On Taylor Expansion Methods For Multivariate Integral .
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESOn Taylor Expansion Methods for MultivariateIntegral Equations of the Second KindBoriboon Novaprateep, Khomsan Neamprem, and Hideaki KanekoAbstract—A new Taylor series method that the authors originally developed for the solution of one-dimensional integralequations is extended to solve multivariate integral equations.In this paper, the new method is applied to the solution of multivariate Fredholm equations of the second kind. A comparisonis given of the new method and the traditional Taylor seriesmethod of solving integral equations. The new method is adaptedto parallel computation and can therefore be highly efficienton modern computers. The method also gives highly accurateapproximations for all derivatives of the solution up to the orderof the Taylor series approximation. Numerical examples are givento illustrate the efficiency and accuracy of the method.Keywords—Taylor-Series Expansion Method, MultivariateFredholm Integral Equations, Galerkin Method, CollocationMethod.II. I NTRODUCTIONNTEGRAL equations are often matched with mathematicalmodels describing some phenomena in Physics andEngineer [3]. The approximation solution can be foundby several methods [7], [6], [9]. In a recent paper [2], aTaylor-series expansion method to approximate the solution ofa class of Fredholm integral equation of the second kind wasconsidered. Subsequently, the method of [2] was improvedin a recent paper [4]. The Taylor series method developedin [4] is different from the traditional Taylor series methoddiscussed in [1], in which the kernel is expanded in the seriesand the resulting equation is treated within the framework ofthe degenerate kernel method. We will review the traditionalTaylor series method in the next section so as to delineate andhighlight the advantages of the current method. Improvementsmade in [4] over the paper [2] primarily lie in two areas.First, it was generalized so as to be applicable to a widerclass of integral equations. The method discussed in the paper[2] applies only to equations with a convolution type kernelk( t s ) in which it decays rapidly as t s increases toinfinity. The new method in [4] on the other hand can beapplied to any differentiable kernel. Secondly, the method [4]produces more accurate approximation than the one given in[2]. It is also important to note that numerical implementationB. Novaprateep is with the Department of Mathematics, Faculty of Science,Mahidol University, Bangkok, 10400, THAILAND and Centre of Excellencein Mathematics, CHE, Si Ayutthaya RD., Bangkok 10400, THAILAND,e-mail: boriboon.nov@mahidol.ac.th.K. Neamprem is with Department of Mathematics, Faculty of AppliedScience, King Mongkut’s University of Technology North Bangkok, Bangkok,10800, THAILAND and Centre of Excellence in Mathematics, CHE, Si Ayutthaya RD., Bangkok 10400, THAILAND, email: khomsann@kmutnb.ac.th.H. Kaneko is with Department of Mathematics, Department of Mathematicsand Statistics,Old Dominion University, Norfolk, Virginia, 23529-0077, USA,email: hkaneko@odu.edu.Issue 8, Volume 6, 2012of the new method can be carried out in parallel. This is acritical point to note since most of the numerical methodscurrently available for approximating the solution of linearas well as nonlinear equations are designed to approximatethe solution globally all in one calculation. The Galerkinmethod and collocation method are two of the most popularmethods used by practitioners. For a detailed reference tonumerical solutions of integral equations, we recommend abook by Atkinson [1]. The Galerkin and collocation methodsnormally leads to a large scale linear system of equationswhich are computationally expensive to solve since matricesinvolved are most often dense rather than sparse. Therefore,reducing an expensive cost involved in the solution processis always an important issue in approximating the solution ofintegral equations. In addition, this paper provides a methodto calculate a solution at a specific region of interest withoutcalculating the solution over the whole region. The purpose ofthis paper is to extend the idea explored in [4] to multivariateFredholm equations. We anticipate that the current approachwill find new applications in numerical solution of boundaryvalue problems. This will be discussed in a future paper.We also note that recent papers [5] and [8] extend theidea developed in [4] to Volterra integral equations and tononlinear Hammerstein equations, respectively.The article is organized as follows: Section II, we introducea traditional Taylor-Series Method for the Fredholm integralequation of the second kind over an one dimensional spaceR. In section III, we extend idea of the Taylor expansionmethod from [4] to the Fredholm integral equations of thesecond kind over a multidimensional space R n . Section IV,we illustrate on the numerical experiments of the new Taylorexpansion method and show the efficiency and accuracy ofthis new method.II. A T RADITIONAL TAYLOR -S ERIES M ETHODIn this section, we review briefly the traditional Taylorseries method, see [1].The review will serve to distinguish the new Taylor seriesmethod which was developed recently in [4] and identify itsstrength and advantages. Recall that one dimensional Fredholm integral equation of the second kind is given by,Z 1x(t) k(t, s)x(s)ds f (t),0 t 1,(2.1)9010
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESwhere k and f are know functions and x is the function tobe determined. This type of equation arises as an integralreformulation of the two-point boundary value problem. Thetraditional Taylor series approximation method is to expandthe kernel k in Taylor series with respect to the variable t ors. For example, if k is expanded in s variable to the nth order,we have an finite rank approximation kn of k which is givebynX ik(t, a)(s a)i .(2.2)kn (t, s) i si 0Upon substituting (2.2) into (2.1), we obtain an approximationxn (t) f (t) nXci ki (t)RDx(t) RI Dk(t, s)x(s)dsD2 x(t) I D2 k(t, s)x(s)ds.RDr x(t) I Dr k(t, s)x(s)ds(2.3)i where ki (t) : si k(t, a)R1and ci : 0 (s a)i xn (s)ds for i 0, 1, . . . n.00III. A N E XPANSION M ETHOD FOR M ULTIVARIATEE QUATIONIn this section, we apply the Taylor expansion methoddeveloped in [4] to the Fredholm integral equations of thesecond kind proposed over a multidimensional space R n ;Zx(t) k(t, s)x(s)ds f (t),(3.1)Iwhere t, s I, I R is a compact set. We recall the Taylorseries for x(t),nPr x(t) Pr1kk 1 k! D x(t) 1k 1 k! ( t1 Rr (t, h)· (h, h, . . . , h) Rr (t, h){z} k times · · · t )k x(t) · (h, h, . . . , h)n{z} k times(3.2)where h s t, c is a point on the line segment s tand Rr (t, h)/khkr 0 as h 0, h Rn . Here of course,Dx(t) is the gradient of x at t and 2 2xx 2x· · · t 1 t(t) t1 t1 (t) t1 t2 (t)n . .D2 x(t) . 222 x x x tn t1 (t) tn t2 (t) · · · tn tn (t)Rk(t, s) · (h, h, . . . , h)dsDx̄(t) PrRDk(t, s) · (h, h, . . . , h)ds Df (t)1kk 0 k! D x̄(t) I1kk 0 k! D x̄(t) IRPr1kk 0 k! D x̄(t) I f (t).D r k(t, s) · (h, h, . . . , h)ds D r f (t).(3.4)Pr1kk 0 k! D (x(t)R x̄(t)) k(t, s) ·R(h, h, . . . , h)ds f (t) D k(t, s)Rr (s, h)dsPr1D k (x(t)D(x(t) x̄(t)) k 0 k! x̄(t)) RDk(t,IRs) · (h, h, . . . , h)ds Df (t) I Dk(t, s)Rr (s, h)ds.Pr1krD (x(t) x̄(t)) k 0 k!RD (x(t) x̄(t)) D r k(t, s) ·R(h, h, . . . , h)dsI D r f (t) I D r k(t, s)Rr (s, h)ds.(3.5)(x(t) x̄(t)) IFor example, with n 2 and r 2, equations(3.5) represent a system of 6 equations in 6 unknowns222 x xx(t), t(t), 2 tx1 (t), t 1 t(t), 2 tx2 (t). As a simplex(t), t122illustration, with n 2 and r 1,RRR1 I k(t, s)ds I k(t, s)(s t)ds I k(t, s)(s t)ds RRR I kt1 (t, s)ds 1 I kt1 (t, s)(s t)ds I kt1 (t, s)(s t)ds RRR I kt2 (t, s)ds I kt2 (t, s)(s t)ds 1 I kt2 (t, s)(s t)ds 1 R x(t) x̄(t)k(t, s)(D 2 x)(c)(s t)2 ds2! I x R x̄2 x)(c)(s t)2 ds (t) t(t) 1k(t, s)(Dt t111 2! I R x x̄122(t) t (t)kt2 (t, s)(D x)(c)(s t) ds t2I2!2 where c is a point between s and t.In general, we have the following theorem regarding theaccuracy of the current expansion method.Theorem 3.1: Let x(t) be the solution of equation (3.1) andx̄(t) be the solutionof equation (3.4). Suppose that the kernelRk is such that I Dti 1 k(t, s)ds for all i r 1. ThenDifferentiating (3.1) r times, we obtainIssue 8, Volume 6, 2012PrDirect substitution of (3.2) into (3.3) and comparing it with(3.4), we obtain a set of equations which yields the followingerror estimate of the current method.Note that the coefficients ci in equation (2.3) is independentof t value and thus an approximation is obtained for x(t).Any property of the derivatives of the solution is not derivedfrom this analysis. x(t) x̄(t) D r x̄(t) i 0, 1, . . . , n.x(s) D r f (t).Multiplying (2.3) by (s a)i and integrating over (0, 1),we obtain the following linear system whose solution givesan approximation xn ;Z 1n Z 1Xci kj (s)(s a)i ds f (s)(s a)i ds, (2.4)(3.3)Now, ignoringerror term in (3.2) for a moment andPr the1Dk x(t) · (h, h, . . . , h) for x(s) in (3.1)substituting k 0 k!and (3.3) [to simplify our notation, we write (h, h, . . . , h) (h, h, . . . , h)] , we obtain an approximation x̄(t) for the {z }k timessolution x(t) of (3.1) by solving the following equations:i 0j 0 Df (t) D 2 f (t).902kDti (x(t) x̄(t))k Mhr 1 ,(r 1)!
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESTABLE IC OMPUTATIONAL RESULTS WITH r 2 FOR EXAMPLE 4.1.where h s t,Rand M max0 i,j r 1 I Dtj k(t, s)(Di 1.01.01.01.01.01.0It is helpful to compare the current expansion method withmore traditional numerical methods, such as the Galerkinmethod and the collocation method, to identify its strength.Suppose that meas(I) 1 and we apply the current expansionmethod with r 4. That is, we apply the Taylor series of order4 in (3.2). Then the resulting system (3.4) represents 14 linear1). Thisequations in 14 unknowns, with its error term O( 5!means that we expect the accuracy of approximation x̄(t) ofx(t) as well as all of its derivatives of order less than 5 to beapproximately 0.008. Now, in order to attain the same orderof accuracy using the Galerkin or collocation method with,say, the linear spline basis, we must establish a 440 440system, since its order of accuracy of linear spline is O(d 2 ), where I α Γ Iα and I maxα Γ Iα and d meas(I),and it requires a basis function with 4 degree of freedom overeach Iα . Thus the comparison shows that the current Taylorexpansion method, when an approximation is desired for thesolution as well as for its derivatives over a specific regionof interest, provides a much more efficient way of calculatingthem than the traditional Galerkin and the collocation methods.IV. N UMERICAL R ESULTSIn this section, we present five numerical examples. Thecomputer programs are run on a personal computer with2.8GHz CPU and 8GB memory.Example 4.1: Consider the equationZ 1Z 1x(s1 , s2 ) k(s1 , s2 , t1 , t2 )x(t1 , t2 )dt1 dt2 f (s1 , s2 ),00[s1 , s2 ] [0, 1] [0, 1]where k(s1 , s2 , t1 , t2 ) s1 s2 t1 t2and f (s1 , s2 ) is chosen so that the isolated solutionx(s1 , s2 ) s1 2s2 .The numerical results can see in Table I.In Example 4.1, the solution is taken to be a polynomialof degree 2. Thus, the current method with r 2 delivers anapproximation which is exact to the machine 000Abs. Error1.6653e-165.5511e-171.1102e-160.0000e 160.0000e 002.2204e-160.0000e 005.5511e-171.1102e-162.2204e-160.0000e 000.0000e 000.0000e 001.1102e-162.2204e-160.0000e 002.2204e-160.0000e 004.4409e-160.0000e 002.2204e-164.4409e-160.0000e 000.0000e 000.0000e 001.1102e-162.2204e-164.4409e-160.0000e 000.0000e 000.0000e 00Example 4.2: Consider the equationZ 1Z 1x(s1 , s2 ) k(s1 , s2 , t1 , t2 )x(t1 , t2 )dt1 dt2 f (s1 , s2 ),00[s1 , s2 ] [0, 1] [0, 1]where k(s1 , s2 , t1 , t2 ) exp(s1 s2 t1 t2 ) and f (s1 , s2 )is chosen so that the isolated solution x(s1 , s2 ) s41 s42 .The numerical results can see in TABLE II.In the next example, the exponential kernel is used,exp(s1 s2 t1 t2 ), and the solution is taken to be apolynomial of degree 4. Thus, the current method with r 4delivers an approximation with the high accuracy.Issue 8, Volume 6, .20.40.60.81.0903
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESTABLE IIC OMPUTATIONAL RESULTS WITH r 4 FOR EXAMPLE 40961.00001.00161.02561.12961.40962.0000TABLE IIIC OMPUTATIONAL RESULTS WITH r 5 FOR EXAMPLE 4.3.Abs. .84150.94900.99760.98430.9100r 5Abs. e-51.2216e-43.1838e-46.6880e-4TABLE IVC OMPUTATIONAL RESULTS WITH 6 FOR EXAMPLE 4.3.Next, we show the comparison of numericalexperimentations with several orders of Taylor expansionmethod.Example 4.3: Consider the equationZ 1Z 1x(s1 , s2 ) k(s1 , s2 , t1 , t2 )x(t1 , t2 )dt1 dt2 f (s1 , s2 ),0s10[s1 , s2 ] [0, 1] [0, 1]where k(s1 , s2 , t1 , t2 ) exp( s1 s2 t1 t2 )and f (s1 , s2 ) is chosen so that the isolated solutionx(s1 , s2 ) sin(s1 s2 ).The numerical results can see in TABLE III and TABLE VI.From Table III and VI, we found that the numerical errorsdeceased when order of Taylor expansion increased for eachpoint. Moreover, we may see it from from Fig. 1 to Fig. 0.99750.98400.9093r 6Abs. e-64.8117e-65.3417e-53.5252e-5Example 4.4: Consider the equationZ 1Z 1x(s1 , s2 ) k(s1 , s2 , t1 , t2 )x(t1 , t2 )dt1 dt2 f (s1 , s2 ),00[s1 , s2 ] [0, 1] [0, 1]where k(s1 , s2 , t1 , t2 ) exp( s1 s2 t1 t2 ) and f (s1 , s2 )is chosen so that the isolated solution x(s1 , s2 ) exp( 2s1 s22 ).The numerical results can see in TABLE VII - TABLE XII.In Example 4.4, the numerical approximation for x(s1 , s2 )and its first and second derivatives with r 8 are shownIssue 8, Volume 6, 2012904
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESTABLE VC OMPUTATIONAL RESULTS WITH r 7 FOR EXAMPLE 8400.9093r 7Approx.Abs. 0.98401.1619e-50.90933.1942e-5r 5r 6r 7r 832.52Errors2x 101.510.500Fig. 1.0.10.20.30.40.5s20.60.70.80.91The comparison of the numerical errors with fixed s1 0.00. 41.4x 10r 5r 6r 7r 81.210.8Errors1 43.50.6TABLE VIC OMPUTATIONAL RESULTS WITH 8 FOR EXAMPLE 00.99750.98400.9093r 8Abs. e-89.1357e-85.8804e-81.5824e-60.200Fig. 2.0.10.20.30.40.5s20.60.70.80.9 48x 10r 5r 6r 7r 87654321000.10.20.30.40.5s0.60.70.80.92in Table VII to XII. We point out that in the examplethe numerical errors of x(s1 , s2 ) and its first and secondderivatives are the same at the each point.Finally, we extend our expansion technique with orderr 3 and r 4 to three dimensional Fredholm equationsshown below.Example 4.5: Consider the equationZ 1Z 1Z 1x( s ) k( s, t )x( t )dt1 dt2 dt3 f ( s ),000Issue 8, Volume 6, 20121The comparison of the numerical errors with fixed s1 0.50.Errors10.4Fig. 3.The comparison of the numerical errors with fixed s1 1.00. t [t1 , t2 , t3 ], s [s1 , s2 , s3 ] [0, 1] [0, 1] [0, 1]where k( s, t ) exp ( s1 s2 s3 t1 t2 t3 )and f ( s ) is chosen so that the isolated solutionx( s ) exp ( s1 s2 s3 ).The numerical results can see in TABLE XIII and TABLEXIV.9051
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESTABLE VIIC OMPUTATIONAL RESULTS WITH r 8 FOR EXAMPLE 0.751.00TABLE IXC OMPUTATIONAL RESULTS WITH r 8 FOR EXAMPLE 4.4. (CONT.)x(s1 , s2 40.07710.0498s1Abs. 750.750.751.001.001.001.001.00TABLE VIIIC OMPUTATIONAL RESULTS WITH r 8 FOR EXAMPLE 4.4. 3-0.1642-0.2707-0.2543-0.2108-0.1542-0.0996Abs. e-61.1408e-63.1583e-62.3462e-5From TABLE XIII and TABLE XIV, it shows the similarresults as two dimensional case. That is the numerical errorsdeceased when order of Taylor expansion increased for eachpoint.V. C ONCLUSIONIn this paper, we have developed the new Taylor seriesmethod and then applied it to find the numerical solution ofIssue 8, Volume 6, 1.00 x(s1 , s2 )/ 1157-0.0996Abs. e-61.1408e-63.1583e-62.3462e-5TABLE XC OMPUTATIONAL RESULTS WITH r 8 FOR EXAMPLE 4.4. (CONT.) x(s1 , s2 )/ 000.250.500.751.000.000.250.500.751.00 2 x(s1 , s2 )/ 2160.30840.1991Abs. e-61.1408e-63.1583e-62.3462e-5multivariate Fredholm integral equations of the second kind.This new method finds an approximation of the solution ofthe solution pointwise and therefore lends itself to numericalcomputations which can be done in parallel. The methodalso computes the derivatives of the solution concurrently.The results of several numerical experiments have shown theefficiency and accuracy of this new method.906
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCESTABLE XIC OMPUTATIONAL RESULTS WITH r 8 FOR EXAMPLE 4.4. 01.001.00 2 x(s1 , s2 )/ s1 1.00TABLE XIIIC OMPUTATIONAL RESULTS WITH r 3 FOR E XAMPLE 10.21080.23130.1991Abs. e-61.1408e-63.1583e-62.3462e-5TABLE XIIC OMPUTATIONAL RESULTS WITH r 8 FOR EXAMPLE 4.4. 6Abs. e-61.1408e-63.1583e-62.3462e-5ACKNOWLEDGMENTThis research is supported by the Centre of Excellence inMathematics, the Commission on Higher Education, Thailand.R EFERENCES[1] K. E. Atkinson, The Numerical Solution of Integral Equations of theSecond Kind, Cambridge University Press, Cambridge, UK. (1997).Issue 8, Volume 6, 0.13500.08160.0492r 3Abs. BLE XIVC OMPUTATIONAL RESULTS WITH r 3 FOR E XAMPLE 4.5. 2 x(s1 , s2 )/ 520.08190.13520.08190.0495r 4Abs. ] Y. Ren, B. Zhang, and H. Qiao, A Simple Taylor-Series ExpansionMethod for a Class of Second Kind Integral Equations, J. Comp. andAppl. Math., Vol. 110, pp. 15–24, (1999).[3] H. Lechoslaw, D. Konrad, Integral Modeling and Simulation inSome Thermal Problems, Proceedings of the 5th IASME/WSEAS Int.Conference on Heat Transfer, Thermal Engineering and Enviroment,Athens, Greece, Augst 25-27, pp. 42–47, (2007).[4] P. Huabsomboon, B. Novaprateep, and H. Kaneko, On Taylor-SeriesExpansion Method for the Second Kind Integral Equations, J. Comp.and Appl. Math., Vol. 234, pp. 1466–1472, (2010).
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES[5] P. Huabsomboon, B. Novaprateep, and H. Kaneko, On Taylor-SeriesExpansion Method for Volterra Integral Equations of the Second Kind,Scientiae Mathematicae Japonicae, Vol. 73, pp. 541–551, (2011).[6] L.Phusanisa, P. Huabsomboon, and H. Kaneko, A ModifiledTaylor Series Method to the Classical Love’s Equation, RecentResearches in Mathematical in Electrical Engineering and ComputerScience,Angers,France, November 17-19, pp. 68–72, (2011).[7] M.Tavassoli Kajani and N.Akbari Shehni, Solutions of two-dimensionalintegral equations systems by using differential transform method,Applications of Mathematics and computer Engineeriong,PuertoMorelos, Mexico, January 29-31, pp. 74–77, (2011).[8] P. Huabsomboon, B. Novaprateep, and H. Kaneko, Taylor-SeriesExpansion Methods for Nonlinear Hammerstein Equations, ScientiaeMathematicae Japonicae, to appear in (2012).[9] B. Novaprateep, K. Neamprem and H. Kaneko, An Expansion Methodsfor Multivariate Fredholm Integral Equations, Latest Advances inInformation Science and Application, Singapore City, Singapore, May11–13, pp 91–94, (2012).Issue 8, Volume 6, 2012908
Integral Equations of the Second Kind Boriboon Novaprateep, Khomsan Neamprem, and Hideaki Kaneko AbstractŠA new Taylor series method that the authors orig-inally developed for the solution of one-dimensional integral equations is extended to solve multivariate integral equations. In this
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