Jizeng Wang,Lei Zhangand Youhe Zhou

1　Introduction

An integral equation is a functional equation with unknown function under the integration sign [Baker (1977)]. Integral equations arise in a great many branches ofscience, including potential theory, acoustics, elasticity, fluid mechanics, radiativetransfer and even theory of population.

Recently,using multiresolution techniques and wavelets to develop numerical schemesfor the solution of differential and integral equations has become increasinglypopular [Beylkin, Coifman, and Rokhlin (1991); Alpert, Beylkin, Coifman, and Rokhlin(1993);Amaratunga,Williams,Qian,and Weiss(1994);Sweldens(1995);Chen,Micchelli,and Xu(1997);Kaneko,Noren,and Novaprateep(2003);Wang(2001);Zhou,Wang,and Zheng(1999);Zhou,Wang,Wang,and Liu(2011);Liu,Wang,and Zhou(2013);Liu,Wang,Zhou and Wang(2013)].Although in applications,many mathematical tools have been demonstrated valid,yet wavelet applications to the solution of nonlinear integral equations with singular kernels arising in different areas of mechanics,physics and engineering have been very limited.It was Belykin et al.[Beylkin,Coifman,and Rokhlin(1991)]who first proposed a wavelet method to solve the integral equations.Their work focused on the sparse discretization of linear integral operators,which leading to a fast numerical algorithm.The reason of the sparse and accurate discretization is mainly due to the compact support and vanishing moment properties of wavelets.Later on,works by Alpert et al.[Alpert,Beylkin,Coifman,and Rokhlin(1993)]and Chen et al.[Chen,Micchelli,and Xu(1997)]continuously attempted to develop a fast wavelet algorithm for the linear second kind integral equations.For the nonlinear integral equations with continuous kernel,several wavelet methods have been proved available,which includes the Galerkin methods based on the Legendre wavelets[Mahmoudi(2005)]and the B-spline wavelets[Sahu and Ray(2013)],and collocation method based on the Haar wavelet[Babolian and Shahsavaran(2009)].However,only a class of simple nonlinear integral equations with common nonlinear terms in the form of polynomials was studied.For the nonlinear integral equations with singular kernels,efficient methods with high accuracy are still very limited[Liang,Liu and Che(2001);Xiao,Wen and Zhang(2006);Gao and Jiang(2007);Galperin,Kansa,Makroglou and Nelson(2000);Panigrahi and Nelakanti(2012)].

Coiflets are the most efficient wavelets in constructing one-point quadrature formula with very high precision[Sweldons and Piessens(1994);Wang(2001);Maleknejad,Lot fiand Rostami(2007)].This interesting property is very convenient in dealing with nonlinear differential and integral equations[Liang,Liu and Che(2001);Maleknejad,Lot fiand Rostami(2007);Liu,Wang,and Zhou(2013);Liu,Wang,Zhou and Wang(2013)].For most wavelet Galerkin methods without using this property,solution procedures will inevitably involve complicated integral computations associated with the scaling functions even in solving certain simple nonlinear integral equations[Avudainayagam and Vani(2000);Liu,Qin,Liu and Cen(2010)].Based on this fact,a Coi flet-based algorithm for numerical solution of the second kind integral equations with continuous and weakly singular kernels is proposed in the present study.Numerical examples are considered to verify the efficiency and accuracy of the proposed method.

2　Function approximation based on series expansion of scaling functions

For a functionf(x)∈L2(R),if?(x)represents a Coiflet scaling function [Daubechies(1993)], then we can have[Daubechies(1993)],then we can have[Daubechies(1993)]

Table 1:Coiflet filter coefficients for N=2,4,and 6.

Noting that the Coiflet scaling function ?(x)has a compact support=[0,3N?1],then ?(x)can be constructed by using the filter coefficientsak,k=0,1,2,3...3N-1,in Tab 1[Wang(2001)]in terms of the relation below,

Such a scaling function has the unique property of shifted moments

Considering Eq.(3)one can have

with a degree of accuracy ofN?1.Combining Eqs.(1)and(4)yields

Such an approach to approximation is attractive because of its simplicity and a degree of accuracy ofN?1.Moreover,if it hasf(x)∈Cγ,γ≤N?1,the precision of Eq.(5)immediately becomes[Sweldons and Piessens(1994);Wang(2001)]

And by using Eq.(5),we can derive the following rules[Wang(2001)]:

Rule 1:for the composite function Π[f(x)]off(x),we have

Rule 2:for a derivative or integration operator D,we have

As we know,the wavelet theory is established on the whole real line.For many applications,the functions involved are usually defined on a bounded interval.In order to apply wavelets in these applications,some modifications will have to be made.Several constructions of wavelets on a bounded interval have become available[Cohen,Daubechies and Vial(1993);Sweldens(1995)].However,all these constructions are mathematically difficult,and the resulting wavelets are complicated to be applied to numerical analysis.We thus need to find a simple alternative solution.To be specific,let us consider the case of a unit interval[0,1].Given a functionf(x)on[0,1],the most obvious approach is to setf(x)=0 outside[0,1],and then using wavelet theory on the whole real line.However,for a general functionf(x)this padding with 0 usually introduces discontinuities at the endpoints 0 and 1,for instance the simple functionf(x)=1,x∈[0,1].Just as Cohen et al.[Cohen,Daubechies and Vial(1993)]and Sweldens[Sweldens(1995)]have pointed out that,because wavelets are effective for detecting singularities,the presence of artificial discontinuities is likely to introduce significant errors.Another approach is to consider the function to be periodic with period one,f(x+1)=f(x).However,unless the behavior of the functionf(x)at 0 matches that at 1 the periodic version off(x)has singularities there.A simple function likef(x)=x,x∈[0,1],gives a good illustration of this.A third method,which works if the bases functions are symmetric,is to use reflection across the edges.This preserves continuity,but introduces discontinuities in the first derivative.

What is really needed is that functions do not introduce discontinuities in up to a certain order of derivative at the endpoints 0 and 1.

Consider a functiong(x)∈L2[0,1],we define

By applying Taylor expansion,the function can continue at endpoints 0,1 as

where δ＞0,i=0,1,2,...,M.It can be seen that such a boundary extension treatment does not introduce discontinuities at endpoints 0 and 1.However,in practical applications,the derivatives at endpoints sometimes are unknown or even do not exist at all.And therefore we have to apply equidistant numerical difference to approximate or replace(when they do not exist.)them via the discrete points in[0,1]as

wherep0,i,j,p1,i,jare parameters associated with numerical difference andgk=g(k/2n),i=1,2,...,m.Hence Eq.(10)becomes

or

or

where

and

Thus,after applying the rules of scaling transform and Taylor expansion,we obtain the modi fied approximation series(13).It can be seen that such approximation manner does not introduce discontinuity up toMth order derivatives at the endpoints 0 and 1.

When we use the Coiflets-like bases to the numerical example at the end of the paper,here,forM=3,m=3,we take the 4-pointMalkoffformula of numerical difference:

wherei=0,1,2,3,j=0,1,2,3 and from here onward,such a numerical difference formula will be used in all the relative numerical examples of this paper.

3　Nonlinear integral equations with continuous kernel

We consider the nonlinear Fredholm integral equations of the second kind with continuous kernel as below

wheref,Kare given smooth functions.Two approaches have been considered here.

3.1　Approach 1:Galerkin method with L2(R)Coiflet bases

Iff,K,andqall belong toL2(R),the functions in Eq.(17)can be approximated by the Coiflet scaling bases onL2(R)as

De fine

Then we have

The Galerkin discretization scheme is applied to Eq.(21),giving the system of nonlinear algebraic equations

3.2　Approach 2:Galerkin method with modified Coiflets bases

Considering Eq.(17)iff,K,andqall belong toL2[0,1],then we have

De fine

which can be obtained easily by the definition(19),thus Eq.(26)becomes

Let Φn,i(x),i=0,1,···,2nbe the weighted functions,multiplying both sides of Eq.(27)by them and then taking integration from 0 and 1,we can obtain

By solving Eq.(29)using Newton-Raphson method,the solution{fk,k=0,1,···,2n}can be obtained readily.

4　Integral equations with weakly singular kernel

We consider the Fredhlom integral equations of the second kind with weakly singular kernel given by

wheres(x,y)satis fies the conditions

in whichMis a positive constant,?1＜α≤0 andris nonnegative integer.First,we will use integration by parts to change the form of the original Eq.(30)to make the singular kernel become very smooth,then solve the resulting equation,eventually convert the solution back to that of the original Eq.(30).The details are as follows.

De fine

andIk(x,y)=hk(x,y)?hk(x,1),k=1,2,3,···,in which

Considering Eq.(31),it can be easily known thatsk(x,y),Ik(x,y)∈ Ck?1.Using integration by parts toEq.(30)can be reduced to

Because we haveI1(x,1)=0,then

Repeat the integration by parts toand consider the fact ofIk(x,1)=0,we have

Letg(x)=f(J)(x),and

then we have

Thus Eq.(35)can be changed into

Substitutingx=t1,t2,t3,···,tJ∈ [0,1]into Eq.(38),ti/= ρ/2n,where ρ is an arbitrary integer,we have

in whichi=1,2,···,J.In order to get the expression off(k)(0),k=1,2,···,we change the formation of Eq.(39)to

Rewriting Eq.(40)to a matrix form,we can obtain

in which A={ai,j},F={f0j},B={bi},and

In order to solve Eq.(41)with unknown F,we denote the adjoint matrix of the matrix A asad jA={ci,j,i,j=1,2,···,J}.Then

Substituting Eq.(43)into Eq.(38),it yields

where

Thus we obtain the transformed Eq.(44)after smoothing treatment.

Using expansion formula(13)to unknown functiong(x),it yields

And consider the definition(36),it gives

Use expansion formula(13)to Eq.(44)again,we have

whereyj=j/2n,gj=g(yj).Then we have

Considering Eq.(49),and for the variablex,using the Eq.(13),we have

Let Φn,?(x),?=0,1,···,2nbe the weighted functions,multiplying both sides of Eq.(50)by them and then taking integration from 0 and 1,we can obtain

then

Also,it can be changed into matrix form as

Solving the Eq.(54)we can obtain the solutiong={gk}T.

5　Numerical examples

Six numerical examples are considered in this section,which include two linear integral equations with continuous kernels(Examples 1 and 2);two nonlinear integral equations with continuous kernel(Example 3 and 4);two nonlinear integral equations with weakly singular kernel(Example 5 and 6).All these examples were solved by using the scaling function withN=6 andM1=7.

Example 1

As the first example,we consider[Liang,Liu and Che(2001);Xiao,Wen and Zhang(2006)]

with exact solution:f(x)=cos(2πx).

Equation(56)has been solved by Liang et al.[Liang,Liu and Che(2001)]and Xiao et al.[Xiao,Wen and Zhang(2006)]by using different methods.Liang et al.[Liang,Liu and Che(2001)]have shown that using the Galerkin method of Daubechies wavelets in solving the integral equations has almost the same accuracy as that of noncontinuous multiwavelets.For the error defined by ε=||Exact solution–Approximation solution||2,Liang et al.[Liang,Liu and Che(2001)]shows that ε=O(10?4)when the resolution leveln=5 andO(10?7)when the resolution leveln=8.When Eq.(56)is solved by using the Galerkin method of periodic Daubechies wavelets[Xiao,Wen and Zhang(2006)],the maximum error becomesO(10?7)when the resolution leveln=5.

Table 2 shows the numerical results for example 1 based on our approach 1 whenn=3,4,5.It can be seen from Tab.2 that numerical results obtained by the proposed method can reach much higher precision,with maximum absolute error on the order ofO(10?16),than the results obtained by other methods[Liang,Liu and Che(2001);Xiao,Wen and Zhang(2006)].

Example 2

in whichA=130sin(1)?202cos(1),B=130cos(1)+202sin(1)?240.Exact solution:f(x)=x.As the kernelhas no definition wheny＜ 0,the approach 1 in this case is no longer valid.Thus we use the approach 2 based on the modifiedL2[0,1]Coiflets-like bases to solve Eq.(57).Table 3 gives the results whenn=3,4,5.It can be seen that,the proposed approach 2 can also have very high accuracy.The absolute error is on the order ofO(10?8)forn=5.

Table 3:Absolute Errors for Example 2.

Example 3

Exact solution:y(x)=x3.Table 4 shows the numerical results for example 3 based on approach 1 whenn=3,4,5.This nonlinear integral equation with continuous kernel also has been solved by the authors of reference[Galperin,Kansa,Makroglou and Nelson(2000)]based on the trapezoidal formula and other techniques associated with variable transformations.For the results with 40 grid points obtained by using the second order Korobov transformation[Galperin, Kansa, Makroglou and Nelson (2000))]and the third order sidi transformation[Galperin,Kansa,Makroglou and Nelson(2000)],the maximum absolute error is on the order ofO(10?6).However as shown in Tab.4,our results have the maximum absolute error on the order ofO(10?7)whenn=4,corresponding to 16 grid points.

Table 4:Absolute Errors for Example 3.

Example 4

whena=1,b=3,andq(x)=ex-(x+2e3x)/9,the exact solution can bef(x)=ex.This nonlinear integral equation has been solved in references[Mahmoudi(2005);Babolian and Shahsavaran(2009)]by numerical methods based on the Legendre and Haar wavelets,respectively.The maximum absolute error is aboutO(10?2)andO(10?3)when the resolution level isn=5 in[Babolian and Shahsavaran(2009)]andn=3 in[Mahmoudi(2005)].Table 5 lists the numerical results for example 4 based on our approach 1 whenn=3,4,5.However,as shown in Tab.5,the results by using the proposed approach 1 have the maximum absolute errorO(10?6)forn=3,which is obviously much better.

Whena=1/2,b=2,andq(x)=7x/8.The exact solution is simplyf(x)=x.This nonlinear integral equation has also been numerically solved by Sahu et al.[Sahu and Ray(2013)]by using the semi-orthogonal linear B-spline wavelets.They have shown that the maximum absolute error is aboutO(10?5)when the resolution level isn=4[Sahu and Ray(2013)].Table 6 lists the absolute errors of the numerical solutions obtained by using the proposed approach 1 whenn=3,4,and 5,respectively.When the resolution leveln=3,our results exhibit the absolute error nearlyO(10?13),being several orders of magnitude smaller.

Table 5:Absolute Errors for Example 4 with a=1,b=3.

Example 5

Table 6:Absolute Errors for Example 4 with a=1/2,b=2.

Exact solution:f(x)=x.This is an integral equation of second kind with weakly singular kernel.Equation(60)has been solved by the methods of reference[Liang,Liu and Che(2001);Panigrahi and Nelakanti(2012)].When the error is defined as ε=||Exact solution–Approximation solution||2,Liang et al.[Liang,Liu and Che(2001)]have shown that this error associated with their results is on the order ofO(10?4)when the resolution leveln=5 andO(10?7)when the resolution leveln=8.In addition,this error ε becomesO(10?2)when 8 grid points are used in reference[Panigrahi and Nelakanti(2012)].To solve this equation by our method,we defineJ=3,t1=0.2,t2=0.6,t3=0.9.Table 7 gives the numerical results whenn=3,4,5.Interestingly,for the present method,the maximum absolute error isO(10?14)when only 8 grid points,corresponding ton=3,have been used.

Table 7:Absolute Errors for Example 5.

Example 6

Table 8:Absolute Errors for Example 6.

6　Conclusions

We have proposed the Coiflet-based methods to deal with the nonlinear integral equations with weakly singular kernels.It can be seen from the numerical examples that the proposed approach is very efficient and accurate comparing with several existing methods.

Acknowledgement:This research is supported by grants from the National Natural Science Foundation of China(11472119,11121202),and the National Key Project of Magneto-Constrained Fusion Energy Development Program(2013GB 110002).

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