Meixner s Polynomial Method for Solving Integral Equations

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1 Article Interntionl Journl of Modern Mthemticl Sciences, 15, 13(3): Interntionl Journl of Modern Mthemticl Sciences Journl homepge: ISSN:166-86X Florid, USA Meixner s Polynomil Method for Solving Integrl Equtions Syed Tuseef Mohyud-Din*, Muhmmd Tufil, Muhmmd Usmn Deprtment of Mthemtics, Fculty of Sciences, HITEC University, Txil Cntt Pkistn * Author to whom correspondence should e ddressed; E-Mil: syedtuseefs@hotmil.com Article history: Received 1 Mrch 15, Received in revised form 31 July 15, Accepted 1 August 15, Pulished 15 August 15. Astrct: This mnuscript witnesses modifiction in the efficient Glerkin weighted residul numericl method is proposed with Cheyshev polynomils s tril functions y inserting Meixner s polynomils insted of the trditionl Cheyshev polynomils. The modified version which is clled the Meixner s polynomils Method (MPM)is highly ccurte nd is tested on liner nd nonliner integrl equtions nd systems. Couple of exmples is given to elucidte the solution procedure. Comprison of numericl results explicitly reflects the very high level of ccurcy. Keywords: Meixner s polynomils, Glerkin weighted residul numericl method, Integrl equtions, MAPLE 13. Mthemtics Suject Clssifiction Code (1): 74S5; 4C4; 65Lxx 1. Introduction Integrl equtions [1] re useful in descriing the vrious phenomen in disciplines. Mny prolems of mthemticl physics cn e strted in the form of integrl equtions. These equtions lso occur s reformultions of other mthemticl prolems such s prtil differentil equtions nd ordinry differentil equtions. Therefore, the study of integrl equtions nd methods for solving them re very useful in ppliction. In recent yers, there hs een growing interest in the Volterr integrl equtions rising in vrious fields of physics nd engineering [], e.g., potentil theory nd Dirichlet

2 9 prolems, electrosttics, the prticle trnsport prolems of strophysics nd rector theory, contct prolems, diffusion prolems, nd het trnsfer prolems. β(x) u(x) = f(x) + K(x, t) dt, (1) α(x) where α(x) nd β(x) re function in x. Fredholm integrl equtions [3] rise in mny scientific pplictions. It ws lso shown tht Fredholm integrl equtions cn e derived from oundry vlue prolems. Erik Ivr Fredholm ( ) is est rememered for his work on integrl equtions nd spectrl theory. Fredholm ws Swedish mthemticin who estlished the theory of integrl equtions nd plyed mjor role in the estlishment of opertor theory. u(x) = f(x) + K(x, t) dt, () where nd re constnts. Similrly, Ael in 183 investigted the motion of prticle tht slides down long smooth unknown curve, in verticl plne, under the influence of the grvity. The prticle tkes the time f(x) to move from the highest point of the verticl height x to the lowest point on the curve. The Ael s prolem is derived to find the eqution of tht curve. Ael s integrl eqution is erliest exmple of n integrl eqution [4]. Ael s integrl eqution hs enormous ppliction in pplied prolems including microscopy, seismology, rdio stronomy, electron emission, tomic scttering, rdr rnging, plsm dignostic, X-rys rdiogrphy, fluid mechnics, io-mechnics, electromgnetic theory nd opticl fier evlution, see [5] nd references therein. The stndrd form of integrl eqution is f(x) = x u(t) (x ρ t ρ ) α dt, x >, < α 1. (3) Recently, there re mny pproches developed to find the exct nd pproximte solutions of integrl equtions, Yousefi et l., used CAS wvelets [6] nd Legendre wvelet [7] methods to find the solutions of liner nd nonliner Fredholm integrl equtions, Bizr nd Erhimi [8] implemented Cheyshev wvelets pproch for nonliner systems of Volterr integrl equtions, Usmn et l. developed some new lgorithms to seek the exct solutions of liner nd nonliner Ael s integrl eqution [9] nd generlized Ael s integrl equtions [1], Moreover; In [11], Brunner et l., introduced clss of methods depending on some prmeters to otin the numericl solution of Ael integrl eqution of the second kind, Vritionl itertion method [1], Homotopy perturtion method [13-14] nd Adomin s decomposition method [15] re effective nd convenient for solving integrl equtions. Kuthen [16] pplied liner multistep methods to otin the numericl solution of singulr nonliner Volterr integrl eqution. Also Kils nd Sigo used n symptotic method [17]to otin numericlly the solution of nonliner Ael Volterr integrl eqution. Orsi used Product Nystro m method [18], s numericl method, to otin the solution of nonliner Volterr integrl eqution, Fettis use the Guss Jcoi qudrture rule [19] to determine the numericl form of the solution of Ael eqution, Hung

3 93 et l. [] used the Tylor expnsion of the unknown function nd otined n pproximte solution, ltter on Piessens nd Vereten [1] nd Piessens [] uses Cheyshev polynomils to developed n pproximte solution to Ael eqution, Yousefi uses Legendte wvelets [3] presented numericl method for the solution of Ael integrl eqution. When input signl is with noisy error, Murio et l. [4] suggested stle numericl solution. Furthermore, Grz et l. [5] nd Hll et l. [6] used the wvelet method to invert the inversion of noisy Ael eqution. Inspired nd motivted y the ongoing reserch in this re, we present new, simple pproch for solving the integrl equtions including Volterr integrl [3], Fredholm integrl [3], Integro- Differentil [3], nonliner Ael s integrl [3], wekly singulr [3] nd system of Integrl [3] equtions. In the proposed scheme, we use efficient Glerkin weighted residul numericl method is proposed with Meixner s polynomils s tril functions. It is to e highlighted tht suggested lgorithm is extremely simple ut highly effective nd my e extended to other singulr prolems of diversified physicl nture. Moreover, this new scheme is cple of reducing the computtionl work to tngile level while still mintining very high level of ccurcy.. Meixner s polynomils Method (MPM) In mthemtics Meixner s polynomils (lso clled discrete Lguerre polynomil) re fmily of discrete orthogonl polynomils introduced y Josef Meixner (1934). The recurrence reltion is ( 1)k n! x! k! г(x δ+1) k!(n k)! k!(x k) г(x δ n+k+1) g k M n (x) = n k (4) M = 1, M 1 = 1 x 1, M 3 = 1 8 x x + 17 x 6, M = 1 4 x 9 x +, 4 M 4 = 1 16 x x x 33 8 x + 4,

4 94 Fig. 1. Grphicl representtion of first five polynomils in [,1] 3. Methodology Integrl Eqution of the 1 st Kind: Consider the integrl eqution of the 1 st kind is given s β(x) f(x) = λ K(x, t)u(t)dt, x. (5) α(x) where u(t) is the unknown function, to e determined, K(x, t), the kernel, is continuous or discontinuous nd squre integrle function, f(x) eing the known function. Now we use the technique of Glerkin method [7], to find n pproximte solution ũ(x) of Eq. (8). For this, we ssume tht n u (x) = k= k K k (x) = U T K(x), (6) where U T = [, 1,, ] T, nd V(x) = [K (x), K 1 (x), K (x), ] T. where K k (x) re Krvchuk polynomils of degree k defined in Eq. (6) nd k re unknown prmeters, to e determined. Sustituting Eq. (5) into Eq. (6), we get β(x) f(x) = λ K(x, t)u T K(t)dt, x. (7) α(x) Then the Glerkin equtions re otined y multiplying oth sides of Eq. (7) y K = K j (x), j =,1,, n, nd then integrting with respect to x from to, we hve φ(x) x dx = ( K(x, t)k(t)k (t)dt )dx, fk K K K 1 K K K K n K fk 1 K K 1 K 1 K 1 K K 1 K n K 1 where φ(x) = fk, K(t)K (t) = K K K 1 K K K K n K. [ fk n ] [ K K n K 1 K n K K n K n K n ] Since in ech eqution, there re two integrls. The inner integrnd of the left side is function of x, nd t, nd is integrted with respect to t from to x. As result the outer integrnd ecomes function of xonly nd integrtion with respect to x from to yields constnt. Thus for ech j =,1,,, n we hve liner eqution with n + 1 unknowns k, k =,1,,, n. (8)

5 95 Finlly the Eq. (8) cn e rewrite s K k,j = χ, (9) x where K k,j = ( K(x, t)k(t)k (t)dt)dx, k, j =,1,, n χ = f(x) K (x)dx j = φ(x) dx, j =,1,, n Now the unknown prmeters k re determined y solving the eqution (9) nd sustituting these vlues of prmeters in Eq. (6), we get the pproximte solution ũ(x) of the integrl eqution (5). Integrl Eqution of the nd Kind: Consider the integrl eqution of the nd kind is β(x) f(x) = c(x)u(x) + λ K(x, t)u(t)dt, x. (1) α(x) where u(t) is the unknown function, to e determined, k(x, t), the kernel, is continuous or discontinuous nd squre integrle function, f(x) nd u(x) eing the known function nd λ is constnt. Proceeding s efore K k,j = χ, (11) where K k,j = c K(t)K (t)dx, χ = f(x) x ( K(x, t)k(t)k (t)dt )dx, k, j =,1,, n K (x)dx j = φ(x) dx, j =,1,, n. Now the unknown prmeters k re determined y solving the eqution (11) nd sustituting these vlues of prmeters in Eq. (6), we get the pproximte solution u (x) of the integrl eqution (5). 4. Numericl Applictions In this section, we pply the proposed technique to construct pproximte nd nlyticl solutions of liner nd nonliner integrl equtions. Numericl results re very encourging Wekly Singulr Volterr Integrl Eqution Consider the following wekly singulr volterr Integrl eqution [3] u(x) = x x9 4 + x 1 3 (x t) 4 u(t)dt. The exct solution of Eq. (1) is u(x) = x. According to the proposed technique, consider the pth solution n u(x) = k= α k M k (x). (13) Consider nd order Meixner Polynomils, i.e. for k =, nd we pply the proposed technique to solve Eq. (13) with k =. We hve Eq. (13) is (1)

6 96 u(x) = k= α k M k (x) = k= C T M(x), (14) where C = [α, α 1, α ] T, nd k= C T M k (x) = x x9 4 + M(x) = [M, M 1, M ] T. Putting Eq. (14) into Eq. (1), we otined x 1 3 (x t) 4 k= C T M k (t) dt. Multiplying oth sides y R j (x), j =,1,. Then integrting over [,1] we hve Eq. (18) is 1 k= 1 C T M k (x)r j (x) dx = [x [ x 1 3 (x t) 4 k= 14 x9 4] for j =,1,. The mtrix form of Eq. (19) is given s R j (x)dx (15) C T M k (t) dt] R j (x)dx, (16) [ α ] [ α ] = [.33 ] α fter solving we get α = 1, α 1 = 18, α = 4. Consequently, we hve the exct solution is u(x) = x.this is the exct solution. α Fig. : Comprison of Exct nd Approximte solutions of u(x) of wekly singulr Volterr Integrl eqution using Meixner polynomil of Eq. (1) 4.. System of Wekly Singulr Integrl Eqution Consider the system of wekly singulr integrl eqution of 1 st Kind [3] x5 4(3x + 39) x9 5(57x 133) = x5 4(3x 39) x9 5(57x + 133) = x 1 (x t) 3 4 x 1 (x t) 1 5 u(t) + 1 (x t) 1 5 u(t) + 1 (x t) 3 4 v(t)dt, (17) v(t)dt. (17) The exct solution of Eq. (17) is u(x) = x 3 + x, v(x) = x 3 x.according to the proposed technique, consider the tril solution n n k=, u(x) = k= α k M k (x), v(x) = β k M k (x)

7 97 Tle 1: Comprison of the Exct Solution nd Approximte Solution of system of wekly singulr integrl eqution using Meixner polynomil of Eq. (17) when k = 3 x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E E E E E E E E E E E E E E E E E E E-6.6E E E-5 Fig. 3. ()-(): Comprison of Exct nd Approximte solutions of u(x) nd v(x) respectively of Volterr integrl eqution of nd Kind using Meixner s Polynomils of Eq. (17) 4.3. Liner Volterr Integrl Eqution Consider the liner Volterr integrl eqution [3] x u(x) = + x + sin(x) + cos(x) (x t) u(t)dt. (18) The exct solution of Eq. (18) is u(x) = (x t).according to the proposed technique, consider the tril solution n u(x) = k= α k M k (x). Consider 3 rd order Meixner s Polynomils, i.e. for k = 3 nd we pply the proposed technique to solve it tles -4 shows the error nlysis of exct nd pproximte solution for k= 3,, 5 respectively.

8 98 Tle : Comprison of the Exct Solution nd Approximte Solutions of Volterr integrl eqution using Meixner polynomil of Eq. (18) when k= 3 x Exct Solution Approximte Solution Error E E E E E E E E E E E- Tle 3: Comprison of the Exct Solution nd Approximte Solutions of Volterr integrl eqution using Meixner polynomil of Eq. (18) when k= x Exct Solution Approximte Solution Error E E E E E E E E E E E- Fig. 4: Comprison of Exct nd Approximte solutions of u(x) of volterr integrl eqution using Meixner polynomil of Eq. (18)

9 99 Tle 4: Comprison of the Exct Solution nd Approximte Solutions of Volterr integrl eqution using Meixner polynomil of Eq. (18) when k= 5 x Exct Solution Approximte Solution Error E E E E E E E E E E E System of Volterr Integrl Eqution Consider the system of Volterr integrl eqution of nd kind [3] x u(x) = 1 x + sin (x) + (u(t) + v(t)), v(x) = 1 x x sin (x) + (tu(t) + tv(t)), (19) (19) The exct solution of Eq. () is u(x) = 1 + sin (x). v(x) = 1 sin (x). According to the proposed technique, consider the tril solution n n k=. u(x) = k= α k M k (x). v(x) = β k M k (x) Consider nd order Meixner s Polynomils, i.e. for k =, nd we pply the proposed technique u(x) = k= α k M k (x) = k= C T M k (x), v(x) = k= β k M k (x) = D T M k (x) where C = [α, α 1, α ] T, D = [β, β 1, β ] T, M(x) = [M, M 1, M ] T. k=, Tles 5-7 shows the error nlysis of exct nd pproximte solution for k=,, 5 respectively.

10 3 Tle 5: Comprison of Exct nd Approximte solutions of u(x) nd v(x) respectively of volterr integrl eqution of nd kind using Meixner polynomil of Eq. () when k= x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E- 1.E E-.E E-.E E+ 6.E E- 8.E E- 1.1E E- 1.E E- 1.E E- 4.E E+ 5.E E+.E- Tle 6: Comprison of Exct nd Approximte solutions of u(x) nd v(x) respectively of volterr integrl eqution of nd kind using Meixner polynomil of Eq. (19) when k= x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E E E-3.346E E E E E E E E E E-3 3.9E E-3.4E E E E-3.318E E E-1

11 31 Tle 7: Comprison of Exct nd Approximte solutions of u(x) nd v(x) respectively of volterr integrl eqution of nd kind using Meixner polynomil of Eq. (19) when k= 5 x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E E E E E E E E E E E E E E E E E-47.56E E E E E-48 Fig. 5 ()-(): Comprison of Exct nd Approximte solutions of u(x) nd v(x) respectively of volterr integrl eqution of nd kind using Meixner polynomil of Eq. (19) 4.5. System of Fredholm Integrl Eqution Consider the system of Fredholm integrl eqution [3] 1 u(x) = ( π ) x + x tn 1 (x) + (xu(t) xv(t)dt)dt. () 1 1 v(x) = ( 3π ) + x + 6 tn 1 (x) + (tu(t) tv(t))dt. () 1 The exct solution of Eq. (3) is u(x) = x tn 1 (x), v(x) = x + tn 1 (x).according to the proposed technique, tle 8-1 shows the error nlysis of exct nd pproximte solution for k=, 5, 5 respectively.

12 3 Tle 8: Comprison of Exct nd Approximte solutions of u(x) nd v(x) respectively of volterr integrl eqution of nd kind using Meixner polynomil of Eq. () when k= x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E- 1.E E-.E E-.E E+ 6.E E- 8.E E- 1.1E E- 1.E E- 1.E E- 4.E E+ 5.E E+.E- Tle 9: Comprison of the Exct Solution nd Approximte Solution of Fredholm integrl eqution of nd Kind using Meixner polynomil of Eq. () when k= 5 x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E E E E E E E-8.414E E E E E E E E E E-9.148E E E E E-7 Fig. 6 ()-(): Comprison of Exct Approximte solutions of u(x) nd v(x) respectively of Fredholm integrl eqution of nd Kind using Meixner polynomil of Eq. ()

13 33 Tle 1: Comprison of the Exct Solution nd Approximte Solution of Fredholm integrl eqution of nd Kind using Meixner polynomil of Eq. () when k= 5 x u(x) v(x) u(x)_pprox. v(x)_pprox. Error in u(x) Error in v(x) E E E E E E E E E E E E E E E E E E E E E E Mixed Fredholm-Volterr Integrl Eqution Consider the Mixed Volterr-Fredholm integrl eqution [3] x π u(x) = x + x cos (x) + tu(t)dt + xu(t)dt, (1) The exct solution of Eq. (4) is u(x) = sin (x). According to the proposed technique, tle 11-1 shows the error nlysis of exct nd pproximte solution for k= 3, 3 respectively. Tle 11: Comprison of the Exct Solution nd Approximte Solution of Volterr-Fredholm integrl eqution using Meixner polynomil of Eq. (4) x Exct Solution Approximte Solution Error E E E E E E E E E E E-

14 34 Tle 1: Comprison of the Exct Solution nd Approximte Solution of Volterr-Fredholm integrl eqution using Meixner polynomil of Eq. (1) x Exct Solution Approximte Solution Error E E E E E E E E E E E-18 Fig. 7: Comprison of Approximte nd exct solution of Volterr-Fredholm integrl eqution using Meixner polynomil of Eq. (1) 4.7. Nonliner Ael s Integrl Eqution Consider the nonliner Ael s integrl eqution [3] 3 4 x 3( 4x + 9x ) = x 1 (x t) 1 3 u (t)dt. () The exct solution of Eq. () is u(x) = (1 x).according to the proposed technique, we get the exct solution for k =. 5. Conclusions A proposed technique sed on Glerkin weighted residul numericl method is proposed with Modified Lommel s polynomils is developed nd pplied to otin exct nd pproximte solutions of liner nd nonliner integrl equtions. Tle 1-3 nd Figure nd 3 shows the efficiency of the proposed

15 35 technique, s we increse n, i.e., order of polynomil solution hve less error. The proposed lgorithm is extremely simple, highly effective nd is of utmost ccurcy. References [1] M. Rhmn, Integrl equtions nd their pplictions, WIT press, (7). [] Adul J. Jerri, Introduction to integrl equtions with pplictions, John Wiley & Sons Inc., (1999). [3] A. M. Wzwz, Liner nd nonliner integrl equtions method nd pplictions, Springer Heidelerg Dordrecht London, New York, (11). [4] A. Jerri, Introduction to integrl eqution with ppliction, Wiley New York, (1999). [5] V. Singh, R. Pndey nd O. Singh, New stle numericl solutions of singulr integrl eqution of Ael type y using normlized Bernstein polynomil, Applied Mthemticl Sciences, 3 (5)(9): [6] S. Yousefi, A. Bniftemi, Numericl solution of Fredholm integrl equtions y using CAS wvelets. Applied mthemtics nd computtion, 183(1)( 6): [7] S. Yousefi, M. Rzzghi. Legendre wvelets method for the nonliner Volterr Fredholm integrl equtions. Mthemtics nd Computers in Simultion, 7(1)( 5): 1 8. [8] J. Bizr, H. Erhimi, Cheyshev wvelets pproch for nonliner systems of Volterr integrl equtions. Computers nd Mthemtics with Applictions, 63(3)(1): [9] M. Usmn, T. Zuir, I. Rshid, N. Khn, U. Iql nd S. T. Mohyud-Din, A new lgorithm for liner nd nonliner Ael s integrl equtions, Interntionl Journl of Modern Mthemticl Sciences, 8()( 13): [1] M. Usmn, I. Rshid, T. Zuir, U. Iql, N. Khn nd S. T. Mohyud-Din, A new lgorithm for liner generlized Ael s integrl equtions, Interntionl Journl of Modern Mthemticl Sciences, 8(1)( 13): 6 [11] H. Brunner, M.R. Crisci, E. Russo, A. Recchio, A fmily of methods for Ael integrl equtions of the second kind, Journl of Computtionl nd Applied Mthemtics. 34() (1991): [1] X. Ln, Vritionl itertion method for solving integrl equtions, Computers nd Mthemtics with Applictions, 54(7) (7): [13] M. El-Shhed, Appliction of He s homotopy perturtion method to Volterr s integrodifferentil eqution, Interntionl Journl of Nonliner Sciences nd Numericl Simultion, 6 ()(5): [14] A. Goli, B. Kermti, Modified homotopy perturtion method for solving Fredholm integrl equtions, Chos Solitons & Frctls, 37 (5)(6):

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