Mthemtil nd Computtionl Applitions Artile The Regulriztion-Homotopy Method for the Two-Dimensionl Fredholm Integrl Equtions of the First Kind Ahmet Altürk Deprtment of Mthemtis, Amsy University, Ipekkoy, Amsy 5, Turkey; hmet.lturk@msy.edu.tr; Tel.: +9-358-26-6 Ademi Editor: Fzl M. Mhomed Reeived: 24 Februry 216; Aepted: 22 Mrh 216; Published: 3 Mrh 216 Abstrt: In this work, we onsider two-dimensionl liner nd nonliner Fredholm integrl equtions of the first kind. The ombintion of the regulriztion method nd the homotopy perturbtion method, or shortly, the regulriztion-homotopy method is used to find solution to the eqution. The pplition of this method is bsed upon onverting the first kind of eqution to the seond kind by pplying the regulriztion method. Then the homotopy perturbtion method is employed to the resulting seond kind of eqution to obtin solution. A few emples inluding liner nd nonliner equtions re provided to show the vlidity nd pplibility of this pproh. Keywords: Fredholm integrl equtions; regulriztion method; homotopy perturbtion method MSC: 45A5; 65R2; 65J2 1. Introdution Integrl equtions pper in mny sientifi pplitions with very wide rnge from physil sienes to engineering. An immense mount of work hs been done on solving them. The literture is very dense on the subjet. Mny nlytil nd numeril tehniques hve been onstruted so fr nd it is still epnding [1 4]. In prtiulr, Fredholm integrl equtions of the first kind pper in mny physil nd engineering pplitions. There re numerous rtiles nd books on the investigtion of nlytil nd numeril solutions of one dimensionl Fredholm integrl equtions of the first kind [1 3]. In generl, integrl equtions re lssified s either first or seond kind depending on where the unknown funtion u( ppers. If it ppers only inside the integrl sign, it is lled n integrl eqution of the first kind, otherwise, it is lled n integrl eqution of the seond kind. The pperne of the unknown funtion only inside the integrl sign introdues some diffiulties. These, for instne, inlude pplying known useful methods introdued for solving the seond kind of equtions to the first kind. To overome this, one either hs to modify the eisting tehniques, trnsform the integrl eqution, or onstrut new method if it is possible. The regulriztion method is method tht trnsforms the integrl eqution of the first kind into the seond kind. We will mke use of this tehnique in the subsequent setions. First kind Fredholm integrl equtions re usully onsidered to be ill-posed problems. Tht mens, solutions my not eist nd if it eists, it my not be unique [3,5,6]. The one dimensionl liner nd nonliner Fredholm integrl equtions re of the form b f ( = λ K(, tu(t dt, (1 216, 21, 9; doi:1.339/m2129 www.mdpi.om/journl/m
216, 21, 9 2 of 1 nd b f ( = λ K(, tf(u(t dt, (2 respetively. In these equtions f (, the kernel K(, t, nd onstnt prmeter λ re given. The independent vrible is tken from losed nd bounded region. F(u( is nonliner funtion of u( nd the desired funtion is u(. There re some nlytil nd numeril pprohes to find et or pproimte solutions for Equtions (1 nd (2 in the literture. The one tht we prtiulrly fous on in this rtile is the regulriztion-homotopy method introdued by A. Wzwz in [7]. We investigte this method further nd show tht it is pplible to the two-dimensionl Fredholm integrl equtions of the first kind (see the net setion. The two-dimensionl liner Fredholm integrl equtions hs the following form: f (, t = λ nd the nonliner eqution hs the form: f (, t = λ K(, t, y, zu(y, z dy dz, (3 K(, t, y, zf(u(y, z dy dz. (4 In these equtions f (, t, the kernel K(, t, y, z, nd onstnt prmeter λ re given. F(u(, t is nonliner funtion of u(, t nd the desired funtion is u(, t. Reserh on the two-dimensionl se hs been getting more ttention reently [8 15]. The min gol in this work is to etend the regulriztion-homotopy method introdued in [7] for one dimensionl Fredholm integrl equtions of the first kind to two-dimensionl Fredholm integrl equtions of the first kind. This method n lso pplied for obtining numeril solutions of the Fredholm integrl equtions of the first kind. Motivted by [14 16], one possible pplition re ould be imge restortion nd denoising. 2. The Regulriztion Method The regulriztion method ws first introdued by A. N. Tikhonov [17,18], nd D. L. Phillips [2]. The pplition of the regulriztion method trnsforms the first kind integrl equtions into the seond. The detils for one dimensionl se n be found in [2,17 19]. We insted fous on the two-dimensionl se. The regulriztion method for the two-dimensionl Fredholm integrl equtions of the first kind ws introdued in [13]. We now briefly eplin the method. Like in one dimensionl se, the regulriztion method trnsforms the first kind of eqution: nd the nonliner eqution: to the seond kind of eqution: f (, t = f (, t = αu α (, t = f (, t K(, t, y, zu(y, z dy dz (5 K(, t, y, zf(u(y, z dy dz (6 K(, t, y, zu α (y, z dy dz (7 nd αu α (, t = f (, t K(, t, y, zf(u α (y, z dy dz, (8
216, 21, 9 3 of 1 respetively, where α is smll positive prmeter. Notie tht one ould epress Equtions (7 nd (8 s nd u α (, t = 1 α f (, t 1 α u α (, t = 1 α f (, t 1 α K(, t, y, zu α (y, z dy dz (9 K(, t, y, zf(u α (y, z dy dz, (1 respetively. It ws shown in [2] tht the solution of Eqution (9 or (1 s α pprohes u(, t whih is the solution of Eqution (5 or (19. In other words, u(, t = lim α u α (, t. We now stte some eistene nd uniqueness results from the opertor theory [1,21]. Let nd the integrl opertor Au(, t = A : C([, b] [, d] C([, b] [, d] d b K(, t, y, zu(y, z dy dz [, b], t [, d]. (11 Theorem 1. Let K : C([, b] [, d] [, b] [, d] R be ontinuous, then the opertor (11 is bounded with the norm: Proof. See [21]. A = m [,b],t [,d] d b K(, t, y, z dy dz. (12 Theorem 2. Let A be bounded opertor on C([, b] [, d] with A < 1 nd I denotes the identity opertor. Then I A hs bounded inverse on C([, b] [, d], whih is given by the Neumnn series (I A 1 = A k (13 k= nd stisfies (I A 1 1 1 A. (14 Proof. See [1]. We lso wnt to note tht for ny α >, Eqution (1 n be written in opertor form s u Au = f (15 With this nottion, theorem 2 ensures tht A < 1 is suffiient ondition for eistene nd uniqueness of the solution of Eqution (15 [21]. 3. The Homotopy Perturbtion Method In this setion, we investigte the pplition of the homotopy perturbtion (HPM to the two-dimensionl Fredholm integrl equtions of the first kind. The HPM is oupling of perturbtion method nd homotopy in topology. To see the bsi ide behind the HPM, let us onsider n eqution of the form: L(u =, (16
216, 21, 9 4 of 1 where L is ny integrl opertor. Then onve homotopy with n embedding prmeter p [, 1] n be defined by H(u, p = (1 pf(u + pl(u, (17 where F(u is funtionl opertor with known solutions. It is then esy to see tht H(u, p = (18 implies H(u, = F(u nd H(u, 1 = L(u One n infer from Equtions (17 nd (18, s the the embedding prmeter monotonilly inreses from to 1, the trivl problem (F(u = deforms the originl problem (L(u = [22]. For more detiled informtion on the HPM, we refer the reder to [23,24]. 4. The Regulriztion-Homotopy Method We investigte the first kind liner eqution: nd the nonliner eqution: f (, t = f (, t = K(, t, y, zu(y, z dy dz (19 K(, t, y, zf(u(y, z dy dz. In wht follows we fous on eplining the regulriztion-homotopy method for the liner se. We just mke note bout the nonliner se sine it will be treted similrly. We finlly give n lgorithm bout how to pply the method. We rell from Setion 2 tht the regulriztion method trnsform Eqution (19 to the following eqution: u α (, t = 1 α f (, t 1 α K(, t, y, zu α (y, z dy dz (2 Sine the im of this rtile is to etend the homotopy-regulriztion method introdued in [7], we onstrut the homotopy s follows: where F(u α = u α (, t, H(u α, p = (1 pf(u α + pl(u α =, (21 L(u α = u α (, t 1 α f (, t + 1 α K(, t, y, zu α (y, z dy dz nd p [, 1] is n embedding prmeter monotonilly inreses from to 1 [7]. The homotopy perturbtion method llows writing s power series in p nd setting p = 1, i.e., u α = u α, + pu α,1 + p 2 u α,2 +... (22 u α = lim p 1 n= p n u α,n (23
216, 21, 9 5 of 1 Now, if we epnd Eqution (21, we obtin or [ (1 pu α (, t + p u α (, t 1 α f (, t + 1 α [ u α (, t + p 1 α f (, t + 1 α Substituting Eqution (22 into (24 nd ombining like terms, we get ] K(, t, y, zu α (y, z dy dz = ] K(, t, y, zu α (y, z dy dz = (24 p : u α, (, t =, p 1 : u α,1 (, t = 1 f (, t, α p 2 : u α,2 (, t = 1 α. p n+1 : u α,n+1 (, t = 1 α K(, t, y, zu α,1 (y, z dy dz, K(, t, y, zu α,n (y, z dy dz, n 1 (25 Thus, we obtin formul for the omponents of the solution. If we substitute these omponents into Eqution (23, we obtin solution if it eists. Tht is, Eqution (23 holds if solution eists. We note tht the nonliner equtions will be treted similrly. We first mke hnge of vribles nd then trnsform the nonliner eqution into liner eqution so tht the bove lgorithm n be pplied. At the end, we reintrodue the originl vrible nd s result we obtin the solution for the nonliner eqution. Although there re some modifitions of HPM (MHPM whih were introdued nd pplied for solving two-dimensionl Fredhom integrl equtions in [25 27], we will not investigte the MHPM further. We will insted limit our fous to etend the regulriztion-homotopy method. We now summrize how to pply the regulriztion-homotopy method to Eqution (5. For Eqution (19, we first mke hnge of vribles to trnsform the nonliner eqution into liner one. We then pply the following steps: Apply the regulriztion method to trnsform the liner Fredholm integrl equtions of the first kind into seond kind, Apply the homotopy perturbtion method to find n pproimte solution, Let the regulriztion prmeter α to obtin solution. 5. Illustrtive Emples We note tht we ssume the kernel k(, t, y, z is seprble, i.e., k(, t, y, z = g(, th(y, z. We lso require the funtion f (, t involve omponents mthed by g(, t. This is neessry ondition for solution to eist [19]. Emple 1: Consider the following liner Fredholm integrl eqution of the first kind [28]: t = The regulriztion method trnsforms Eqution (26 to u α (, t = 1 α t 1 α te y+z u(y, z dy dz. (26 te y+z u α (y, z dy dz. (27
216, 21, 9 6 of 1 From Eqution (24 we hve ( 1 u α (, t = p α t 1 α Following the steps in Eqution (25, we get p : u α, (, t =, p 1 : u α,1 (, t = 1 α t, te y+z u α (y, z dy dz. (28 p 2 : u α,2 (, t = 1 α = t α 2, p 3 : u α,3 (, t = 1 α = t α 3, p 4 : u α,4 (, t = 1 α = t α 4,. te y+z u α,1 (y, z dy dz, te y+z u α,2 (y, z dy dz, te y+z u α,3 (y, z dy dz, (29 Thus, the pproimte solution beomes u α (, t = 1 α t ( 1 1 α + 1 α 2 1 α 3 +... = t α + 1. Letting α, we obtin the et solution s u(, t = t. There re other solutions to this eqution. For instne, u(, t = 2 t 2 (e 2 2, 3 t 3 (6 2e 2, 4 t 4 (9e 24 2... This is epeted beuse Fredholm integrl equtions of the first kind re often ill-posed problems. Tht mens solutions my not eist nd if it eists it my not be unique. Emple 2: Consider the following liner Fredholm integrl eqution of the first kind [1]: 1 2 (e2 1e +y = The regulriztion method trnsforms Eqution (3 to u α (, y = 1 2α (e2 1e +y 1 α e +y+s+t u(s, t ds dt. (3 e +y+s+t u(s, t ds dt. (31
216, 21, 9 7 of 1 Now, to onstrut homotopy let ( 1 u α (, y = p 2α (e2 1e +y 1 α p : u α, (, y =, p 1 : u α,1 (, y = 1 2α (e2 1e +y, e +y+s+t u α (s, t ds dt p 2 : u α,2 (, y = 1 e +y+s+t u α α,1 (s, t ds dt, = 1 8α 2 (e2 1 3 e +y, p 3 : u α,3 (, y = 1 e +y+s+t u α,2 (s, t ds dt, α = 1 32α 3 (e2 1 5 e +y, p 4 : u α,4 (, y = 1 α Thus, the pproimte solution beomes. e +y+s+t u α,3 (s, t ds dt, = 1 128α 4 (e2 1 7 e +y, u α (, y = 1 2α (e2 1e +y( 1 1 2 2 α (e2 1 2 + 1 2 4 α 2 (e2 1 4 1 2 6 α 3 (e2 1 6 +... = 2(e2 1e +y 2 2 α + (e 2 1 2. (32 (33 Letting α, we obtin the et solution s u(, y = 2e+y e 2 1. There re other solutions to this eqution. For instne, u(, y = 9(e2 1e 2+2y 8e 3+3y 2(e 3 1 2, (e 2 1(e 2 + 1 2,... Hving infinitely mny solutions for this eqution is quite norml beuse it is n ill-posed problem. Emple 3: Consider the following nonliner Fredholm integrl eqution of the first kind [1]: 6(1 + y = 1 1 1 + y (1 + s + tu2 (s, t ds dt. (34 We first trnsform the nonliner Eqution (34 to liner eqution by using the hnge of vrible v(s, t = u 2 (s, t (35
216, 21, 9 8 of 1 so tht Eqution (34 beomes 6(1 + y = 1 1 (1 + s + tv(s, t ds dt. (36 1 + y One we obtin solution to Eqution (36, then reversing Eqution (35, i.e., u(s, t = ± v(s, t, we obtin the desired solutions. The regulriztion method trnsform Eqution (36 to v α (, y = Now, to onstrut homotopy let 6α(1 + y 1 α ( v α (, y = p 6α(1 + y 1 α 1 + y (1 + s + tv α(s, t ds dt. (37 1 + y (1 + s + tv α(s, t ds dt, (38 p : v α, (, y =, p 1 : v α,1 (, y = 6α(1 + y, p 2 : v α,2 (, y = (1 + s + tv α(1 + y α,1 (s, t ds dt ( 3 + 2 log(2 = 6α 2, (1 + y 6 p 3 : v α,3 (, y = (1 + s + tv α(1 + y α,2 (s, t ds dt ( 3 + 2 log(2 2, = 6α 3 (1 + y 6 p 4 : v α,4 (, y = (1 + s + tv α(1 + y α,3 (s, t ds dt ( 3 + 2 log(2 3, = 6α 4 (1 + y 6 (39 Thus, the pproimte solution beomes v α (, y = = ( 1 1 ( 3 + 2 log(2 6α(1 + y α 6 (1 + y(6α + 3 + 2 log(2. Letting α, we obtin the et solution s. v(, y = + 1 ( 3 + 2 log(2 2 1 ( 3 + 2 log(2 3 +... α 2 6 α 3 6 (1 + y(3 + 2 log(2 Sine u(, y = ± v(, y,. (4 = ± (1 + y(3 + 2 log(2 These re etly the sme solutions obtined in [1]. There re other solutions s well.
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