A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

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Abstract

In this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

Abstract

In this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

1 Introduction

Integral equations have motivated a large amount of research work in recent years. Integral equations find its applications in various fields of mathematics, science and technology has been studied extensively both at the theoretical and practical level. In particular, integral equations arise in fluid mechanics, biological models, solid state physics, kinetics in chemistry etc. In most of the cases, it is difficult to solve them, especially analytically [1]. Analytical solutions of integral equations, however, either does not exist or are difficult to find. It is precisely due to this fact that several numerical methods have been developed for finding solutions of integral equations.

Consider the Fredholm integral equation:

y(x)=f(x)+01k(x,t)y(t)dt0x,t1

where f (x) and the kernels k(x,t) are assumed to be in L2(R) on the interval 0 ≤ x,t ≤ 1. We assume that Eq.(1) has a unique solution y to be determined. There are several numerical methods for approximating the solutionof Fredholm integral equations are known and many different basic functions have been used. Such as, Lepik et al. [2] applied the Haar Wavelets. Maleknejad et. al [3] applied a combination of Hybrid taylor and block-pulse functions, Rationalized haar wavelet [4], Hermite Cubic splines [5]. Muthuvalu et al. [6] applied Half-sweep arithmetic mean method with composite trapezoidal scheme. In previous paper [35], we compared with [5] for the numerical solution of Fredholm integral equations.

Wavelets theory is a relatively new and an emerging tool in applied mathematical research area. It has been applied in a wide range of engineering disciplines; particularly, signal analysis for waveform representation and segmentations, time-frequency analysis and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms [7, 8]. Since 1991 the various types of wavelet method have been applied for the numerical solution of different kinds of integral equations [2]. In the previous work, system analysis via Haar wavelets was led by Chen and Hsiao [9], who first derived a Haar operational matrix for the integrals of the Haar function vector and put the applications for the Haar analysis into the dynamic systems. Recently, Haar wavelet method is applied for different type of problems. Namely, Bujurke et al. [10, 11, 12] used the single term Haar wavelet series for the numerical solution of stiff systems from nonlinear dynamics, nonlinear oscillator equations and Sturm-Liouville problems. Shiralashetti et al. [131415,16, 18] applied for the numerical solution of Klein?Gordan equations, multi-term fractional differential equations, singular initial value problems,nonlinear Fredholm integral equations, Riccati and Fractional Riccati Differential Equations. Shiralashetti et al. [17] have introduced the adaptive gird Haar wavelet collocation method for the numerical solution of parabolic partial differential equations.

At present, field of a graph theory is a dynamic field in both theory and applications. Graphs can be utilized as a modeling tool to solve a many problems of practical importance. Structural graph theory is naturally a very wide field itself, and the current state of art is more advanced in some parts than in others.

A graph G consists of a finite nonempty set V of n points(vertex) together with a prescribed set X of m unordered pairs of distinct points of V. Each pair x = u,v of points in X is a edge of G, and x is said to join u and v. We write x = uv and say that u and v are adjacent points (sometimes denoted u adj v).

Let G be a simple connected graph with a vertex set V(G) = {v1,v2, · · · ,vn} and edge set E(G) = {e1,e2, · · · ,en}, where |V(G)| = n and |E(G)| = m. The degree of a vertex v in G is the number of edges incident to it and denoted by degG(v). The distance between the vertices vi and v j in G is equal to the length of the shortest path joining them and is denoted by d (vi,vj|G).

The Wiener index W =W(G) of a graph G is defined as the sum of the distances between all pairs of vertices of G, that is

W=W(G)=1i<jnd(ui,vj|G).

This index was putforward by Wiener [21] in 1947 for approximating the boiling points of alkanes. The effect of approximation was surprisingly good. From that point forward, the Wiener index has attracted the attention of chemists.

Recently several modifications of the Wiener index were put forward, of which we mention the following: the hyper–Wiener index [22, 23]:

WW=WW(G)=12u<vd(u,v)2+d(uv)

the Harary index [24]:

H(G)=u<v1d(u,v)

It is worth noting that all the above structure–descriptors are either special cases of, or are simply related to the graph invariant Wl , defined as [25, 26]

Wλ=Wλ(G)=k1d(G,k)kλ

where d (G,k) is the number of pairs of vertices of the graph G whose distance is k , and where λ is some real (or complex) number. Evidently,

W=W1WW=12W2+12W1Ha=W2RW=W1.

Another related quantity is the Hosoya polynomial. In the present work, a comparison of a Haar wavelet and Hosoya polynomial methods for the numerical solution of Fredholm integral equations is proposed.

2 Properties of Haar wavelet and Hosoya polynomial

2.1 Haar wavelet

The scaling function h1(x) for the family of the Haar wavelets is defined as

h1(x)=1forx[0,1)0otherwise

The Haar wavelet family for x 2 [0,1) is defined as,

hi(x)=1forxα,β1forxβ,γ0otherwise

where α=km,β=k+0.5m,γ=k+1m

In the above definition the integer, m = 2l , l = 0,1,⋯,J, indicates the level of resolution and integer k = 0,1,⋯,m-1 is the translation parameter. Maximum level of resolution is J. The index i in (6) is calculated using, i = m +k +1: In case of minimal values m = 1 and k = 0 then i = 2: The maximal value of i is N = 2J+1.

Let us define the collocation points xj=j0.5K,j=1,2,...,N,discretize the Haar function hi(x) and the corresponding Haar coefficient matrix H (i, j) = (hi(xj)), which has the dimension N × N.

If J = 3 )N = 16,

Eq. 6 generates the Haar matrix,

H(16,16)=1111111111111111111111111111111111111111000000000000000011111111111100000000000000001111000000000000000011110000000000000000111111000000000000000011000000000000000011000000000000000011000000000000000011000000000000000011000000000000000011000000000000000011

As Haar Wavelets are orthogonal; this mean that any square integrable function over [0,1] can be expanded into Haar wavelets series as:

f(x)=i=1aihi(x)

where ai’s are Haar wavelet coefficients.

If f (x) be piecewise constant, then sum can be terminated to finite term, that is

f(x)=i=1Naihi(xi)=aTH
aT=(a1,a2,...,aN),H=(h1(x),h2(x),...,hN(x))T.

2.2 Hosoya polynomial

The Hosoya polynomial of a graph is a generating function about distance distributing, introduced by Hosoya [27] in 1988. For a connected graph G, the Hosoya polynomial denoted by H (G,λ) is defined as

H(G,λ)=k0d(G,k)λk

where d (G,k) is the number of pairs of vertices of G that are at distance k and λ is the parameter.

Comparing (4) and (10) we see that both Wl and H(λ) are determined by the numbers d (G,k) , k = 1,2, . . . . Indeed, from knowing H (G,l) one can deduce Wl (G) and vice versa. Thus, from the Hosoya polynomial it is possible to calculate a large variety of distance–based molecular structure–descriptors.

The connection between the Hosoya polynomial and the Wiener index is elementary [27, 28]:

W(G)=H(G,1)

where H'(G,λ) is the first derivative of H (G,l) . The hyper–Wiener index can be computed from the first and second derivatives as [29]

WW(G)=H(G,1)+12H(G,1).

The path on n vertices (or of length n -1) is the graph with n vertices-say, 1,2, .,n and with n -1 edges, such that vertices i and i+1 are adjacent, i = 1,2,...,n-1. For any positive integer n we denote path as Pn, then Hosoya polynomial of path is:

H(P1,λ)=k0d(P1,k)λk=1H(P2,λ)=k0d(P2,k)λk=λ+2H(P3,λ)=k0d(P3,k)λk=λ2+2λ+3...H(Pn,λ)=k0d(Pn,k)λk=n+(n1)λ+(n2)λ2+...+[n(n2)]λn2+[n(n1)λn1]

For more information about the Hosoya polynomial one can refer [30, 31, 32, 33, 34].

A function f(x) ϵ L2[0,1] is expanded as:

f(x)=i=1nciH(Pi,x)=CTHP(x),

where C and HP(x) are n × 1 matrices given by:

C=[c1,c2,...,cn]T,

and

HP(x)=[H(P1,x),H(P2,x),...,H(Pn,x)]T.

3 Method of Solution

3.1 Haar Wavelet Method(HWM)

Let us consider the equation

y(x)01k(x,t)y(t)dt=f(x),x,t[0,1]

and seek the solution in the form

y(x)=i=1Naihi(x)

where ai’s are Haar wavelet coefficients.

Substituting (15) in (14), we get

i=1Naihi(x)i=1NaiGi(x)=f(x),

where

Gi(x)=01K(x,t)hi(t)dt

Substituting the collocation points in Eq. (16) , we get a system of algebraic equations

i=1Naihi(xl)aiGi(xl)=f(xl),l=1,2,...,N

Matrix form of this system of Eq. (17) is as follows,

a(HG)=F,

Substituting the Coefficients ai in Eq. (15), we get y = (y(xl)) is the approximate solution

y=aH

where, F = f(f(xl)),G = (Gi(xl))

In the present case

Gi(xl)=01K(x,t)hi(t)dt=x+0.5fori=114m2fori>1

where K(x,t) = x + t.

3.2 Hosoya Polynomial Method (HPM)

Consider the Fredholm integral equation,

y(x)=f(x)+01K(x,t)u(t)dt,0xt1

To solve Eq. (19), the procedure is as follows:

Step 1: We first approximate y(x) as truncated series defined in Eq. (11) That is,

y(x)=CTHP(x)

where C and HP(x)are defined similarly to Eqs. (12) and (13).

Step 2: Then substituting Eq. (20) in Eq. (19), we get,

CTHP(x)=f(x)+01K(x,t)[CTHP(t)]dt

Step 3: Substituting the collocation point xi=i0.5n,i=1,2,...,nin Eq. (21). Then we obtain,

CTHP(xi)=f(xi)+CT01K(xi,t)HP(t)dt
CT(HP(xi)Z)=f,whereZ=01K(xi,t)HP(t)dt

Step 4: Now, we get the system of algebraic equations with unknown coefficients.

CTK=f,whereK=(HP(xi)Z)

Solving the above system of equations, we get the Hosoya coefficients ‘C’ and then substituting these coefficients in Eq. (20), we get the required approximate solution of Eq. (19).

4 Numerical Examples

In this section, we consider the some of the demonstrate the capability of the method and error function is presented to verify the accuracy and efficiency of the following numerical results:

Errorfunction=ye(xi)ya(xi)=i=1n(ye(xi)ya(xi))2

where ye and ya are the exact and approximate solution respectively.

Example 1.Consider Fredholm integral equation of the second kind [20],

y(x)=x6log(x)+01(x+t)y(t)dt,0x1.

which has the exact solution y(x) = x6log(x)+0.3096x+0.1752.

Table 1

Numerical result of Example 1.

xExact solutionHWM (N = 8)Abs. Error (HWM)HPM (n = 8)Abs. Error (HPM)
0.06250.19450.20461.00e-020.19471.94e-04
0.18750.23320.24531.20e-020.23342.36e-04
0.31250.27090.28501.41e-020.27112.78e-04
0.43750.30490.32111.62e-020.30523.19e-04
0.56250.33110.34951.83e-020.33153.61e-04
0.68750.34850.36892.04e-020.34894.03e-04
0.81250.36700.38952.25e-020.36754.45e-04
0.93750.42160.44622.45e-020.42214.86e-04

Example 2. Consider the Fredholm integral equation of the second kind [2],

y(x)=x2+01(x+t)y(t)dt,0x1.

which has the exact solution y(x) = x2-5x-(17/6). We solved the Eq. (24) by Hosoya polynomial method for n = 3, we get the Hosoya coefficient C1=496,C2=7,C3=1. Substituting these coefficients in Eq. (20) we obtain,

y(x)=496HP(x1)+(7)HP(x2)+(1)HP(x3)y(x)=x25x(17/6)

Solved by analyticaly, which gives the exact solution of this problem.

Example 3. Consider the Fredholm integral equation of the second kind [3],

y(x)=exp(x)+(1exp(1))x1+01(x+t)y(t)dt,0x1.

which has the exact solution y(x) = exp(x).

Table 2

Numerical result of Example 3.

xExact solutionHWM (N = 8)Abs. Error (HWM)HPM (n = 8)Abs. Error (HPM)
0.06251.06451.08922.47e-021.06455.50e-09
0.18751.20621.23622.99e-021.20626.56e-09
0.31251.36681.40213.52e-021.36687.63e-09
0.43751.54881.58934.04e-021.54888.69e-09
0.56251.75511.80084.57e-021.75519.77e-09
0.68751.98872.03975.09e-021.98871.08e-08
0.81252.25352.30985.62e-022.25351.19e-08
0.93752.55362.61516.14e-022.55361.30e-08
Table 3

Error Analysis.

Haar Wavelet Method (HWM)
Hosoya Polynomial Method (HPM)
NExample 1Example 2Example 3nExample 1Example 3
82.46e-027.2e-022.17e-0231.25e-011.83e-02
166.70e-031.7e-021.55e-0244 6.89e-022.15e-03
322.10e-034.3e-033.90e-0366 5.64e-036.86e-06
649.25e-041.3e-029.82e-0488 4.86e-041.30e-08

Example 4. Consider the Fredholm integral equation of the second kind [19],

y(x)=e2x+(13)+0113e2x(53)ty(t)dt,0x1.

which has the exact solution y(x)=e2x.Applying the proposed method to solve Eq. (26) for n = 8. We obtain the approximate solution y(x) as shown in table 4. Error analysis is shown in figure 1.

Table 4

Numerical result of Example 4.

xExact solutionMethod [19] (m = 128Abs. Error (Method [19])HPM (n = 8)Abs. Error (HPM)
01.000001.000044.00e-050.999997.39e-06
0.11.221401.221476.72e-051.221406.61e-07
0.21.491821.491802.46e-051.491821.35e-09
0.31.822111.822186.12e-051.822114.61e-08
0.42.225542.225513.09e-052.225541.29e-07
0.52.718282.718226.18e-052.718286.17e-08
0.63.320113.320175.30e-053.320111.73e-07
0.74.055194.055299.00e-054.055191.31e-07
0.84.953034.953062.75e-054.953031.25e-07
0.96.049646.049613.74e-056.049649.70e-07
17.389057.389014.60e-057.389058.99e-06
Fig. 1

Download Figure

Fig. 1

Error analysis of Example 4.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00035

Example 5. Consider the Fredholm integral equation of the second kind [4],

y(x)=exex+11x+1+01exty(t)dt,0x1.

which has the exact solution y(x) = ex. Applying the proposed method to solve Eq. (27) for N = 8. We obtain the approximate solution y(x) as shown in table 5. Error analysis is shown in figure 2.

Table 5

Numerical result of Example 5.

xExact solutionMethod [4] (k = 32)Abs. Error (Method [4])HPM (n = 8)Abs. Error (HPM)
01.000001.016421.64e-021.000003.79e-06
0.11.105171.116271.10e-021.105171.82e-07
0.21.221401.225934.52e-031.221404.30e-07
0.31.349851.346373.48e-031.349854.56e-07
0.41.491821.478641.31e-031.491824.88e-07
0.51.648721.623912.48e-021.648725.39e-07
0.61.822111.840041.79e-021.822115.80e-07
0.72.013752.020827.06e-032.013756.34e-07
0.82.225542.219366.18e-032.225546.90e-07
0.92.459602.437422.21e-022.459607.18e-07
12.718282.676904.13e-022.718281.10e-06
Fig. 2

Download Figure

Fig. 2

Error analysis of Example 5.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00035

5 Conclusion

A newly developed method Hosoya polynomial method (HPM) [35] is applied for the numerical solution of Fredholm integral equations and compared with the Haar Wavelet Method (HWM)). Both the methods reduces an integral equation into a set of algebraic equations. Numerical results of the Hosoya polynomial method (HPM) gives higher accuracy with exact ones and existing method (HWM). Some illustrative examples are tested through the error analysis. This shows that, the accuracy improves with increasing the n number of vertices of polynomial for better accuracy. Error analysis justifies the comparative study of a new developed method is effective, validity and applicability with existing method.

Communicated by Juan L.G. Guirao

Acknowledgements

The authors thank for the financial support of UGC’s UPE Fellowship vide sanction letter D. O. No. F. 14-2/2008(NS/PE), dated-19/06/2012 and F. No. 14-2/2012(NS/PE), dated 22/01/2013.

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[1]

A.M. Wazwaz. A First Course in Integral Equations. WSPC. New Jersey. 1997.

[2]

U. Lepik, E. Tamme, Application of the Haar wavelets for solution of linear integral equations, in: Dynamical Systems and Applications, Antala. Proce. (2004) 494-507.

[3]

K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions, App. Math. Comp. 149 (2004) 799-806.

[4]

K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving linear integral equations, App. Math. Comp. 160 (2005) 579-587.

[5]

K. Maleknejad, M. Yousefi, Numerical solution of the integral equation of the second kind by using wavelet bases of hermite cubic splines, App. Math. Comp. 183 (2006) 134-141.

[6]

M. S. Muthuvalu, J. Sulaiman, Half-Sweep Arithmetic Mean method with composite trapezoidal scheme for solving linear fredholm integral equations, App. Math. Comp. 217 (2011) 5442-5448.

[7]

G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Commun. Pure Appl. Math. 44 (1991) 141-183.

[8]

C. K. Chui, Wavelets: A Mathematical Tool for Signal Analysis, SIAM, Philadelphia, PA, 1997.

[9]

C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEEE Proc. Pt. D. 144 (1) (1997) 87-94.

[10]

N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, Numerical solution of stiff systems from non-linear dynamics using single term haar wavelet series, Inter. Jour. Nonlin. Dynam. 51 (2008) 595-605.

[11]

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