J
2
M
Error of  functionu(x)
Error of  functionv(x)
2
8
5.76
E
-
3
4.23
E
-
3
3
16
1.47
E
-
3
1.10
E
-
3
4
32
2.72
E
-
4
2.82
E
-
4
5
64
9.36
E
-
5
7.09
E
-
5(a) Exact and Haar wavelet solution ofu(x) at J=5. (b) Exact and Haar wavelet solution of v(x) at J=5.
(a)
(b)


## 5. Conclusion

In this work the Haar wavelet method for solution of linear systems integral equations is proposed. A method of solution which is applicable for different kind of integral equations, Fredholm and Volterra systems of integral equations, is worked out. The solution is based on the collocation techniques which are proposed. The elaborated method is very simple and, as it follows from the test problems, high precision of results can be obtained with a small number of calculation points. The calculations show that by doubling the number of the calculation points the error function decreases. This result follows also from analytical considerations. It should be noted that in the case of Haar wavelets we have to solve systems of linear equations with a smaller condition number as for other methods based on piecewise constant approximation.



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*Source: 418909-2014-07-21.xml*