Haar Wavelet Method for the System of Integral Equations

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.

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: 𝐔(𝑥)−¹₀𝐊(𝑥,𝑡)𝐔(𝑡)𝑑𝑡=𝐅(𝑥),0<𝑥<1, (1) and the system of linear Volterra integral equations (SLVIEs) of the second kind: 𝐔(𝑥)−^𝑥₀𝐊(𝑥,𝑡)𝐔(𝑡)𝑑𝑡=𝐅(𝑥),0<𝑥<1, (2) where 𝐔(𝑥)=𝑢₁(𝑥),𝑢₂(𝑥),…,𝑢_𝑛(𝑥)^𝑇,𝐅(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑛(𝑥)^𝑇,𝐊(𝑥,𝑡)=𝑘_𝑖𝑗(𝑥,𝑡),𝑖,𝑗=1,2…,𝑛. (3)

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 interval 𝑡∈[0,1]. The Haar wavelet family is ℎ_𝑖(𝑥)=1for𝑡∈𝜏₁,𝜏₂−1for𝑡∈𝜏₂,𝜏₃0elsewhere. (4) Here the notations are as follows: 𝜏₁=𝑘𝑚,𝜏₂=𝑘+(1/2)𝑚,𝜏₃=𝑘+1𝑚. (5)

The integer =2^𝑗,   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=2^𝐽. The index 𝑖=1 corresponds to the scaling function of the Haar wavelet ℎ₁(𝑡)≡1.

Simple calculations show the following.

We have that ¹₀ℎ_𝑖(𝑡)ℎ_𝑙(𝑡)1𝑑𝑡=𝑚for𝑖=𝑙0for𝑖≠𝑙; (6) consequently, the functions ℎ_𝑖(𝑡) are orthogonal.

Next we discretize the functions ℎ_𝑖(𝑥) by dividing the interval 𝑥∈[0,1] into 2𝑀 parts of equal length Δ𝑥=1/(2𝑀) and introduce the collocation points: 𝑥_𝑙=𝑙−(1/2)2𝑀,𝑙=1,2,…,2𝑀. (7)

Following Chen and Hsiao [15, 16] the coefficients matrix 𝐻_𝑖𝑙=ℎ_𝑖(𝑡_𝑙) 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: 𝑢(𝑥)=^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖(𝑥), (8) where 𝑎 is the wavelet coefficients. The discrete form of this equation is 𝑢𝑥_𝑙=^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖𝑥_𝑙=^2𝑀_𝑖=1𝑎_𝑖𝐻_𝑖𝑙 (9) 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): 𝑢(𝑥)−¹₀𝑘₁₁(𝑥,𝑡)𝑢(𝑡)𝑑𝑡−¹₀𝑘₁₂(𝑥,𝑡)𝑣(𝑡)𝑑𝑡=𝑓₁(𝑥),𝑣(𝑥)−¹₀𝑘₂₁(𝑥,𝑡)𝑢(𝑡)𝑑𝑡−¹₀𝑘₂₂(𝑥,𝑡)𝑣(𝑡)𝑑𝑡=𝑓₂(𝑥),𝑥∈[0,1]. (10)

Let 𝑢(𝑥)=^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖(𝑥),𝑣(𝑥)=^2𝑀_𝑖=1𝑏_𝑖ℎ_𝑖(𝑥). (11)

Replacing (12) into (10) and (11) we find ^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖(𝑥)−^2𝑀_𝑖=1𝑎_𝑖𝐺_1𝑖(𝑥)−^2𝑀_𝑖=1𝑏_𝑖𝐺_2𝑖(𝑥)=𝑓₁(𝑥),_2𝑀𝑖=1𝑏_𝑖ℎ_𝑖(𝑥)−^2𝑀_𝑖=1𝑎_𝑖𝐺_3𝑖(𝑥)−^2𝑀_𝑖=1𝑏_𝑖𝐺_4𝑖(𝑥)=𝑓₂(𝑥), (12) where 𝐺_1𝑖(𝑥)=¹₀𝑘₁₁(𝑥,𝑡)ℎ_𝑖(𝑡)𝐺𝑑𝑡,_2𝑖(𝑥)=¹₀𝑘₁₂(𝑥,𝑡)ℎ_𝑖(𝑡)𝐺𝑑𝑡,_3𝑖(𝑥)=¹₀𝑘₂₁(𝑥,𝑡)ℎ_𝑖(𝑡)𝐺𝑑𝑡,_4𝑖(𝑥)=¹₀𝑘₂₂(𝑥,𝑡)ℎ_𝑖(𝑡)𝑑𝑡. (13)

Next we will evaluate the wavelet coefficients 𝑎_𝑖 and 𝑏_𝑖 in the following way.

Collocation Method. Satisfying (12) only at the collocation points (7) we get a system of linear equations: ^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖𝑡_𝑙−𝐺_1𝑖𝑥_𝑙−𝑏_𝑖𝐺_2𝑖𝑥_𝑙=𝑓₁𝑥_𝑙,_2𝑀𝑖=1𝑏_𝑖ℎ_𝑖𝑡_𝑙−𝐺_4𝑖𝑥_𝑙−𝑎_𝑖𝐺_3𝑖𝑥_𝑙=𝑓₂𝑥_𝑙,𝑙=1,2,…,2𝑀. (14) Therefore the matrix form of this system is given by 𝑎𝐻−𝐺₁−𝑏𝐺₂=𝐹₁,𝑏𝐻−𝐺₄−𝑎𝐺₃=𝐹₂,where𝐆_𝑖𝑙=𝐆_𝑖𝑥_𝑙,𝐅_𝑙=𝐟𝑥_𝑙. (15)

Now, we can present the following problems.

Example 1. Consider the following SLFIEs [3, 16]: 𝑢(𝑥)−¹₀(𝑥−𝑡)³𝑢(𝑡)𝑑𝑡−¹₀(𝑥−𝑡)²𝑣(𝑡)=1𝑑𝑡−2011530𝑥+3𝑥²−13𝑥³,𝑣(𝑥)−¹₀(𝑥−𝑡)⁴𝑢(𝑡)𝑑𝑡−¹₀(𝑥−𝑡)³𝑣(𝑡)1𝑑𝑡=−−3041360𝑥+𝑥20²+23𝑥12³−13𝑥⁴. (16) Carrying out the integration in (13) we obtain 𝐺₁(𝑥)=𝑥1+𝑥²−1+6𝑥²4for𝑖=17−24𝑚𝑥+24𝑘+𝑘²−2𝑘𝑚𝑥+𝑚²𝑥²32𝑚⁴𝐺for𝑖>1,₂(𝑥)=13+()−𝑥−1𝑥for𝑖=11+2𝑘−2𝑚𝑥4𝑚³𝐺for𝑖>1,₃(𝑥)=15((−)+𝑥𝑥−1)(1+𝑥𝑥−1))for𝑖=1(1+2𝑘−2𝑚𝑥3+8𝑘²()()×+𝑘8−16𝑚𝑥+8𝑚𝑥−1+𝑚𝑥16𝑚⁵⁻¹𝐺for𝑖>1,₄(𝑥)=𝑥1+𝑥²−1+6𝑥²4for𝑖=17−24𝑚𝑥+24𝑘+𝑘²−2𝑘𝑚𝑥+𝑚²𝑥²32𝑚⁴for𝑖>1. (17)
We apply collocation method, and vectors 𝑎_𝑖 and 𝑏_𝑖 can be calculated from (15); the functions 𝑢(𝑥) and 𝑣(𝑥) are evaluated from (11).
Computations were carried out for different values of 𝐽. These results were compared with the exact solution 𝑢(𝑥)=𝑥² and 𝑣(𝑥)=−𝑥+𝑥²+𝑥³.
The accuracy of the results (see Table 1 and Figure 1) was estimated by the error function for 𝑢(𝑥) and 𝑣(𝑥): 𝑒_𝐽=max_{1≤𝑙≤2𝑀}𝑢𝑥_𝑙−𝑢_𝑒𝑥𝑥_𝑙,𝑒_𝐽=max_{1≤𝑙≤2𝑀}𝑣𝑥_𝑙−𝑣_𝑒𝑥𝑥_𝑙, (18) where 𝑥_𝑙 is defined by (7).

Table 1 

Error of 𝑢(𝑥) and 𝑣(𝑥) of Example 1.

(a)

(b)

Figure 1 

(a) Exact and Haar wavelet solution of 𝑢(𝑥) at 𝐽=5. (b) Exact and Haar wavelet solution of 𝑣(𝑥) at 𝐽=5.

Example 2. Consider the following SLFIEs given by [10, 12]: 𝑢(𝑥)+¹₀𝑒^𝑥−𝑡𝑢(𝑥)𝑑𝑡+¹₀𝑒^(𝑥+2)𝑡𝑣(𝑥)𝑑𝑡=2𝑒^𝑥+𝑒^𝑥+1−1,𝑣(𝑥)+𝑥+1₁₀𝑒^𝑥𝑡𝑢(𝑥)𝑑𝑡+¹₀𝑒^𝑥+𝑡𝑣(𝑥)𝑑𝑡=𝑒^𝑥+𝑒^−𝑥+𝑒^𝑥+1−1.𝑥+1 (19) Carrying out the integration in (13) we obtain 𝐺₁(𝑥)=(−1+𝑒)𝑒^−1+𝑥𝑒for𝑖=1_{−((1+𝑘)/𝑚)+𝑥}−1+𝑒^1/2𝑚2𝐺for𝑖>1,₂(𝑥)=−1+𝑒^2+𝑥−𝑒2+𝑥for𝑖=1_{𝑘(2+𝑥)/𝑚}−1+𝑒^{(2+𝑥)/2𝑚2}𝐺2+𝑥for𝑖>1,₃(𝑥)=−1+𝑒^𝑥𝑥−for𝑖=1𝑥−𝑒^{𝑘𝑥/𝑚}−1+𝑒^𝑥/2𝑚2/𝑥4𝑚²𝐺for𝑖>1,₄(𝑥)=(−1+𝑒)𝑒^𝑥for𝑖=1−𝑒^{(𝑘/𝑚)+𝑥}−1+𝑒^1/2𝑚2for𝑖>1. (20)
We apply collocation method, and vectors 𝑎_𝑖 and 𝑏_𝑖 can be calculated from (15); the functions 𝑢(𝑥) and 𝑣(𝑥) are evaluated from (11).
Computations were carried out for different values of 𝐽. These results were compared with the exact solution (see Table 2 and Figure 2) 𝑢(𝑥)=𝑒^𝑥 and 𝑣(𝑥)=𝑒^−𝑥.

Table 2 

Error of 𝑢(𝑥) and 𝑣(𝑥) of Example 2.

(a)

(b)

Figure 2 

(a) Exact and Haar wavelet solution of 𝑢(𝑥) at 𝐽=5. (b) Exact and Haar wavelet solution of 𝑣(𝑥) at 𝐽=5.

Example 3. Consider the following SLFIEs given by [10]: 𝑢(𝑥)−¹₀𝑥+𝑡3𝑢(𝑡)𝑑𝑡−¹₀𝑥+𝑡3𝑣(𝑡)𝑥𝑑𝑡=+1817,𝑣(𝑥)−36₁₀𝑥𝑡𝑢(𝑡)𝑑𝑡−¹₀𝑥𝑡𝑣(𝑡)𝑑𝑡=𝑥²−1912+1. (21) Carrying out the integration in (13) we obtain 𝐺_𝑗(𝑥)=16+𝑥3−1for𝑖=1𝑗=1,2,3,412𝑚².for𝑖>1 (22) We apply collocation method, and vectors 𝑎_𝑖 and 𝑏_𝑖 can be calculated from (15); the functions 𝑢(𝑥) and 𝑣(𝑥) are evaluated from (11).
Computations were carried out for different values of 𝐽 (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 of 𝑢(𝑥) and 𝑣(𝑥) of Example 3.

(a)

(b)

Figure 3 

(a) Exact and Haar wavelet solution of 𝑢(𝑥) at 𝐽=5. (b) Exact and Haar wavelet solution of 𝑣(𝑥) at 𝐽=5.

4. System of Volterra Integral Equations

Let us consider system of linear Volterra integral equations (SLVIEs) (2) at (𝑖,𝑗=1,2,𝑛=2): 𝑢(𝑥)−^𝑥₀𝑘₁₁(𝑥,𝑡)𝑢(𝑡)𝑑𝑡−^𝑥₀𝑘₁₂(𝑥,𝑡)𝑣(𝑡)𝑑𝑡=𝑓₁(𝑥),𝑣(𝑥)−^𝑥₀𝑘₂₁(𝑥,𝑡)𝑢(𝑡)𝑑𝑡−^𝑥₀𝑘₂₂(𝑥,𝑡)𝑣(𝑡)𝑑𝑡=𝑓₂(𝑥),𝑥∈[0,1]. (23)

Its discrete form is 𝑢𝑥_𝑙−^𝑥𝑙₀𝑘₁₁𝑥_𝑙𝑢(𝑡)−,𝑡𝑑𝑡_𝑥𝑙0𝑘₁₂𝑥_𝑙,𝑡𝑣(𝑡)𝑑𝑡=𝑓₁𝑥_𝑙,𝑣𝑥_𝑙−^𝑥𝑙₀𝑘₂₁𝑥_𝑙𝑢(𝑡)−,𝑡𝑑𝑡_𝑥𝑙0𝑘₂₂𝑥_𝑙,𝑡𝑣(𝑡)𝑑𝑡=𝑓₂𝑥_𝑙,𝑥∈[0,1], (24) where 𝑥_𝑙 defined in (7) are the collocation points.

Let 𝑢(𝑥)=^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖(𝑥),𝑣(𝑥)=^2𝑀_𝑖=1𝑏_𝑖ℎ_𝑖(𝑥). (25) Replacing (25) with (24) we find ^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖𝑥_𝑙−^2𝑀_𝑖=1𝑎_𝑖𝐺_11𝑖𝑥_𝑙−^2𝑀_𝑖=1𝑏_𝑖𝐺_12𝑖𝑥_𝑙=𝑓₁𝑥_𝑙,_2𝑀𝑖=1𝑏_𝑖ℎ_𝑖𝑥_𝑙−^2𝑀_𝑖=1𝑎_𝑖𝐺_21𝑖𝑥_𝑙−^2𝑀_𝑖=1𝑏_𝑖𝐺_22𝑖𝑥_𝑙=𝑓₂𝑥_𝑙. (26)

The matrices (𝐺_𝑠𝑟)_𝑖𝑙=(𝐺_𝑠)_𝑖(𝑥_𝑙),   𝑠,𝑟=1,2, are now defined as 𝐺_11𝑖𝑙=^𝑥𝑙₀𝑘₁₁𝑥_𝑙,𝑡ℎ_𝑖(𝑡)𝐺𝑑𝑡,_12𝑖𝑙=^𝑥𝑙₀𝑘₁₂𝑥_𝑙,𝑡ℎ_𝑖(𝑡)𝐺𝑑𝑡,_21𝑖𝑙=^𝑥𝑙₀𝑘₂₁𝑥_𝑙,𝑡ℎ_𝑖(𝑡)𝐺𝑑𝑡,_22𝑖𝑙=^𝑥𝑙₀𝑘₂₂𝑥_𝑙,𝑡ℎ_𝑖(𝑡)𝑑𝑡. (27)

By computing these integrals the following cases should be distinguished: 𝐺_{𝑠𝑟𝑖𝑙}=0if𝑥_𝑙<𝜏_{1𝑥𝑙𝜏1}𝑘_𝑠𝑟𝑥_𝑙,𝑡𝑑𝑡if𝜏₁≤𝑥_𝑙<𝜏_2𝜏2𝜏1𝑘_𝑠𝑟𝑥_𝑙,𝑡𝑑𝑡−^𝑥𝑙_𝜏2𝑘_𝑠𝑟𝑥_𝑙,𝑡𝑑𝑡if𝜏₂≤𝑥_𝑙<𝜏_3𝜏2𝜏1𝑘_𝑠𝑟𝑥_𝑙,𝑡𝑑𝑡−^𝜏3_𝜏2𝑘_𝑠𝑟𝑥_𝑙,𝑡𝑑𝑡if𝑥_𝑙≥𝜏₃. (28)

Next we will evaluate the wavelet coefficients 𝑎_𝑖 and 𝑏_𝑖 in the following way.

Collocation Method. Satisfying (27) and (28) only at the collocation points (7) we get a system of linear equations ^2𝑀_𝑖=1𝑎_𝑖ℎ_𝑖𝑡_𝑙−𝐺_11𝑖𝑥_𝑙−𝑏_𝑖𝐺_12𝑖𝑥_𝑙=𝑓₁𝑥_𝑙,_2𝑀𝑖=1𝑏_𝑖ℎ_𝑖𝑡_𝑙−𝐺_22𝑖𝑥_𝑙−𝑎_𝑖𝐺_21𝑖𝑥_𝑙=𝑓₂𝑥_𝑙,𝑙=1,2,…,2𝑀. (29) The matrix form of this system is 𝑎𝐻−𝐺₁₁−𝑏𝐺₁₂=𝐹₁,𝑏𝐻−𝐺₂₂−𝑎𝐺₂₁=𝐹₂. (30)

Example 4. Consider the following (SLVIEs) [17]: 𝑢(𝑥)−^𝑥₀𝑥𝑡𝑢(𝑡)𝑑𝑡−^𝑥₀(𝑥+𝑡)𝑣(𝑡)𝑑𝑡=𝑒^2𝑥−12𝑥²+14𝑥+1+𝑒^−2𝑥1𝑥+4−341𝑥−4,𝑣(𝑥)−^𝑥₀(𝑥−𝑡)𝑢(𝑡)𝑑𝑡−^𝑥₀(𝑥+𝑡)²𝑣(𝑡)𝑑𝑡=𝑒^−2𝑥2𝑥²5+𝑥+4−14𝑒^2𝑥−12𝑥². (31) Carrying out the integration in (28) we obtain 𝐺_11𝑖𝑙=0for𝑥_𝑙<𝜏₁12𝑥³_𝑙−𝑥_𝑙𝜏²₁for𝜏₁≤𝑥_𝑙<𝜏₂−12𝑥_𝑙𝑥²_𝑙+𝜏²₁−2𝜏²₂for𝜏₂≤𝑥_𝑙<𝜏₃−12𝑥_𝑙𝜏²₁−2𝜏²₂+𝜏²₃for𝑥_𝑙≥𝜏₃,𝐺_12𝑖𝑙=0for𝑥_𝑙<𝜏₁12𝑥_𝑙−𝜏₁3𝑥_𝑙+𝜏₁for𝜏₁≤𝑥_𝑙<𝜏₂−3𝑥²_𝑙2−𝜏²₁2−𝑥_𝑙𝜏₁−2𝜏₂+𝜏²₂for𝜏₂≤𝑥_𝑙<𝜏₃−𝜏²₁2+𝜏²₂−𝜏²₃2−𝑥_𝑙𝜏₁−2𝜏₂+𝜏₃for𝑥_𝑙≥𝜏₃,𝐺_21𝑖𝑙=0for𝑥_𝑙<𝜏₁12𝑥_𝑙−𝜏₁²for𝜏₁≤𝑥_𝑙<𝜏₂12−𝑥²_𝑙+𝜏²₁−2𝑥_𝑙×𝜏₁−2𝜏₂−2𝜏²₂for𝜏₂≤𝑥_𝑙<𝜏₃12𝜏²₁−2𝜏²₂+𝜏²₃−2𝑥_𝑙𝜏₁−2𝜏₂+𝜏₃for𝑥_𝑙≥𝜏₃,𝐺_22𝑖𝑙=0for𝑥_𝑙<𝜏₁8𝑥³_𝑙3−13𝑥_𝑙+𝜏₁³for𝜏₁≤𝑥_𝑙<𝜏₂13−8𝑥³_𝑙−𝑥_𝑙+𝜏₁³+2𝑥_𝑙+𝜏₂³for𝜏₂≤𝑥_𝑙<𝜏₃13−𝑥_𝑙+𝜏₁³+2𝑥_𝑙+𝜏₂³−𝑥_𝑙+𝜏₃³for𝑥_𝑙≥𝜏₃. (32)
We apply collocation method, and vectors 𝑎_𝑖 and 𝑏_𝑖 can be calculated from (30); the functions 𝑢(𝑥) and 𝑣(𝑥) are evaluated from (25).
Computations were carried out for different values of 𝐽 (see Table 4 and Figure 4). These results were compared with the exact solution 𝑢(𝑥)=𝑒^2𝑥 and 𝑣(𝑥)=𝑒^−2𝑥.
The accuracy of the results was estimated by the error function for 𝑢(𝑥) and 𝑣(𝑥): 𝑒_𝐽=max_{1≤𝑙≤2𝑀}𝑢𝑥_𝑙−𝑢_𝑒𝑥𝑥_𝑙,𝑒_𝐽=max_{1≤𝑙≤2𝑀}𝑣𝑥_𝑙−𝑣_𝑒𝑥𝑥_𝑙, (33) where 𝑥_𝑙 is defined by (7).

Table 4 

Error of 𝑢(𝑥) and 𝑣(𝑥) of Example 4.

(a)

(b)

Figure 4 

(a) Exact and Haar wavelet solution of 𝑢(𝑥) at 𝐽=5. (b) Exact and Haar wavelet solution of 𝑣(𝑥) at 𝐽=5.

Example 5. Consider the following (SLVIEs) [4, 18]: 𝑢(𝑥)=−1−𝑥cos(𝑥)²(𝑥)(𝑥)1+2cos+sin1−2(𝑥)+𝑥+cos_𝑥0())𝑢(𝑡)+((𝑥𝑣(𝑡𝑣(𝑥)(𝑥)((sin𝑥−𝑡−11−𝑡cos))))𝑑𝑡,=−𝑥+sin+^𝑥₀(𝑢(𝑡)+(𝑥−𝑡)𝑣(𝑡))𝑑𝑡. (34) Carrying out the integration in (28) we obtain 𝐺_11𝑖𝑙=0for𝑥_𝑙<𝜏₁1−cos𝑥_𝑙−𝜏₁−𝑥_𝑙+𝜏₁for𝜏₁≤𝑥_𝑙<𝜏₂−1−cos𝑥_𝑙−𝜏₁+2cos𝑥_𝑙−𝜏₂+𝑥_𝑙+𝜏₁−2𝜏₂for𝜏₂≤𝑥_𝑙<𝜏₃−cos𝑥_𝑙−𝜏₁+2cos𝑥_𝑙−𝜏₂−cos𝑥_𝑙−𝜏₃+𝜏₁−2𝜏₂+𝜏₃for𝑥_𝑙≥𝜏₃,𝐺_12𝑖𝑙=0for𝑥_𝑙<𝜏₁𝑥_𝑙−𝜏₁+12cos𝑥_𝑙−𝑥²_𝑙+𝜏²₁for𝜏₁≤𝑥_𝑙<𝜏₂−𝑥_𝑙−𝜏₁+2𝜏₂+12cos𝑥_𝑙𝑥²_𝑙+𝜏²₁−2𝜏²₂for𝜏₂≤𝑥_𝑙<𝜏₃12−2𝜏₁−2𝜏₂+𝜏₃+cos𝑥_𝑙𝜏²₁−2𝜏²₂+𝜏²₃for𝑥_𝑙≥𝜏₃,𝐺_21𝑖𝑙=0for𝑥_𝑙<𝜏₁𝑥_𝑙−𝜏₁for𝜏₁≤𝑥_𝑙<𝜏₂−𝑥_𝑙−𝜏₁+2𝜏₂for𝜏₂≤𝑥_𝑙<𝜏₃−𝜏₁+2𝜏₂−𝜏₃for𝑥_𝑙≥𝜏₃,𝐺_22𝑖𝑙=0for𝑥_𝑙<𝜏₁12𝑥_𝑙−𝜏₁²for𝜏₁≤𝑥_𝑙<𝜏₂12−𝑥²_𝑙+𝜏²₁−2𝑥_𝑙𝜏₁−2𝜏₂−2𝜏²₂for𝜏₂≤𝑥_𝑙<𝜏₃12𝜏²₁−2𝜏²₂+𝜏²₃−2𝑥_𝑙𝜏₁−2𝜏₂+𝜏₃for𝑥_𝑙≥𝜏₃. (35)
We apply collocation method, and vectors 𝑎_𝑖 and 𝑏_𝑖 can be calculated from (29); the functions 𝑢(𝑥) and 𝑣(𝑥) are evaluated from (25).
Computations were carried out for different values of 𝐽 (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 of 𝑢(𝑥) and 𝑣(𝑥) of Example 5.

(a)

(b)

Figure 5 

(a) Exact and Haar wavelet solution of 𝑢(𝑥) at 𝐽=5. (b) Exact and Haar wavelet solution of 𝑣(𝑥) at 𝐽=5.

Example 6. Consider the following (SLVIEs): 𝑢(𝑥)=1−𝑥²+^𝑥₀𝑣(𝑡)𝑑𝑡,𝑣(𝑥)=𝑥+^𝑥₀𝑢(𝑡)𝑑𝑡. (36) Carrying out the integration in (28) we obtain 𝐺_12𝑖𝑙=0for𝑥_𝑙<𝜏₁𝑥_𝑙−𝜏₁for𝜏₁≤𝑥_𝑙<𝜏₂−𝑥_𝑙−𝜏₁+2𝜏₂for𝜏₂≤𝑥_𝑙<𝜏₃−𝜏₁+2𝜏₂−𝜏₃for𝑥_𝑙≥𝜏₃,𝐺_21𝑖𝑙=0for𝑥_𝑙<𝜏₁𝑥_𝑙−𝜏₁for𝜏₁≤𝑥_𝑙<𝜏₂−𝑥_𝑙−𝜏₁+2𝜏₂for𝜏₂≤𝑥_𝑙<𝜏₃−𝜏₁+2𝜏₂−𝜏₃for𝑥_𝑙≥𝜏₃. (37)
We apply collocation method, and vectors 𝑎_𝑖 and 𝑏_𝑖 can be calculated from (29); the functions 𝑢(𝑥) and 𝑣(𝑥) are evaluated from (25).
Computations were carried out for different values of 𝐽 (see Table 6 and Figure 6). These results were compared with the exact solution 𝑢(𝑥)=1 and 𝑣(𝑥)=2𝑥.

Table 6 

Error of 𝑢(𝑥) and 𝑣(𝑥) of Example 6.

(a)

(b)

Figure 6 

(a) Exact and Haar wavelet solution of 𝑢(𝑥) at 𝐽=5. (b) Exact and Haar wavelet solution of 𝑣(𝑥) at 𝐽=5.

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.