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Research ArticleSolving the Linear Integral Equations Based onRadial Basis Function Interpolation

Huaiqing Zhang Yu Chen and Xin Nie

The State Key Laboratory of Transmission Equipment and System Safety and Electrical New TechnologyChongqing University Chongqing 400044 China

Correspondence should be addressed to Huaiqing Zhang zhanghuaiqingcqueducn

Received 2 January 2014 Revised 14 May 2014 Accepted 15 May 2014 Published 1 June 2014

Academic Editor Saeid Abbasbandy

Copyright copy 2014 Huaiqing Zhang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The radial basis function (RBF) method especially the multiquadric (MQ) function was introduced in solving linear integralequations The procedure of MQ method includes that the unknown function was firstly expressed in linear combination formsof RBFs then the integral equation was transformed into collocation matrix of RBFs and finally solving the matrix equationand an approximation solution was obtained Because of the superior interpolation performance of MQ the method can acquirehigher precision with fewer nodes and low computations which takes obvious advantages over thin plate splines (TPS) method Inimplementation two types of integration schemes as the Gauss quadrature formula and regional split technique were put forwardNumerical results showed that the MQ solution can achieve accuracy of 1119864 minus 5 So the MQ method is suitable and promising forintegral equations

1 Introduction

In mechanics electromagnetic theory thermal radiation sci-ence and other fields the mathematical models of science orengineering problems can be attributed to the integral equa-tions Comparing to the differential equation the integralequation embodies the initial value or boundary informationjust in a brief form Moreover the numerical integral bringssmaller relative error than the numerical differentiation So itcan obtain higher accuracy and the research on numericalsolution of integral equations is of great importance Thegeneral form for linear integral equation is

119891 (119875) = 120583int

Ω

119896 (119875 119876) 119891 (119876) d119876 + 119892 (119875) 119875 119876 isin Ω (1)

In the above 119892(119875) is free function 119896(119875 119876) is kernel function119891(119876) is the unknown function Ω is integral region and 120583

is the parameter where the 119892(119875) 119896(119875 119876) and 120583 are knownThe numerical methods for solving integral equations mainlyinclude the undetermined coefficient approximationmethodthe direct numerical integration method the successiveapproximation method and the approximation method forspecific kernel function

Currently a variety of new numerical methods wereproposed for instance the combination method of Galerkinand collocation with Haar wavelet function for the first kindof linear Fredholm equation [1] the Galerkin method withDaubechies wavelet for the second kind of linear Volterraequations [2] however the calculation error of waveletmethods is relatively large The Sinc-collocation method[3] and the modified block pulse function [4] are for onedimension problemsTheChebyshev collocationmethod [5]block pulse method [6] and polynomial approximation withsmooth kernel function [7] are for solving two-dimensionalFredholm or Volterra equations In the above unknown func-tion approximation methods the polynomial Chebyshevpolynomials or wavelet function were adopted respectivelyHowever the polynomial interpolation is unstable due tothe strong rigidity of the polynomial function space theinterpolation error is relatively large for wavelets the suitableinterpolation basis functions for two or multidimensionalproblems are hard to construct and so on The interpolationbasis function with high accuracy and convenience formultidimensions is particularly necessary

And fortunately the radial basis function (RBF) notonly is insensitive to the space dimensions but also has

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 793582 8 pageshttpdxdoiorg1011552014793582

2 Journal of Applied Mathematics

high interpolation accuracy The radial basis functions werefirst developed by Hardy [8] in 1971 as a multidimensionalscattered interpolation method in modeling of the earthrsquosgravitational field It was not recognized by most of the aca-demic researchers until Franke [9] published a review paperin the evaluation of two-dimensional interpolation methodsThe RBF method in solving integral equations was initiallyproposed in 2006 [10ndash13] The related research attracted a lotof attention recentlyTherefore this paper attempted to adoptthe RBF method for solving linear integral equations TheRBF and interpolation principle were described in Section 2And in Section 3 the RBF method for solving linear integralequation and the numerical integration scheme for coeffi-cients matrix were proposed And finally the RBF methodwas applied in one- and two-dimensional integral equationsits performance was analyzed and compared

2 Radial Basis Function andInterpolation Principle

Radial basis function B 119877+ rarr 119877 (domain 119877119889) is definedas the function of distance 119903 = ||119909 minus 119909

119895|| The commonly

used RBF contains the thin plate splines function (B(119903) =

1199032 ln 119903) the Gaussian function (B(119903) = exp(minus120573119903

2)) Hardyrsquos

multiquadric (MQ) functions and so forth Franke hasproved that the MQmethod has the superior comprehensiveperformance in 29 kinds of scattered data interpolationmethods So in this paper we choose the MQ as the basisfunction which is compared with thin plate splines (TPS)function The widely used MQ function has the followingexpression

120601 (x) = radicx minus c2 + 1205722 (2)

In the above c means the center of basis function and 120572 isthe shape parameter which is generally associated with thedistance between adjacent centers So 120572 = 120573||c

119894minus c119895|| and 120573

is also called the shape parameterThe principle of RBF interpolation regards the unknown

function as linear combination of radial functions So theapproximation function can be obtained after calculating thecoefficients For a series of known data points (119909

119894 119891(119909119894)) 119894 =

1 2 119873 the approximation function can be constructed byRBF as

119891 (119909) =

119873

sum

119895=1

120582119895120601119895 (119909) =

119873

sum

119895=1

120582119895radic(119909 minus 119888

119895)2

+ 1205722 (3)

Then substituting the interpolation data points into the aboveequation we obtain

119891 (119909119894) =

119873

sum

119895=1

120582119895120601119895(119909119894) =

119873

sum

119895=1

120582119895radic(119909119894minus 119888119895)2

+ 1205722 (4)

With its abstract matrix form

[Φ119889] [120582] = [f

119889] (5)

where [f119889] = [119891(119909

1) 119891(119909

1) 119891(119909

119873)]119879 is the known data

[120582] = [1205821 1205822 120582

119873]119879 is the coefficients and [Φ

119889] is

the coefficient matrix whose element calculation formula isΦ119898119899

= radic(119909119898

minus 119888119899)2+ 1205722

The MQ function is globally defined which results in afull resultant coefficient matrix And consequently the coef-ficientmatrix in (5) is nonsingular andusually ill conditionedThe accuracy of the MQ solution depends heavily on theshape parameter In general for a fixed number of centers119873smaller shape parameters producemore accurate approxima-tions but they also are associated with a poorly conditionedinterpolation matrix Also the condition number grows with119873 for fixed values of the shape parameter120572The choice of thisoptimal value is still under intensive investigationThereforesolving (5) can acquire

[120582] = [Φ119889]minus1

[f119889] (6)

The approximation value 119891(119909) at any point 119909 is

119891 (119909) = [Φ (119909)] [120582] = [Φ (119909)] [Φ119889]minus1

[f119889] (7)

3 Solving Linear Integral Equations Based onthe RBF Interpolation

The basic idea of RBF method for solving integral equationis to employ a linear combination of RBF to approximate theunknown function thus the integral equation is transformedinto the combination form of RBFs and their coefficientsThen the weight coefficients of RBF can be calculatedby using the collocation type weighted residual methodAnd finally an approximate representation of the unknownfunction can be obtained Specifically the RBF method forsolving linear integral equations is proposed in the following

31 RBF Interpolation to Approximate One-Dimensional Lin-ear Integral Equation The general form of one-dimensionallinear integral equation is

119891 (119909) = 120583int

119887|119909

119886

119896 (119909 119905) 119891 (119905) d119905 + 119892 (119909) 119909 isin [119886 119887] (8)

Here 119887 | 119909 represents the upper limit of integralThe constant119887 or variable 119909 is respectively corresponding to Fredholm orVolterra equation In the following discussion we choose theupper limit 119887 for demonstration In RBF approximation theunknown function119891(119909) is expressed as a combination of RBFThus the approximate formula (9) is obtained

119873

sum

119894=1

120582119894120601119894 (119909) = 120583int

119887

119886

119896 (119909 119905)

119899

sum

119894=1

120582119894120601119894 (119905) d119905 + 119892 (119909) 119909 isin [119886 119887]

(9)

Journal of Applied Mathematics 3

Furthermore for any collocation point 119909119889in [119886 119887] by

using the linear property of integral we obtain the specificexpression of the above equation as

119873

sum

119894=1

120582119894120601119894(119909119889) minus 120583

119873

sum

119894=1

120582119894int

119887

119886

119896 (119909119889 119905) 120601119894 (119905) d119905 = 119892 (119909

119889)

119909119889isin [119886 119887]

(10)

when choosing 119872 collocation points hence forming thecollocation equations in matrix form as

[Φ minus 120583Κ] [120582] = [G] (11)

where [G] = [119892(1199091) 119892(1199091) 119892(119909

119872)]119879 and [120582] =

[1205821 1205822 120582

119873]119879 The elements in matrix [B] and [K] are

defined

Φ119898119899

= radic(119909119898

minus 119888119899)2+ 1205722

119870119898119899

= int

119887

119886

119896 (119909119898 119905) radic(119905 minus 119888

119899)2+ 1205722d119905 = int

119887

119886

ℎ (119905) d119905(12)

The notations 119909119898

and 119888119899represent correspondingly the

collocation point and RBF center while the element 119870119898119899

isthe definite integral of function ℎ(119905) which can be calculatedthrough the following Gauss quadrature formula

int

1

minus1

ℎ (120585) d120585 asymp

119876

sum

119902=1

119882119902ℎ (120585119902) (13)

The integral range can be transformed with formula 119905 = (119887 +

119886)2 + (119887 minus 119886)1205852 = 119901(120585) so the final calculation formula is

119870119898119899

= int

119887

119886

119896 (119909119898 119905) 120601119899 (119905) d119905

=119887 minus 119886

2int

1

minus1

119896 [119909119898 119901 (120585)] 120601119899 [119901 (120585)] d120585

asymp119887 minus 119886

2

119876

sum

119902=1

119882119902119896 [119909119898 119901 (120585119902)] 120601119899[119901 (120585119902)]

(14)

32 RBF Interpolation to Approximate Two-DimensionalLinear Integral Equation Two-dimensional linear integralequation has the general form

119891 (119909 119910) = 120583int

Ω

119896 (119909 119910 119904 119905) 119891 (119904 119905) d119904 d119905 + 119892 (119909 119910)

(119909 119910) isin Ω

(15)

The corresponding RBF approximation is

119873

sum

119894=1

120582119894120601119894(119909119889 119910119889)

= 120583

119873

sum

119894=1

120582119894int

Ω

119896 (119909119889 119910119889 119904 119905) 120601

119894 (119904 119905) d119904 d119905 + 119892 (119909119889 119910119889)

(16)

And similarly we have the interpolation matrix [Φ minus

120583Κ][120582] = [G] The differences lie in the distance formulaand the definite integral of element 119870

119898119899= intΩ

119896(119909119898 119910119898

119904 119905)120601119899(119904 119905)d119904 d119905 According to the domain type two inte-

gration schemes were put forward for regular and irregularsituation

321 Definite Integral for Regular Domain When the integraldomain Ω is regular in [119886 119887] times [119888 119889] the two-dimensionalGauss quadrature formula is adopted with integral rangetransformation 119905 = (119887 + 119886)2 + (119887 minus 119886)1205852 = 119901(120585) 119904 =

(119889 + 119888)2 + (119889 minus 119888)1205782 = 119902(120578) We have

119870119898119899

= int

119889

119888

int

119887

119886

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

asymp(119887 minus 119886) (119889 minus 119888)

4

1198991

sum

119894=1

1198992

sum

119895=1

119882119894119882119895119896 [119909119898 119910119898 119901 (120585119894) 119902 (120578

119895)] 120601119899

times [119901 (120585119894) 119902 (120578

119895)]

(17)

322 Definite Integral for Irregular Domain When the inte-gral regionΩ is irregular we can employ the similar numeri-cal integration scheme as

119870119898119899

= int

Ω

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

= sum

119902

119882119902119870(119909119898 119910119898 119904119902 119905119902) 120601119899(119904119902 119905119902)

(18)

However the traditional regional split technique was adoptedin this paper Firstly the irregular region was split geo-metrically Then the entire domain definite integral can beobtained by adding up all the integral of subregions which isindicated as the following

119870119898119899

= int

Ω

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

= sum

119894

int

Ω119894

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905 asymp sum

119894

119868119894

(19)

where Ω119894represents the 119894th subregion Specifically the trian-

gular element is chosen because of its strong adaptability inconforming two-dimensional irregular domainThe triangu-lation can be implemented automatically by the PDE toolboxinMatlabThen let the vertex coordinates denote by (119909

119894119895 119910119894119895)

119895 = 1 2 3 so the element area 119878119894and the three midpoints

(119909119894119888119895

119910119894119888119895

) of element boundary were easily obtained Andfinally the quadratic interpolation formula was applied tocalculate the definite integral as

119868119894=

119878119894

3

3

sum

119895=1

119896 (119909119898 119910119898 119909119894119888119895

119910119894119888119895

) 120601119899(119909119894119888119895

119910119894119888119895

) (20)

33 The Implementation Procedure of RBF for Solving LinearIntegral Equation In summary the implementation proce-dure of RBF for solving linear integral equation was listed asthe following


(1) Setting the centers c of RBFs and shape parameters 120572then setting the collocation points x

119889 the numbers of

RBFs and collocation points are respectively 119873 and119872 in general Note that to simplify and speed up thecomputations we can identify the set of centers withthe set of collocation points

(2) Substituting approximate solution based on RBF intothe linear integral equation then the collocationconditions are imposed to form the coefficientmatrix

(3) Matrix elements calculation the Gauss quadra-ture formula is adopted for one-dimensional andtwo-dimensional regular domain problems whilefor irregular domain case the numerical integra-tion scheme based on regional split technique wasemployed

(4) Coefficient calculation the equation [Φ minus 120583Κ][120582] =

[G] was solved by SVD method and the [120582] wasobtained

(5) RBF solution for unknown function then can beachieved as

119891 (x) asymp 119891 (x) = [Φ (x)] [120582] = [Φ (x)] [Φ minus 120583K]minus1

[G] (21)

34 Error Analysis Utilizing the same way in error analysisas [13] let the operator 119870 119877

1rarr 1198771 for one-dimensional

case be defined as

119870[119891 (119909)] = int

119887|119909

119886

119896 (119909 119905) 119891 (119905) d119905 (22)

Now we can rewrite (8) in abstract form as

(119868 minus 120583119870)119891 = 119892 (23)

Let 119870 be a bounded operator with 120583||119870|| lt 1 and then theoperator 119868 minus 120583119870 is a contraction operator by the Banachcontraction mapping principle (8) has a unique solution Insuch a case the geometric series theorem implies that theoperator 119868 minus 120583119870 has a bounded inverse with

10038171003817100381710038171003817(119868 minus 120583119870)

minus110038171003817100381710038171003817le

1

1 minus 119870 (24)

So the error of the presentedRBFmethod ismainly basedupon the RBF interpolation error which emerges in (3) andthe error caused by performing the numerical integrationscheme over domain which appears in (18) For sufficientlylarge nodes 119873 the error of the RBF interpolation is domi-nated over the integration error and thence increasing thenumber of nodes in the numerical integrationmethod has nosignificant effect on the error

4 Numerical Experiments and Analysis

In this section the RBFmethod is applied to some numericalexamples involving one-dimensional and two-dimensionalintegral equations on the regular and nonrectangular regionsTwo types of RBF as multiquadric (MQ) function and thin

plate splines (TPS) function were adopted simultaneouslyin the following simulations Their parameters were set asthe same except that the additional shape parameter wasprovided for MQ In order to measure the accuracy of themethod the maximum absolute error (MAE) the maximumrelative error (MRE) and the root mean squared error(RMSE) have been used respectively The RMSE has thefollowing definition as

RMSE = radic1

119898119873

119898119873

sum

119896=1

[119891119890(119909119896) minus 119891(119909

119896)]2 (25)

where 119891119890(119909119896) 119891(119909

119896) denote the exact and approximation

value at point 119909119896and symbol 119898

119873is the total testing points

Example 1 Consider the following Fredholm integral equa-tion in [1]

int

1

0

radic119909 + 119905119891 (119905) d119905 = 119892 (119909)

119892 (119909) = minus16

10511990972

+16

1051199092(119909 + 1)

32

minus8

35119909(119909 + 1)

32+

2

7(119909 + 1)

32

0 le 119909 le 1

(26)

The exact solution for this equation is 119891(119909) = 1199092

Setting RBF centers with interval ℎ = 02 the collocationpoints are the same as the center points 119873 = 6 For MQthe shape parameter 120573 = 12 Then the 20-point Gaussquadrature formula is adopted in matrix element calculationAnd finally the numerical result was shown in Table 1Comparing to [1] where theHaar wavelet was adopted as basefunction for the scaling 119895 = 4 the RMSE MAE and MREwere 41774119864 minus 2 1007119864 minus 1 and 38046 and for 119895 = 6the corresponding errors were 20457119864 minus 2 51024119864 minus 2 and18715 And in TPS method the errors were 13403119890 minus 122625119864 minus 1 and 15979 But for the MQ method theaccuracy was significantly improved as 57774119864minus5 10512119864minus

4 and 04112 respectively whose accuracy was improvednearly 3 orders of magnitude

Then the shape parameter was analyzed with discreteinterval of 01 the RMSE and the condition number of matrix[Φ minus 120583Κ] were calculated in case of ℎ = 02 It is obviousto see in Figure 1 that the optimal shape parameter lies inthe interval of [10 20] showed in red dashed box And inthe whole range of [01100] the MQ method can maintainthe RMSE at the level of 1119864 minus 3 The curve of conditionnumber was shown in Figure 2 and it fluctuates in the rangeof [1119864 + 16 1119864 + 18] when 120573 is greater than 20 So the MQmethod is stable when the shape parameter varies in a relativewide range

And finally the RBFrsquos center spacingwas analyzed in caseof 120573 = 12 The RMSE curve was shown in Figure 3 It can


Table 1 Comparison between RBF methods and Haar wavelet methods

119909119905

Exact Haar wavelet with 119895 = 4 Haar wavelet with 119895 = 6 Thin plate splines Multiquadric01 001 004804645706772 002871498616505 001369148850547 00100411246814902 004 002853148488475 004216759916577 minus002391498932119 00400360682442603 009 005136013092094 007818735894931 014584667811187 00900471235052004 016 015024502143722 014464852362516 028759641578216 01600543236304505 025 021756512590070 023821128159978 018330827120676 02500411217959006 036 037282520080419 037175451578151 013743517284623 03600185211132607 049 054342797321883 050029535575412 050098212437262 04900193060050208 064 063155345618083 066060990603746 086624590306761 06400618035268109 081 080994050257895 080342706047881 083621252785184 08100890978960310 100 089929178048446 094897579805255 077598833938371 099989488295933

RMSE 41775119864 minus 002 20457119864 minus 002 13403119864 minus 001 57774119864 minus 005

0 10 20 30 40 50 60 70 80 90 100

RMSE

10minus1

10minus2

10minus3

10minus4

10minus5

120573

Figure 1 The RMSE curve versus the shape parameter 120573

120573

0 10 20 30 40 50 60 70 80 90 100

Con

ditio

n nu

mbe

r

108

1010

1012

1014

1016

1018

1020

Figure 2 The condition number curve versus the shape parameter120573

be seen clearly that the optimal center spacing was ℎ = 02

and 00833 But for mostly center spacing the MQ methodcan maintain RMSE at the level of 1119864 minus 3 As for conditionnumber whose curve also fluctuates in the range of [1119864 + 131119864 + 20] it was shown in Figure 4

RMSE

0 005 01 015 02 025

10minus1

10minus2

10minus3

10minus4

10minus5

h

Figure 3 The RMSE curve versus the RBFrsquos center spacing ℎ

Con

ditio

n nu

mbe

r

0 005 01 015 02 0251013

1014

1015

1016

1017

1018

1019

1020

h

Figure 4 The condition number curve versus the RBFrsquos centerspacing ℎ

Example 2 Consider the one-dimensional Volterra integralequation in [4]

int

119909

0

cos (119909 minus 119905) 119891 (119905) d119905 = 119909 sin (119909) 0 le 119909 le 1 (27)

The exact solution is 119891(119909) = 2 sin(119909)


Table 2 Error comparison between MQ and TPS

ℎMQ TPS

RMSE MAE MRE Cond RMSE MAE MRE Cond025 13784119890 minus 4 13784119890 minus 4 62050119890 minus 4 36563119890 + 08 23237 80507 280041198901 28490119890 + 03

020 16127119890 minus 5 99905119890 minus 5 59363119890 minus 5 54846119890 + 09 11164119890 minus 1 18176119890 minus 1 20428 47439119890 + 02


005 96970119890 minus 5 18010119890 minus 4 89035119890 minus 3 14434119890 + 16 19299119890 minus 2 83868119890 minus 2 62500119890 minus 1 82771119890 + 05

0025 34048119890 minus 4 49369119890 minus 4 24357119890 minus 2 31722119890 + 17 19065119890 minus 1 11247 67714119890 minus 01 57935119890 + 09

Firstly setting the RBF center spacing ℎ = 01 thecollocation points and centers are the same as 119873 = 11The shape parameter 120573 = 10 for MQ and the 20-pointGauss quadrature formula is adopted The total test pointsare 101 with equidistant ℎ

119905= 001 We focus on the center

spacing variation the approximation solutions were shownin Figure 5 for TPS method Generally the TPS can acquirebetter approximation as the center spacing decreases whichmeans that the solution of ℎ = 005 was better than that ofℎ = 01 However numerical simulation also indicated thatkeeping on decreasing ℎ may result in unstable solution

The comparison between MQ and TPS was shown inTable 2

Therefore we can conclude from Table 2 that the MQmethod takes obvious advantages over TPS whose accuracyis improved about 3 or 4 orders of magnitude And the MQmethod is also much stable for wide range of center spacingℎ The MQ especially can acquire high precision even withlarger center spacing as ℎ = 025 Compared to themethod in[4] the MQ also showed its superiority the highest precisionof modified block pulse functions method is about the levelof 1119864 minus 2

Example 3 Consider the two-dimensional integral equationin [14]

119891 (119909 119910) = 119906 (119909 119910) + int

1

0

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

119906 (119909 119910) = 119909 cos (119910) minussin (1) (3 + sin (1))

6

0 le 119909 119910 le 1

(28)

The exact solution is 119891(119909 119910) = 119909 cos(119910)

Setting the RBF center spacing ℎ = 01 the collocationpoints and centers are 119873 = 121 The shape parameter is120573 = 35 for MQ and the 20-point Gauss quadrature formulawas also adopted in both 119909 and 119910 directionsThe total testingpoints are 441 with equidistant ℎ

119905= 005 The RMSE

MAE MRE and condition number for MQ are 26583119864 minus 655309119864 minus 6 16077119864 minus 4 and 80241119864 + 18 respectivelyHowever the above values for TPS are 12273119864minus3 99312119864minus

3 34146119864minus1 and 19795119864+3 And comparing to the rationalHaar functionmethod in [14] whose accuracy was about level1119864 minus 2 and 1119864 minus 3 the MQ acquires much higher accuracy

0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05f(x)

x

Exacth = 02

h = 01

h = 005

Figure 5 The TPS solutions for different RBF center spacing ℎ

than others The error surfaces of MQ and TPS were shownin Figure 6

Then analyzing the shape parameter of MQwith discretestep of 10 the RMSE and the condition number werecalculated in case of ℎ = 01 The local optimal shapeparameters in this instancewere120573 = 7 35 58 and 64 revealedin Figure 7 The overall RMSE of MQ method was about thelevel of 1119864minus4 and 1119864minus5 And correspondingly the curve ofcondition number was similar to that in Figure 2 except thatthe curve fluctuates in the range of [1119864 + 18 1119864+ 20] when 120573

is greater than 10

Example 4 Consider the two-dimensional integral equationin an irregular domain in [13]

119891 (119909 119910) minus int

Ω

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

= 119909 cos (119910) minus 02804057157

Ω = (119909 119910) isin R2 0 le 119910 le 1

radic025 minus (119910 minus 05)2

le 119909 le radic1 minus 4(119910 minus 05)2

(29)



minus6minus5minus4minus3minus2minus1012

0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method

minus10minus8minus6minus4minus2024

0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method

Figure 6 The error surfaces for MQmethod and TPS method

0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3


RBF centers and collocation points

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y

Figure 8 RBF centers and collocation points

Firstly setting the RBF centers and collocation points asFigure 8 the total number was 119873 = 35 The shape parameterwas 120572 = 120573ℎ = 23 for MQ and the regional splitting withtriangular

Then the PDE toolbox in Matlab was utilized in trian-gulation The above irregular region Ω was split into 2272triangular elements with 1225 nodes In simplicity the testing

points were chosen at the element nodes And the quadraticformof interpolationwas employed in calculating the definiteintegral of subregions in (20) The results showed that theRMSE MAE MRE and condition number are 26260119864 minus 530345119864minus5 15783119864minus3 and 33527119864+14 However the abovevalues for TPSmethod are 39563119864minus3 19925119864minus2 86492119864minus

1 and 43509119864 + 2 The error curves of MQ and TPS areshown in Figure 9 Comparing to [13] where 33 collocationpointswere selected theRMSE is 196119864minus4 and forGAweightfunction the RMSE is 165119864minus4 So the RBFmethod proposedin this paper can obtain higher accuracy

5 Conclusion

In this paper the RBF method specifically the MQ wasintroduced in solving linear integral equationsThe followingconclusions can be drawn

(1) The procedure of MQ method includes that theunknown function was firstly expressed in linearcombination forms of RBFs then the integral equa-tion was transformed into collocation matrix ofRBFs and finally solving the matrix equation and anapproximation solution was obtained

(2) Due to the excellent interpolation performance ofMQ the method can acquire higher precision withfewer nodes and low computations Numerical resultsshowed that the MQ method can achieve solutionaccuracy of 1119864minus5 Numerical results showed that theMQ method takes obvious advantages over TPS Inaddition the method was very convenient for solvinghigher dimensional integral equations because theonly modification in implementation is the distanceformula

(3) One of the key issues in MQ method is the definiteintegral calculation Two types of integration schemesas the Gauss quadrature formula and regional splittechnique were put forward Specifically the formeris adopted in one-dimensional and two-dimensionalregular domain problems and the latter is employedin irregular domain case


0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method

Figure 9 The error curves for MQmethod and TPS method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Fundamental ResearchFunds for the Central Universities (no CDJZR11150008)

References

[1] M Rabbani K Maleknejad N Aghazadeh and R Molla-pourasl ldquoComputational projection methods for solving Fred-holm integral equationrdquo Applied Mathematics and Computa-tion vol 191 no 1 pp 140ndash143 2007

[2] J Saberi-Nadjafi M Mehrabinezhad and H Akbari ldquoSolvingVolterra integral equations of the second kind by wavelet-Galerkin schemerdquo Computers and Mathematics with Applica-tions vol 63 no 11 pp 1536ndash1547 2012

[3] K Maleknejad R Mollapourasl and M Alizadeh ldquoConver-gence analysis for numerical solution of Fredholm integralequation by Sinc approximationrdquoCommunications in NonlinearScience and Numerical Simulation vol 16 no 6 pp 2478ndash24852011

[4] K Maleknejad and B Rahimi ldquoModification of Block PulseFunctions and their application to solve numerically Volterraintegral equation of the first kindrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 16 no 6 pp 2469ndash2477 2011

[5] E Babolian S Abbasbandy and F Fattahzadeh ldquoA numericalmethod for solving a class of functional and two dimensionalintegral equationsrdquoApplied Mathematics and Computation vol198 no 1 pp 35ndash43 2008

[6] E Babolian K Maleknejad M Mordad and B Rahimi ldquoAnumericalmethod for solving Fredholm-Volterra integral equa-tions in two-dimensional spaces using block pulse functionsand an operational matrixrdquo Journal of Computational andApplied Mathematics vol 235 no 14 pp 3965ndash3971 2011

[7] W-J Xie and F-R Lin ldquoA fast numerical solution method fortwo dimensional Fredholm integral equations of the second

kindrdquo Applied Numerical Mathematics vol 59 no 7 pp 1709ndash1719 2009

[8] R L Hardy ldquoMultiquadric equations of topography and otherirregular surfacesrdquo Journal of Geophysical Research vol 176 pp1905ndash1915 1971

[9] R Franke ldquoScattered data interpolation test of some methodsrdquoMathematics of Computation vol 38 pp 181ndash200 1982

[10] A Golbabai and S Seifollahi ldquoNumerical solution of the secondkind integral equations using radial basis function networksrdquoApplied Mathematics and Computation vol 174 no 2 pp 877ndash883 2006

[11] A Golbabai and S Seifollahi ldquoAn iterative solution for thesecond kind integral equations using radial basis functionsrdquoApplied Mathematics and Computation vol 181 no 2 pp 903ndash907 2006

[12] A Alipanah and S Esmaeili ldquoNumerical solution of thetwo-dimensional Fredholm integral equations using Gaussianradial basis functionrdquo Journal of Computational and AppliedMathematics vol 235 no 18 pp 5342ndash5347 2011

[13] P Assari H Adibi and M Dehghan ldquoA numerical method forsolving linear integral equations of the second kind on the non-rectangular domains based on the meshless methodrdquo AppliedMathematical Modelling vol 37 pp 9269ndash9294 2013

[14] E Babolian S Bazm and P Lima ldquoNumerical solution ofnonlinear two-dimensional integral equations using rational-ized Haar functionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1164ndash1175 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


high interpolation accuracy The radial basis functions werefirst developed by Hardy [8] in 1971 as a multidimensionalscattered interpolation method in modeling of the earthrsquosgravitational field It was not recognized by most of the aca-demic researchers until Franke [9] published a review paperin the evaluation of two-dimensional interpolation methodsThe RBF method in solving integral equations was initiallyproposed in 2006 [10ndash13] The related research attracted a lotof attention recentlyTherefore this paper attempted to adoptthe RBF method for solving linear integral equations TheRBF and interpolation principle were described in Section 2And in Section 3 the RBF method for solving linear integralequation and the numerical integration scheme for coeffi-cients matrix were proposed And finally the RBF methodwas applied in one- and two-dimensional integral equationsits performance was analyzed and compared

2 Radial Basis Function andInterpolation Principle

Radial basis function B 119877+ rarr 119877 (domain 119877119889) is definedas the function of distance 119903 = ||119909 minus 119909

119895|| The commonly

used RBF contains the thin plate splines function (B(119903) =

1199032 ln 119903) the Gaussian function (B(119903) = exp(minus120573119903

2)) Hardyrsquos

multiquadric (MQ) functions and so forth Franke hasproved that the MQmethod has the superior comprehensiveperformance in 29 kinds of scattered data interpolationmethods So in this paper we choose the MQ as the basisfunction which is compared with thin plate splines (TPS)function The widely used MQ function has the followingexpression

120601 (x) = radicx minus c2 + 1205722 (2)

In the above c means the center of basis function and 120572 isthe shape parameter which is generally associated with thedistance between adjacent centers So 120572 = 120573||c

119894minus c119895|| and 120573

is also called the shape parameterThe principle of RBF interpolation regards the unknown

function as linear combination of radial functions So theapproximation function can be obtained after calculating thecoefficients For a series of known data points (119909

119894 119891(119909119894)) 119894 =

1 2 119873 the approximation function can be constructed byRBF as

119891 (119909) =

119873

sum

119895=1

120582119895120601119895 (119909) =

119873

sum

119895=1

120582119895radic(119909 minus 119888

119895)2

+ 1205722 (3)

Then substituting the interpolation data points into the aboveequation we obtain

119891 (119909119894) =

119873

sum

119895=1

120582119895120601119895(119909119894) =

119873

sum

119895=1

120582119895radic(119909119894minus 119888119895)2

+ 1205722 (4)

With its abstract matrix form

[Φ119889] [120582] = [f

119889] (5)

where [f119889] = [119891(119909

1) 119891(119909

1) 119891(119909

119873)]119879 is the known data

[120582] = [1205821 1205822 120582

119873]119879 is the coefficients and [Φ

119889] is

the coefficient matrix whose element calculation formula isΦ119898119899

= radic(119909119898

minus 119888119899)2+ 1205722

The MQ function is globally defined which results in afull resultant coefficient matrix And consequently the coef-ficientmatrix in (5) is nonsingular andusually ill conditionedThe accuracy of the MQ solution depends heavily on theshape parameter In general for a fixed number of centers119873smaller shape parameters producemore accurate approxima-tions but they also are associated with a poorly conditionedinterpolation matrix Also the condition number grows with119873 for fixed values of the shape parameter120572The choice of thisoptimal value is still under intensive investigationThereforesolving (5) can acquire

[120582] = [Φ119889]minus1

[f119889] (6)

The approximation value 119891(119909) at any point 119909 is

119891 (119909) = [Φ (119909)] [120582] = [Φ (119909)] [Φ119889]minus1

[f119889] (7)

3 Solving Linear Integral Equations Based onthe RBF Interpolation

The basic idea of RBF method for solving integral equationis to employ a linear combination of RBF to approximate theunknown function thus the integral equation is transformedinto the combination form of RBFs and their coefficientsThen the weight coefficients of RBF can be calculatedby using the collocation type weighted residual methodAnd finally an approximate representation of the unknownfunction can be obtained Specifically the RBF method forsolving linear integral equations is proposed in the following

31 RBF Interpolation to Approximate One-Dimensional Lin-ear Integral Equation The general form of one-dimensionallinear integral equation is

119891 (119909) = 120583int

119887|119909

119886

119896 (119909 119905) 119891 (119905) d119905 + 119892 (119909) 119909 isin [119886 119887] (8)

Here 119887 | 119909 represents the upper limit of integralThe constant119887 or variable 119909 is respectively corresponding to Fredholm orVolterra equation In the following discussion we choose theupper limit 119887 for demonstration In RBF approximation theunknown function119891(119909) is expressed as a combination of RBFThus the approximate formula (9) is obtained

119873

sum

119894=1

120582119894120601119894 (119909) = 120583int

119887

119886

119896 (119909 119905)

119899

sum

119894=1

120582119894120601119894 (119905) d119905 + 119892 (119909) 119909 isin [119886 119887]

(9)




119873

sum

119894=1

120582119894120601119894(119909119889) minus 120583

119873

sum

119894=1

120582119894int

119887

119886

119896 (119909119889 119905) 120601119894 (119905) d119905 = 119892 (119909

119889)

119909119889isin [119886 119887]

(10)


[Φ minus 120583Κ] [120582] = [G] (11)

where [G] = [119892(1199091) 119892(1199091) 119892(119909

119872)]119879 and [120582] =

[1205821 1205822 120582


defined

Φ119898119899

= radic(119909119898

minus 119888119899)2+ 1205722

119870119898119899

= int

119887

119886

119896 (119909119898 119905) radic(119905 minus 119888

119899)2+ 1205722d119905 = int

119887

119886

ℎ (119905) d119905(12)





int

1

minus1

ℎ (120585) d120585 asymp

119876

sum

119902=1

119882119902ℎ (120585119902) (13)



119870119898119899

= int

119887

119886

119896 (119909119898 119905) 120601119899 (119905) d119905

=119887 minus 119886

2int

1

minus1

119896 [119909119898 119901 (120585)] 120601119899 [119901 (120585)] d120585


2

119876

sum

119902=1

119882119902119896 [119909119898 119901 (120585119902)] 120601119899[119901 (120585119902)]

(14)


119891 (119909 119910) = 120583int

Ω

119896 (119909 119910 119904 119905) 119891 (119904 119905) d119904 d119905 + 119892 (119909 119910)

(119909 119910) isin Ω

(15)


119873

sum

119894=1

120582119894120601119894(119909119889 119910119889)

= 120583

119873

sum

119894=1

120582119894int

Ω

119896 (119909119889 119910119889 119904 119905) 120601

119894 (119904 119905) d119904 d119905 + 119892 (119909119889 119910119889)

(16)



119898119899= intΩ

119896(119909119898 119910119898




(119889 + 119888)2 + (119889 minus 119888)1205782 = 119902(120578) We have

119870119898119899

= int

119889

119888

int

119887

119886

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905


4

1198991

sum

119894=1

1198992

sum

119895=1

119882119894119882119895119896 [119909119898 119910119898 119901 (120585119894) 119902 (120578

119895)] 120601119899

times [119901 (120585119894) 119902 (120578

119895)]

(17)


119870119898119899

= int

Ω

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

= sum

119902

119882119902119870(119909119898 119910119898 119904119902 119905119902) 120601119899(119904119902 119905119902)

(18)


119870119898119899

= int

Ω

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

= sum

119894

int

Ω119894

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905 asymp sum

119894

119868119894

(19)



119894119895 119910119894119895)


(119909119894119888119895

119910119894119888119895


119868119894=

119878119894

3

3

sum

119895=1

119896 (119909119898 119910119898 119909119894119888119895

119910119894119888119895

) 120601119899(119909119894119888119895

119910119894119888119895

) (20)












[G] (21)



case be defined as

119870[119891 (119909)] = int

119887|119909

119886

119896 (119909 119905) 119891 (119905) d119905 (22)


(119868 minus 120583119870)119891 = 119892 (23)


10038171003817100381710038171003817(119868 minus 120583119870)

minus110038171003817100381710038171003817le

1

1 minus 119870 (24)





RMSE = radic1

119898119873

119898119873

sum

119896=1

[119891119890(119909119896) minus 119891(119909

119896)]2 (25)

where 119891119890(119909119896) 119891(119909





int

1

0

radic119909 + 119905119891 (119905) d119905 = 119892 (119909)

119892 (119909) = minus16

10511990972

+16

1051199092(119909 + 1)

32

minus8

35119909(119909 + 1)

32+

2

7(119909 + 1)

32

0 le 119909 le 1

(26)








119909119905



0 10 20 30 40 50 60 70 80 90 100

RMSE

10minus1

10minus2

10minus3

10minus4

10minus5

120573


120573

0 10 20 30 40 50 60 70 80 90 100

Con

ditio

n nu

mbe

r

108

1010

1012

1014

1016

1018

1020




RMSE

0 005 01 015 02 025

10minus1

10minus2

10minus3

10minus4

10minus5

h


Con

ditio

n nu

mbe

r

0 005 01 015 02 0251013

1014

1015

1016

1017

1018

1019

1020

h



int

119909

0





ℎMQ TPS












119891 (119909 119910) = 119906 (119909 119910) + int

1

0

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

119906 (119909 119910) = 119909 cos (119910) minussin (1) (3 + sin (1))

6

0 le 119909 119910 le 1

(28)



119905= 005 The RMSE



0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05f(x)

x

Exacth = 02

h = 01

h = 005




is greater than 10


119891 (119909 119910) minus int

Ω

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

= 119909 cos (119910) minus 02804057157

Ω = (119909 119910) isin R2 0 le 119910 le 1



(29)




0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method


0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method


0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3



0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y






5 Conclusion






0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119873

sum

119894=1

120582119894120601119894(119909119889) minus 120583

119873

sum

119894=1

120582119894int

119887

119886

119896 (119909119889 119905) 120601119894 (119905) d119905 = 119892 (119909

119889)

119909119889isin [119886 119887]

(10)


[Φ minus 120583Κ] [120582] = [G] (11)

where [G] = [119892(1199091) 119892(1199091) 119892(119909

119872)]119879 and [120582] =

[1205821 1205822 120582


defined

Φ119898119899

= radic(119909119898

minus 119888119899)2+ 1205722

119870119898119899

= int

119887

119886

119896 (119909119898 119905) radic(119905 minus 119888

119899)2+ 1205722d119905 = int

119887

119886

ℎ (119905) d119905(12)





int

1

minus1

ℎ (120585) d120585 asymp

119876

sum

119902=1

119882119902ℎ (120585119902) (13)



119870119898119899

= int

119887

119886

119896 (119909119898 119905) 120601119899 (119905) d119905

=119887 minus 119886

2int

1

minus1

119896 [119909119898 119901 (120585)] 120601119899 [119901 (120585)] d120585


2

119876

sum

119902=1

119882119902119896 [119909119898 119901 (120585119902)] 120601119899[119901 (120585119902)]

(14)


119891 (119909 119910) = 120583int

Ω

119896 (119909 119910 119904 119905) 119891 (119904 119905) d119904 d119905 + 119892 (119909 119910)

(119909 119910) isin Ω

(15)


119873

sum

119894=1

120582119894120601119894(119909119889 119910119889)

= 120583

119873

sum

119894=1

120582119894int

Ω

119896 (119909119889 119910119889 119904 119905) 120601

119894 (119904 119905) d119904 d119905 + 119892 (119909119889 119910119889)

(16)



119898119899= intΩ

119896(119909119898 119910119898




(119889 + 119888)2 + (119889 minus 119888)1205782 = 119902(120578) We have

119870119898119899

= int

119889

119888

int

119887

119886

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905


4

1198991

sum

119894=1

1198992

sum

119895=1

119882119894119882119895119896 [119909119898 119910119898 119901 (120585119894) 119902 (120578

119895)] 120601119899

times [119901 (120585119894) 119902 (120578

119895)]

(17)


119870119898119899

= int

Ω

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

= sum

119902

119882119902119870(119909119898 119910119898 119904119902 119905119902) 120601119899(119904119902 119905119902)

(18)


119870119898119899

= int

Ω

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905

= sum

119894

int

Ω119894

119896 (119909119898 119910119898 119904 119905) 120601

119899 (119904 119905) d119904 d119905 asymp sum

119894

119868119894

(19)



119894119895 119910119894119895)


(119909119894119888119895

119910119894119888119895


119868119894=

119878119894

3

3

sum

119895=1

119896 (119909119898 119910119898 119909119894119888119895

119910119894119888119895

) 120601119899(119909119894119888119895

119910119894119888119895

) (20)












[G] (21)



case be defined as

119870[119891 (119909)] = int

119887|119909

119886

119896 (119909 119905) 119891 (119905) d119905 (22)


(119868 minus 120583119870)119891 = 119892 (23)


10038171003817100381710038171003817(119868 minus 120583119870)

minus110038171003817100381710038171003817le

1

1 minus 119870 (24)





RMSE = radic1

119898119873

119898119873

sum

119896=1

[119891119890(119909119896) minus 119891(119909

119896)]2 (25)

where 119891119890(119909119896) 119891(119909





int

1

0

radic119909 + 119905119891 (119905) d119905 = 119892 (119909)

119892 (119909) = minus16

10511990972

+16

1051199092(119909 + 1)

32

minus8

35119909(119909 + 1)

32+

2

7(119909 + 1)

32

0 le 119909 le 1

(26)








119909119905



0 10 20 30 40 50 60 70 80 90 100

RMSE

10minus1

10minus2

10minus3

10minus4

10minus5

120573


120573

0 10 20 30 40 50 60 70 80 90 100

Con

ditio

n nu

mbe

r

108

1010

1012

1014

1016

1018

1020




RMSE

0 005 01 015 02 025

10minus1

10minus2

10minus3

10minus4

10minus5

h


Con

ditio

n nu

mbe

r

0 005 01 015 02 0251013

1014

1015

1016

1017

1018

1019

1020

h



int

119909

0





ℎMQ TPS












119891 (119909 119910) = 119906 (119909 119910) + int

1

0

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

119906 (119909 119910) = 119909 cos (119910) minussin (1) (3 + sin (1))

6

0 le 119909 119910 le 1

(28)



119905= 005 The RMSE



0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05f(x)

x

Exacth = 02

h = 01

h = 005




is greater than 10


119891 (119909 119910) minus int

Ω

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

= 119909 cos (119910) minus 02804057157

Ω = (119909 119910) isin R2 0 le 119910 le 1



(29)




0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method


0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method


0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3



0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y






5 Conclusion






0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















[G] (21)



case be defined as

119870[119891 (119909)] = int

119887|119909

119886

119896 (119909 119905) 119891 (119905) d119905 (22)


(119868 minus 120583119870)119891 = 119892 (23)


10038171003817100381710038171003817(119868 minus 120583119870)

minus110038171003817100381710038171003817le

1

1 minus 119870 (24)





RMSE = radic1

119898119873

119898119873

sum

119896=1

[119891119890(119909119896) minus 119891(119909

119896)]2 (25)

where 119891119890(119909119896) 119891(119909





int

1

0

radic119909 + 119905119891 (119905) d119905 = 119892 (119909)

119892 (119909) = minus16

10511990972

+16

1051199092(119909 + 1)

32

minus8

35119909(119909 + 1)

32+

2

7(119909 + 1)

32

0 le 119909 le 1

(26)








119909119905



0 10 20 30 40 50 60 70 80 90 100

RMSE

10minus1

10minus2

10minus3

10minus4

10minus5

120573


120573

0 10 20 30 40 50 60 70 80 90 100

Con

ditio

n nu

mbe

r

108

1010

1012

1014

1016

1018

1020




RMSE

0 005 01 015 02 025

10minus1

10minus2

10minus3

10minus4

10minus5

h


Con

ditio

n nu

mbe

r

0 005 01 015 02 0251013

1014

1015

1016

1017

1018

1019

1020

h



int

119909

0





ℎMQ TPS












119891 (119909 119910) = 119906 (119909 119910) + int

1

0

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

119906 (119909 119910) = 119909 cos (119910) minussin (1) (3 + sin (1))

6

0 le 119909 119910 le 1

(28)



119905= 005 The RMSE



0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05f(x)

x

Exacth = 02

h = 01

h = 005




is greater than 10


119891 (119909 119910) minus int

Ω

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

= 119909 cos (119910) minus 02804057157

Ω = (119909 119910) isin R2 0 le 119910 le 1



(29)




0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method


0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method


0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3



0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y






5 Conclusion






0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119905



0 10 20 30 40 50 60 70 80 90 100

RMSE

10minus1

10minus2

10minus3

10minus4

10minus5

120573


120573

0 10 20 30 40 50 60 70 80 90 100

Con

ditio

n nu

mbe

r

108

1010

1012

1014

1016

1018

1020




RMSE

0 005 01 015 02 025

10minus1

10minus2

10minus3

10minus4

10minus5

h


Con

ditio

n nu

mbe

r

0 005 01 015 02 0251013

1014

1015

1016

1017

1018

1019

1020

h



int

119909

0





ℎMQ TPS












119891 (119909 119910) = 119906 (119909 119910) + int

1

0

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

119906 (119909 119910) = 119909 cos (119910) minussin (1) (3 + sin (1))

6

0 le 119909 119910 le 1

(28)



119905= 005 The RMSE



0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05f(x)

x

Exacth = 02

h = 01

h = 005




is greater than 10


119891 (119909 119910) minus int

Ω

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

= 119909 cos (119910) minus 02804057157

Ω = (119909 119910) isin R2 0 le 119910 le 1



(29)




0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method


0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method


0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3



0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y






5 Conclusion






0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎMQ TPS












119891 (119909 119910) = 119906 (119909 119910) + int

1

0

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

119906 (119909 119910) = 119909 cos (119910) minussin (1) (3 + sin (1))

6

0 le 119909 119910 le 1

(28)



119905= 005 The RMSE



0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05f(x)

x

Exacth = 02

h = 01

h = 005




is greater than 10


119891 (119909 119910) minus int

Ω

(119905 sin (119904) + 1) 119891 (119904 119905) d119904 d119905

= 119909 cos (119910) minus 02804057157

Ω = (119909 119910) isin R2 0 le 119910 le 1



(29)




0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method


0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method


0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3



0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y






5 Conclusion






0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 002 02

04 0406 06

0808

11

y x

Erro

r sur

face

(Δf(xy))

times10minus6

(a) MQ method


0 002 02

04 0406 06

0808

11

Erro

r sur

face

(Δf(xy))

y x

times10minus3

(b) TPS method


0 10 20 30 40 50 60 70 80 90 100

RMSE

120573

10minus6

10minus5

10minus4

10minus3



0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

x

y






5 Conclusion






0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 100011001200Nodes number

16

18

2

22

24

26

28

3

32

Erro

r cur

ve (Δ

f(xy))

times10minus5

(a) MQ method

0 100 200 300 400 500 600 700 800 900 1000110012000

0002000400060008

0010012001400160018

002

Nodes number

Erro

r cur

ve (Δ

f(xy))

(b) TPS method




Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Solving the Linear Integral Equations ...downloads.hindawi.com › journals › jam...

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