Composite Mendeleev s Quadratures for Solving a Linear Fredholm Integral Equation of The Second Kind

Transcription

1 Globl Journl of Pure nd Applied Mthemtics. ISSN Volume 12, Number (2016), pp Reserch Indi Publictions Composite Mendeleev s Qudrtures for Solving Liner Fredholm Integrl Eqution of The Second Kind Ilis Suryni Deprtment of Mthemtics, University of Riu, Peknbru 28293, Indonesi M. Imrn Numericl Computing Group, Deprtment of Mthemtics, University of Riu, Peknbru 28293, Indonesi M. D. H. Gml Deprtment of Mthemtics, University of Riu, Peknbru 28293, Indonesi Abstrct This pper discusses composite Mendeleev s qudrtures for solving liner Fredholm integrl eqution of the second kind. The numericl results show tht our estimtion hve good degree of ccurcy. AMS subject clssifiction: Keywords: Liner Fredholm integrl eqution, Mendeleev s qudrtures. 1. Introduction Consider liner Fredholm integrl eqution of the second kind s follows: u(x) = f(x)+ λ k(x, t)u(t)dt, x b. (1.1)

2 39 Ilis Suryni, M. Imrn, nd M. D. H. Gml where k(x, t) nd f(x)re known functions, but u(x) is function to be determined [1, ]. Mny uthors hve considered solving problem (1.1) with different methods such s Tylor s expnsion [6, 8, 1], Adomin s decomposition [2], wvelet [3, 7], Sinc colloction [10, 13], homotopy perturbtion [5] nd qudrture methods [9, 11]. In this pper, we solve problem (1.1) using composite Mendeleev s qudrtures. The derivtion of the method is given in the second section. The numericl pproch nd result of this method respectively is presented by third nd fourth section. In the end of the discussions is given the conclusion. 2. Composite Mendeleev s Qudrtures Mendeleev s qudrtures for pproximting the solution of [12]: nd f(x)dx hs two forms f(x)dx b ( f()+ 3f ( + 23 )) (b ), (2.2) f(x)dx b ( 3f ( + 13 ) ) (b ) + f(b). (2.3) We cll formul (2.2) nd (2.3) respectively the left nd right Mendeleev s qudrture. Suppose the intervl [,b] divided into n subintervls, tht is where The definite integrl = <x 2 <x <x 2n = b, nd h = b n, f(x)dx = x2 x 2i = + ih, f(x)dx cn be written s f(x)dx+ x i = 1, 2,...,n. x 2 f(x)dx+ + x 2n 2 f(x)dx. (2.) If we pply the left Mndeleev s qudrture (2.2) to pproximte the eqution (2.), then we hve f(x)dx h n (f (x 2i 2 ) + 3f(x 2i 1 )), (2.5) where i=1 x 2i 1 = x 2i 2 + 2x 2i, i = 1, 2,...,n. 3

3 Composite Mendeleev s Qudrtures By the sme wy, we use the right Mendeleev s qudrture (2.2), we obtin where f(x)dx h n (3f(x2i 1 ) + f(x 2i)), (2.6) i=1 x2i 1 = 2x 2i 2 + x 2i, i = 1, 2,...,n. 3 Formul (2.5) nd (2.6), we cll the composite left nd right Mendeleev s qudrture. 3. Solving Liner Fredholm Integrl Eqution of The Second Kind with Composite Mendeleev s Qudrtures To solve (1.1), we pproximte the integrl in the rigth-side of (1.1) with the composite Mendeleev s qudrtures (2.5) nd (2.6), we hve u(x) = f(x)+ h n (k(x, t 2j 2 )u(t 2j 2 ) + 3k(x, t 2j 1 )u(t 2j 1 )), (3.7) nd u(x) = f(x)+ h n (3k(x, t2j 1 )u(t 2j 1 ) + k(x, t 2j)u(t 2j )). (3.8) For x = x i, i = 0, 1, 2,...,2n in (3.7) nd (3.8), we get the following system: u(x i ) = f(x i ) + h n (k(x i,t 2j 2 )u(t 2j 2 ) + 3k(x i,t 2j 1 )u(t 2j 1 )), (3.9) nd u(x i ) = f(x i ) + h n (3k(x i,t2j 1 )u(t 2j 1 ) + k(x i,t 2j )u(t 2j )). (3.10) For simplicity, we define the following nottion: u i = u(x i ), k ij = k(x i,t j ), f i = f(x i ), (3.11) so eqution (3.9) nd (3.10) cn be written s u i = f i + h n (k i,2j 2 u 2j 2 + 3k i,2j 1 u 2j 1 ), (3.12)

4 396 Ilis Suryni, M. Imrn, nd M. D. H. Gml nd u i = f i + h n (3ki,2j 1 u 2j 1 + k i,2ju 2j ), (3.13) Eqution (3.12) nd (3.13) re the pproximted solution of the liner Fredholm integrl eqution of the second kind using the composite Mendeleev s qudrtures.. A Numericl Result In this section, we solve the Love integrl eqution using the composite left Mendeleev s qudrture (3.9). Exmple.1. Love s integrl eqution, is defined s follows: u(x) = f(x)+ 1 π 1 1 s s 2 u(t)dt, 1 x 1. (.1) + (x t) 2 We consider this eqution in prticulr cse when s = 1 nd f(x) = 1. The numericl results re show in Tble 1 nd Figure 1. Tble 1: The numericl solutions of (.1) with the composite left Mendeleev s qudrture by vrying n x n = 8 n = 16 n = 32 n =

5 Composite Mendeleev s Qudrtures () (b) (c) (d) Figure 1: The numericl solutions of (.1) with the composite left Mendeleev s qudrture for () n = 8, (b) n = 16, (c) n = 32 nd (d) n = 6 In Tble 1 we present the numericl solutions of (.1) by vrying the number of subintervls, n, in points x =±1, ± 3, ±1 2, ±1, 0. Then we plot ech numericl solution s depicted in Figure 1. From Tble 1 nd Figure 1 we see tht, for sufficiently smll h, we get good ccurcy. References [1] K. E. Atkinson, The Numericl Solution of Integrl Eqution of The Second Kind, Cmbridge University Press, New York. [2] E. Bbolin nd A. Dvri, 200. Numericl implementtion of Adomin decomposition method, Applied Mthemtics nd Computtion, 153, [3] E. Bbolin nd F. Ftthzdeh, Numericl computtion method in solving

6 398 Ilis Suryni, M. Imrn, nd M. D. H. Gml integrl equtions by using Chebyshev wvelet opertionl mtrix of integrtion, Applied Mthemtics nd Computtion, 188, [] J. M. Delves nd J. L. Mohmed, Computtionl Methods for Integrl Eqution, Cmbridge University Press, New York. [5] M. Jvidi nd A. Golbbi, A Numericl solution for solving system of Fredholm integrl equtions system by using homotopy perturbtion method, Applied Mthemtics nd Computtion, 175, [6] K. Mleknejd, N. Aghzdeh nd M. Rbbni, Numericl solution of second kind Fredholm integrl equtions system by using Tylor-series expnsion method, Applied Mthemtics nd Computtion, 175, [7] K. Mleknejd, M. T. Kjni nd Y. Mhmoudi, Numericl solution of liner Fredholm nd Volterr integrl eqution of the second kind by using Legendre wvelets, Applied Mthemtics nd Computtion, 19, [8] K. Mleknejd nd Y. Mhmoudi, 200. Numericl solution of liner Fredholm integrl eqution by using hybrid Tylor nd Block-Pulse functions, Applied Mthemtics nd Computtion, 19, [9] F. Mirzee nd S. Piroozfr, Numericl solution of liner Fredholm integrl equtions vi modified Simpson s qudrture rule, Journl of King Sud University, 23, [10] M. Muhmmd, A. Nurmuhmmd, M. Mori nd M. Sugihr, Numericl solution of integrl equtions by mens of the Sinc colloction method bsed on the double exponentil trnsformtion, Journl of Computtionl nd Applied Mthemtics, 177, [11] J. S. Ndjfi nd M. Heidri, Solving liner integrl equtions of the second kind with repeted modified trpezoid qudrture method, Applied Mthemtics nd Computtion, 189, [12] G. N. Pleshkov, On Mendeleev s qudrture formul, Computtionl Mthemtics nd Mthemticl Physics, 52, [13] J. Rshidini nd M. Zrebni, Convergence of pproximte solution of system of Fredholm integrl equtions, Journl of Mthemticl Anlysis nd Applictions, 333, [1] Y. Ren, B. Zhng nd H. Qio, A simple Tylor-series expnsion method for clss of second kind integrl equtions, Journl of Computtionl nd Applied Mthemtics, 110, 15 2.