Research Article The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels

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1 Applied Mathematics, Article ID 72327, 7 pages Research Article The Approimate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels Qinghua Wu School of Mathematics and Statistics, Central South University, Changsha, Hunan 40083, China Correspondence should be addressed to Qinghua Wu; jackwqh@sinacom Received 9 March 204; Accepted 4 April 204; Published 7 April 204 Academic Editor: Turgut Öziş Copyright 204 Qinghua Wu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited A method for approimating the solution of weakly singular Fredholm integral equation of the second kind with highly oscillatory trigonometric kernel is presented The unknown function is approimated by epansion of Chebychev polynomial and the coefficients are determinated by classical collocation method Due to the highly oscillatory kernels of integral equation, the discretised collocation equation will give rise to the computation of oscillatory integrals These integrals are calculated by using recursion formula derived from the fundamental recurrence relation of Chebyshev polynomial The effectiveness and accuracy of the proposed method are tested by numerical eamples Introduction The Fredholm integral equations of second kind, b K (, t) y () +λ α y (t) dt = f (), a t [a, b], 0<α<, where K(, t) is a continuous function and f() is a given function, have many applications in mathematical physics and engineering, such as heat conduction problem, potential problems, quantum mechanics, and seismology image processing [ 4] Particularly, when K(, ) =0and 0<α<, ()iscalledweaklysingular In most of the cases, the integral equation cannot be done analytically and one has to resort to numerical methods Many numerical methods, such as collocation method and Galerkinmethod,havebeendevelopedtosolve(); for details see [2, 5, 6] These methods are well-established numerical algorithms; however, standard version of these classical methods may suffer from difficulty for computation of (), containing highly oscillatory kernels since the computation of the highly oscillatory integrals by standard quadrature methods is eceedingly difficult and the cost steeply increases with the frequency Furthermore, Galerkin method requires many double integrals when approimating the solution of integral () equation Specially, when the kernel is highly oscillatory, it requires the evaluation of many highly oscillatory double integrals, which can become computationally epensive Recently, for weakly singular Volterra integral equations of the second kind with highly oscillatory Bessel kernels, it was found that the collocation methods are much more easily implemented and can get higher accuracy than discontinuous Galerkin methods under the same piecewise polynomials space; for details see [7 0] In addition, collocation method only involves single integrals which are a little easier to evaluate Motivated by this fact, here we investigate the application of collocation method for the solution of Fredholm integral equation b e iω( t) y () +λ α y (t) dt = f (), a t [a, b], α <, ω Here y() is the unknown function and f() is a given function and assume that λ is not the eigenvalue of integral equation, since for any finite interval [a, b] it can be transformed to interval [, ] by linear transformation In this paper, we only consider the case of [a, b] = [, ] On the other hand, when the unknown function is approimated by Chebychev polynomial, the discretization of (2)

2 2 Applied Mathematics the integral equation (2) by collocation method will give rise to highly oscillatory integral which can be formulated as b e (iω( t)) a t α T j (t) d, [a, b], 0<α<, (3) where T j () denotes the Chebychev polynomials of the first kind In last few years many efficient methods have been devised for the evaluation of oscillatory integral, such as asymptotic method [], Filon-type method [2, 3], Levin s collocationmethod[4], modified Clenshaw-Curtis method, Clenshaw-Curtis-Filon-type method [5], and generalized quadrature rule [6], although some of these methods do not involve Chebyshev polynomial Piessens and Poleunis [7] consider a simpler case of α=0and use Chebyshev polynomials to evaluate the integral by somewhat indirect method involving a truncated infinite series of Bessel functions The evaluation of this infinite summation suggests a defect of this method An alternative procedure which avoids this infinite series is the original Bakhvalo and Vasil eva-legendre work involved in the evaluation of the integral M i (ω) = P i () e iω d, (4) by recurrence relation, where P i () denotes Legendre polynomialsadly,thisrelationprovedtobeunstableintheforwards direction for small ω In[8], Alaylioglu et al proposed a simple alternative approach analogous to Newton-Cotes based formulae This method avoids the instability to a certain degree compared with these two methods mentioned above We used the same idea and generalized it to the case of weakly singular and calculated oscillatory integral (3) This paper is organized as follows: in Section 2 we derive some basic formulae and introduce some mathematical preliminaries of the proposed method In Section 3 we discuss the evaluation of the integrals occurring in collocation equation In Section4 numerical eperiments are conducted to illustrate the performance of the proposed method 2 Fundamental Relations Firstly, by separation of real and imaginary part of y() and e iw( t), we transform the integral equation (2)intoequivalent systems of two linear integral equations of Fredholm in the forms y () +λ cos (ω ( t)) t α y (t) dt sin (ω ( t)) λ t α y 2 (t) dt = f (), y 2 () +λ sin (ω ( t)) t α y (t) dt cos (ω ( t)) +λ t α y 2 (t) dt = f 2 (), where y() = y () + iy 2 (), f() = f () + if 2 (), andi= (5) Assume that y() = (y (), y 2 ()) T and f() = (f (), f 2 ()) T ; then, systems of linear integral equations (5)canbe written in the matri form Iy () +λ K (, t) y (t) dt = f (), (6) where I =( 0 0 ), cos (ω ( t)) t α K (, t) =( sin (ω ( t)) t α sin (ω ( t)) t α ) cos (ω ( t)) t α Set y i () = T()b i, i =,2,whereT() = [T 0 (), T (),,T N ()], b i =[b i0,b i,,b in ] T Hence, unknown functions can be epressed by y () = TB, (8) where y () =( y () () 0 ), T () =(T y 2 () 0 T () ), (9) B=( b b 2 ) Then, the aim is to find Chebyshev coefficients, that is, the matri B We first substitute the Chebyshev collocation points, which are defined by i = cos(iπ/n), i = 0,,N, into (6) and then rearrange a new matri form to determine B: IY +λk = F (0) in which K is the integral part of (6)and y ( I ) 0 I 0 I =( d ), Y =( y ( ) ), 0 0 I y ( N ) f ( 0 ) IK ( 0 ) F =( f ( ) ), K =( IK ( ) ) f ( N ) IK ( N ) (7) () By substituting (8)into(0), the unknown coefficients can be easily computed from this linear algebraic equations and therefore we find the solution of integral equation (2) 3 Evaluation of the Integral I[ω, α, j, ]= (e(iω( t)) / t α )T j (t)dt The discretization of integral equation will lead to the calculation of integral I[ω, α, j, ] Inthissection,wewill derive the recursion formula to compute it efficiently from the fundamental recurrence relation of Chebyshev polynomial

3 Applied Mathematics Approimate the solution of eample 4 with α = 05, = 000, N=4 ω 0 0 Approimate the solution of eample 4 with α = 05, ω = 000, N= (a) (b) 0 0 Approimate the solution of eample 4 with α = 05, = 000, N=6 ω (c) Figure : Eample : the absolute error versus the -coordinate for N=4(a),N =8(b), and N=6(c) with negative data ignored The Chebyshev polynomials are of the form T 0 () = T () = T n () =2T n () T n 2 (), n=2,3,, (2) and the coefficients C i,j of j in T i () canbeeasilycalculated by the means of the recurrence relation C i,j =2C i,j C i 2,j, i 2, j i (3) By epanding the Chebyshev polynomial T i () in terms of powers of, theintegrali[ω, α, j, ] would be transformed into the form of I [ω, α, j, ] = j k=0 C j,k I [ω, α, k, ], (4) where I[ω,α,k,]= (e(iω( t)) / t α )t k dt By separation of real and imaginary part of e iw( t),(4)is transformed into t j I c [ω,α,j,]= α cos (ω ( t)) dt, t t j I s [ω,α,j,]= α sin (ω ( t)) dt, t (5) whose main difficulty now is turning to the evaluation of the following basic integrals: t j M c [ω,α,j,]= α cos (ωt) dt, t t j M s [ω,α,j,]= α sin (ωt) dt t (6)

4 4 Applied Mathematics 0 0 Approimate the solution of eample 42 with α=/3, ω = 000, N=4 0 0 Approimate the solution of eample 42 with α=/3, ω = 000, N= (a) 0 0 Approimate the solution of eample 42 with α=/3, = 000, N=6 ω (b) (c) Figure 2: Eample 2: the absolute error versus the -coordinate for N=4(a),N =8(b), and N=6(c) with negative data ignored With substitution for definite integral and binomial theorem, (6) isequivalentto eplicitly by the incomplete Gamma function Γ(z, α) [9]; that is, M c [ω,α,j,] = j k=0 C k j j k [( ) k + 0 t k α cos (ω ( t)) dt + t k α cos (ω ( t)) dt], 0 (7) t μ cos (ωt) dt = 2 [(iω) μ Γ (μ, iωt) + ( iω) μ Γ (μ, iωt)], t μ sin (ωt) dt = i 2 [(iω) μ Γ (μ, iωt) ( iω) μ Γ (μ, iωt)], (8) (9) where C k j is number of combination and M s[ω,α,j,]can be formulated analogy Efficient evaluation of (8) isbasedonaccuratecalculation of integrals in the formula which can be computed in which μ>0,>0; for details see ([20], pp 25) Once the integral I[ω, α, j, ] is obtained, by substituting into (0), the coefficient matri is derived; then, we can compute the solution of integral by solving these linear algebraic equations

5 Applied Mathematics 5 Table : Approimations for y() + (eiω( t) / t 05 )y(t)dt = e with ω=0 3 cos(π) cos(3π/4) cos(0) y y y y y y Table 2: Approimations for y() + (eiω( t) / t 05 )y(t)dt = e with ω=0 3 cos(π/4) y y y y y y Numerical Eamples In this section, we give some numerical eamples to illustrate the performance of proposed method In all the following eamples, N+isthenumberofmeshpoints, y N i () denotes the approimate solution, where N is the number of terms of the Chebyshev series, and y() denotes the eact solution, respectively All the computations have been performed by using Matlab R202a on a 25 GHz PC with 2 GB of RAM Eample We consider y() + (eiω( t) / t α )y(t)dt = f() and set f() = + (eiω( t) / t α )dt; then,the solution of which is y () = (20) We plot the absolute error y() y() by Matlab 202a internal function ezplot Specially,forN=4,withω=0 3 and α = 05, solve linear algebraic equations (0); it can be found that the approimate solutions are y () = , y 2 () = (2) It is easy to see from Figure that the proposed method is convergingandweonlyneedn=4so we could achieve high accuracy Eample 2 We consider y() + (eiω( t) / t α )y(t)dt = f() and set f() = e + (eiω( t) / t α )e t dt; then,the solution of which is y () =e (22) In this case, for N=4,withω = 000 and α=/3,the approimate solutions are y () = , y 2 () = (23) It is obvious to see from Figure 2 that the proposed method is efficient and could achieve high accuracy with little number of collocation points Eample 3 For a general case, we consider y() + (eiω( t) / t α )y(t)dt = e and set α = 05 Althoughthesolutionofthisequationisunknown,wecan verify the calculation precision of the method to a certain degree by comparing the approimate evaluation at mesh points In this case, for N=4,withω = 000 and α = 05, the approimate solutions are y () = ,

6 6 Applied Mathematics Table 3: Approimations for y() + (eiω( t) / t 05 )y(t)dt = e with ω=0 6 cos(π) cos(3π/4) cos(0) y y y y y y Table 4: Approimations for y() + (eiω( t) / t 05 )y(t)dt = e with ω=0 6 cos(π/4) y y y y y y y 2 () = (24) Acknowledgment This paper was supported by NSF of China (no 37376) References It is also easy to see from Tables, 2, 3, and4 that the presented method is efficient and accurate, although the eact solution is unknown 5 Conclusion In this paper, we eplore quadrature methods for weakly singular Fredholm integral equation of the second kind with oscillatory trigonometric kernels and present collocation methods with Chebyshev series for calculation of the solution For integral equation with highly oscillatory kernels, the standard collocation methods with classical quadrature methods are not suitable for the numerical approimation of the solution of integral equation, since the computation of the highly oscillatory integrals by standard quadrature methods is eceedingly difficult and the cost steeply increases with the frequency Based on the recursion formula derived in Section3, we compute the highly oscillatory integrals occurring in collocation equation, directly and efficiently Numerical eamples demonstrate the performance of algorithm Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper [] V M Aleksandrov and E V Kovalenko, Mathematical method in the displacement problem, Inzhenernyj Zhurnal Mekhanika Tverdogo Tela,vol2,pp77 89,984 [2] H Brunner, On the Numerical Solution of First-Kind Volterra Integral Equations with Highly Oscillatory Kernels, Highly Oscillatory Problems: From Theory to Applications, Isaac Newton Institute, HOP 3-7, 200 [3] R Estrada and R P Kanwal, Singular Integral Equations, Birkhäauser, Boston, Mass, USA, 2000 [4] A D Polyanin and A V Manzhirov, Handbook of Integral Equations, CRC Press, 998 [5] H Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, Cambridge, UK, 2004 [6] I G Graham, Galerkin methods for second kind integral equations with singularities, Mathematics of Computation, vol 39,no60,pp59 533,982 [7] S Xiang and H Brunner, Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels, BIT Numerical Mathematics,vol53,no,pp24 263,203 [8] S Xiang and K He, On the implementation of discontinuous Galerkin methods for Volterra integral equations with highly oscillatory Bessel kernels, Applied Mathematics and Computation,vol29,no9,pp ,203 [9] S Xiang and Q Wu, Numerical solutions to Volterra integral equations of the second kind with oscillatory trigonometric kernels, Applied Mathematics and Computation, vol 223, pp 34 44, 203 [0] Q Wu, On graded meshes for weakly singular Volterra integral equations with oscillatory trigonometric kernels,

7 Applied Mathematics 7 Computational and Applied Mathematics,vol263,pp , 204 [] A Iserles and S P Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT Numerical Mathematics,vol44,no4,pp ,2004 [2] A Iserles and S P Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proceedings of The Royal Society of London A,vol46,no2057,pp ,2005 [3] S Xiang, Efficient Filon-type methods for b a f()eiωg() d, Numerische Mathematik,vol05,no4,pp ,2007 [4] D Levin, Fast integration of rapidly oscillatory functions, Computational and Applied Mathematics,vol67,no, pp 95 0, 996 [5]SXiang,YJCho,HWang,andHBrunner, Clenshaw- Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications, IMA Numerical Analysis,vol 3,no4,pp28 34,20 [6] G A Evans and K C Chung, Some theoretical aspects of generalised quadrature methods, Compleity,vol9, no 3, pp , 2003 [7]RPiessensandFPoleunis, Anumericalmethodforthe integration of oscillatory functions, vol, pp , 97 [8] AAlaylioglu,GAEvans,andJHyslop, TheuseofChebyshev series for the evaluation of oscillatory integrals, The Computer Journal,vol9,no3,pp ,976 [9] M Abramowitz and I A Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, USA, 964 [20] I S Gradshteyn and I M Ryzhik, Table of Integrals, Series, and Products, New York Academic Press, 965

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The Approximate Solution of Non Linear Fredholm Weakly Singular Integro-Differential equations by Using Chebyshev polynomials of the First kind

The Approximate Solution of Non Linear Fredholm Weakly Singular Integro-Differential equations by Using Chebyshev polynomials of the First kind AUSTRALIA JOURAL OF BASIC AD APPLIED SCIECES ISS:1991-8178 EISS: 2309-8414 Journal home page: wwwajbaswebcom The Approximate Solution of on Linear Fredholm Weakly Singular Integro-Differential equations