Exact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method

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1 Exact Solutios for a Class of Noliear Sigular Two-Poit Boudary Value Problems: The Decompositio Method Abd Elhalim Ebaid Departmet of Mathematics, Faculty of Sciece, Tabuk Uiversity, P O Box 741, Tabuki 71491, Saudi Arabia Reprit requests to A E E; halimgamil@yahoocom Z Naturforsch 65a, (21; received Jauary 7, 29 / revised May 23, 29 May problems i applied mathematics ad egieerig are usually formulated as sigular twopoit boudary value problems A well-kow fact is that the exact solutios i closed form of such problems were ot obtaied i may cases I this paper, the exact solutios for a class of oliear sigular two-poit boudary value problems are obtaied to the first time by usig Adomia decompositio method Key words: Adomia Decompositio Method; Noliear Sigular Two-Poit Boudary Value Problems; Exact Solutios 1 Itroductio Some of the most commo problems i applied scieces ad egieerig are usually formulated as sigular two-poit boudary value problems of the form (x α y = f (x,y, x 1 (1 Subject to the boudary coditios y(=a ad y(1=b, (2 where α (,1], A ad B are fiite costats For example, whe α = ad f (x,y =q(xy σ,(1is kow as the geeralized Emde-Fowler equatio with egative expoet ad arises frequetly i applied mathematics (see [1, 2] ad the refereces cited therei Also, whe α = ad f (x,y = x 1/2 y 3 2, (1 is kow as Thomas-Fermi equatio [3], give by the sigular equatio y = x 1/2 y 3/2, which arises i the study of the electrical potetial i a atom Whe α = p {,1,2}, aother example is give by the sigular equatio (x p y = x p f (y, which results from a aalysis of heat coductio through a solid with heat geeratio The fuctio f (y represets the heat geeratio withi the solid, y is the temperature, ad the costat p is equal to, 1, or 2 depedig o whether the solid is a plate, a cylider or a sphere [4] I recet years, the class of sigular two-poit boudary value problems (BVPs modelled by (1 ad (2 was studied by may mathematicias ad a umber of umerical methods [4 14] ad aalytical methods [15 18] have bee proposed Although these umerical methods have may advatages, a huge amout of computatioal work is required i obtaiig accurate approximatios The most importat otice here is that all the previous attempts durig the last two decades were devoted oly to obtai approximate solutios wether umerical or aalytical I this research, the exact solutios for the class (1 ad (2 will be obtaied to the first time by implemetig the Adomia decompositio method (ADM [19 21] The ADM has bee used extesively durig the last two decades to solve effectively ad easily a large class of liear ad oliear ordiary ad partial differetial equatios However, a little attetio was devoted for its applicatio i solvig the sigular two-poit boudary value problems To the best of the author s kowledge, the oly attempt to solve the sigular twopoit boudary value problems by usig Adomia s method has bee doe recetly by Ic ad Evas [16] They obtaied a approximate solutio for oly oe oliear example by usig the ADM-Padé techique Very recetly, approximate solutios of liear sigular two-poit BVPs were obtaied by Bataieh et al [17] usig the modified homotopy aalysis method Also i [18], Kath ad Arua used aother aalytical method, the differetial trasformatio method, to obtai the exact solutios for some liear sigular two-poit BVPs Although these aalytical methods [17, 18] were show to be effective for solvig a / 1 / $ 6 c 21 Verlag der Zeitschrift für Naturforschug, Tübige

2 146 A E Ebaid Exact Solutios for Noliear Sigular Two-Poit Boudary Value Problems few liear examples, their applicability for oliear problems were ot examied So, we aim i this paper to show how to apply the decompositio method to obtai the exact solutios of these oliear sigular two-poit BVPs i a straightforward maer Before doig this, let us begi by itroducig the aalysis of the method i the ext sectio 2 Aalysis of Adomia Decompositio Method I this sectio, the class of sigular BVPs, (1 ad (2, will be hadled more easily, quickly, ad elegatly by implemetig the ADM rather tha by the traditioal methods for the exact solutios without makig massive computatioal work Let us begi our aalysis by writig (1 i a operator form: L xx (y= f (x,y, (3 where the liear differetial operator L xx is defied by L xx []= d (x α ddx dx [] (4 The iverse operator L 1 xx is therefore defied by ( L 1 xx []= x α []dx dx (5 Operatig with L 1 xx o (3, it the follows ( y = y(+ x α f (x,ydx dx (6 Notice that oly y(=a, is sufficiet to carry out the solutio ad the other coditio y(1=b, ca be used to show that the obtaied solutio satisfies this give coditio Assumig that f (x, y =r(xg(y, (6 becomes ( y = y(+ x α r(xg(ydx dx (7 The ADM is based o decomposig y ad the oliear term g(y as y = = y, g(y= = A (y,y 1,,y, (8 where A are specially geerated Adomia s polyomials for the specific oliearity ad ca be foud from the formula [ A = 1 d ]! dλ g( λ i y i, (9 i= λ = Isertig (8 ito (7, it the follows ( y = y(+ x α = = r(xa dx dx (1 Accordig to the stadard ADM, the solutio ca be computed by usig the recurrece relatio y = y(, ( y +1 = x α [r(xa ]dx dx, (11 I the ext sectio, we show that the algorithm (11 is very effective ad powerful i obtaiig the exact solutios for a wide class of oliear sigular twopoit boudary value problems i the form give by (1 ad (2 3 Applicatios ad Exact Solutios 31 Example 31 Firstly, we cosider the oliear sigular equatio [15, 16]: y + 1 2x y = e y ( 1 2 ey, (12 subject to the boudary coditios y(=l2 ad y(1= (13 The umerical solutios of this oliear sigular boudary value problem have bee discussed by El- Sayed [15] ad also by Ic ad Evas [16] I [15] the author established three iterative techiques to obtai the umerical solutios While i [16], Ic ad Evas used the stadard ADM with Padé trasformatio to approximate the solutio These attempts, as metioed above, were oly to obtai approximate solutios Here we show that we are able to obtai the exact solutio by usig the proposed algorithm (11 To do so, we firstly rewrite (12 i the form (x 1/2 y ( 1 = x 1/2 e y 2 ey (14 I order to apply algorithm (11 for this oliear sigular secod-order boudary value problem, we begi by usig formula (9 to calculate the first few

3 A E Ebaid Exact Solutios for Noliear Sigular Two-Poit Boudary Value Problems 147 terms of Adomia s polyomials of the oliear term e y ( 1 2 ey as ( A = e y 1 2 ey, A 1 = 1 2 ey (1 4e y y 1, Table 1 x y(x-adm-padé[5/5](x y(x Φ 1 (x E E E-4 E E-5 162E E-6 177E E E E E-17 A 2 = 1 4 ey [y y 2 8e y (y y 2 ], A 3 = 1 [ (15 12 ey y y 1y 2 + 6y 3 8e y (2y y 1y 2 + 3y 3 ], A 4 = 1 [ 48 ey y y2 1 y y y 1y y 4 32e y (y y 2 1y 2 + 3y y 1 y 3 + 3y 4 ] Now, applyig algorithm (11 o this sigular BVP yields y = l2, y +1 = ( x 1/2 [x 1/2 A ]dx dx, (16 From this recurrece relatio ad the Adomia s polyomials give by (15, we ca easily obtai y = l2, y 1 = x 2, y 2 = x4 2, y 3 = x6 3, y 4 = x8 4, y 5 = x1 5, y = ( 1 x 2, 1 The solutio is ow give by y = = y = y + =1 =1 y = l2 + =1 ( 1 +1 (x 2 = l2 = l2 l(1 + x 2 2 =l( 1 + x 2, ( 1 x 2 (17 (18 which is the exact solutio of (12 ad (13 The umerical results for the absolute errors y(x-adm-padé[5/5](x obtaied i [16] ad y(x Φ 1 (x are show i Table 1 Note that the approximate solutio Φ 1 = 1 = y is obtaied through our algorithm (17 The umerical values i Table 1 show that a good approximatio is achieved usig small values of 1-terms of our algorithm without usig Padé-approximat as made i [16] 32 Example 32 We ext cosider the oliear sigular equatio [15] subject to y + 1 x y = y 3 3y 5, (19 y(=1ady(1= 1 2 (2 We also rewrite (19 i the form (xy = x(y 3 3y 5 (21 This sigular BVP has bee also discussed by El- Sayed [15] by usig the same iterative techiques to obtai the umerical solutios Usig algorithm (11 for this oliear sigular boudary value problem, the solutio ca be elegatly computed by usig the recursive scheme y = 1, y +1 = ( x 1 [xa ]dx dx, (22 The formula (9 gives the first few terms of Adomia s polyomials as A = y 3 3y5, A 1 = 3(1 5y 2 y 2 y 1, A 2 = 3[(1 1y 2 y y (1 5y 2 y 2 ]y, A 3 =(1 3y 2 y (1 1y2 y y 1 y 2 + 3(1 5y 2 y 2 y 3 (23

4 148 A E Ebaid Exact Solutios for Noliear Sigular Two-Poit Boudary Value Problems Table 2 For simplicity, we cosider (26 whe ν = 2 I this x Itegral method [15] y(x Φ 1 (x case, the exact solutio of (26 is give by E-5 111E-16 ( 2 646E-6 111E E E-13 y = 2l E E-1 x ( E E-8 From the recursive relatio (22 ad Adomia s polyomials (23, we get So i this example we cosider the oliear sigular BVP y + 1 x y + 2e y =, (29 y = 1, y 1 = x2 2, y 2 = 3x4 8, y 3 = 5x6 16, y 4 = 35x8 128, y 5 = 63x1 256, y 6 = 231x12 124, y 7 = 429x14 248, Hece, the solutio i closed form is give by (24 subject to y(=2l2 ad y(1= (3 Kumar [11], applied a three-poit fiite differece method based o a uiform mesh to obtai approximate umerical solutio for this oliear sigular boudary value problem However, a huge amout of computatioal work bas bee doe to obtai such solutio We aim here to cofirm that algorithm (11, ot oly used i a straightforward maer, but also leads directly to the exact solutio Before doig so, we rewrite (29 i the form y = 1 x x4 8 5x x x (25 1 = x The umerical results for the absolute errors obtaied via itegral method [15] ad y(x Φ 1 (x are show i Table 2 Here, the approximate solutio Φ 1 = 1 = y is obtaied via algorithm (24 The umerical results i Table 2 show that very small errors are achieved usig oly 1-terms of our algorithm 33 Example 33 Fially, we cosider the oliear sigular boudary value problem [11] y + 1 x y + νe y = (26 This equatio has the exact solutio ( B + 1 y = 2l Bx 2, + 1 where B = (8 2ν ± (27 (8 2ν 2 4ν 2 2ν (xy = xe y (31 Followig the same aalysis of the previous examples, the solutio of this sigular BVP ca be elegatly computed by usig the recursive scheme y = 2l2, y +1 = 2 ( x 1 [xa ]dx dx,, (32 with Adomia s polyomials of the oliear term e y give as A = e y, A 1 = e y y 1, A 2 = 1 2 ey (y y 2, A 3 = 1 6 ey (y y 1y 2 + 6y 3, A 4 = 1 24 ey (y y2 1 y y 1 y y y 4 (33 By usig the recursive relatio (32 ad the first few terms of Adomia s polyomials (33, we ca easily obtai y = 2l2, y 1 = 2x 2, ( x y 2 = x 4 4 ( x 6 = 2, y 3 = 2, 2 3

5 A E Ebaid Exact Solutios for Noliear Sigular Two-Poit Boudary Value Problems 149 Table 3 x Three-poit F-D method [15] y(x Φ 1 (x E-9 E E-1 E E E E-9 632E E-9 186E-9 y 4 = x8 2 = 2 y 6 = x12 3 = 2 y = 2 ( x 8 4 ( x 12 [ ( 1 (x 2, y 5 = 2 6, y 7 = 2 ( x 1, 5 ( x 14 7, ], 1 (34 Now, the solutio ca be put i the closed form: y = = y = y + = 2l2+ 2 = 2l2 2 y =1 =1 =1 ( 1 x 2 ( 1 +1 (x 2 = 2l2 2l(1 + x 2 =2l ( x 2, (35 which is also the exact solutio of (29 ad (3 I Table 3, the umerical results for the absolute errors obtaied by the method i [11] ad y(x Φ 1 (x are show, where Φ 1 is obtaied by algorithm (34 Although the umerical results of the absolute errors obtaied by the method i [11] are very small, our absolute errors are still better without a huge amout of computatioal work as i [11] 4 Remarks Iitially, we ote that the class give by (1 (2 ivolve oly regular-sigular poits So, algorithm (11 is valid i this case oly Hece, for problems with irregular-sigularity, algorithm (11 should be chaged To make this poit as clear as possible, let us cosider a geeral class of sigular two-poit BVPs, y + λ x µ y = f (x,y, µ > 1, (λ is a fiite costat, (36 subject to the boudary coditios give by (2 It should be oted that with µ > 1, the sigular poit x = becomes a irregular-sigular poit I order to establish the ew algorithm, we firstly trasform (36 ito a ew equivalet form give by (x µ y =[µx µ 1 λ ]y + x µ f (x,y (37 The, we defie the iverse operator L 1 xx ( L 1 xx []= x µ as []dx dx (38 Operatig with L 1 xx o (37 ad usig the same aalysis of Sectio 2, we obtai the geeral algorithm y = y(, [ y +1 = x µ ( + x µ ] (µx µ 1 λ y dx dx [x µ r(xa ]dx dx, (39 Fially, we ote that algorithm (39 is valid for the class (36 uder the followig coditios: (i The itegral [ x µ (µxµ 1 λ y dx ] dx,exists 1 (ii The itegral (x µ x [xµ r(xa ]dxdx, exists 1 5 Coclusios Based o Adomia s method, a efficiet approach is proposed i this work to solve effectively ad easily a class of oliear sigular two-poit BVPs Moreover, the proposed approach ot oly used i a straightforward maer, but also requires less computatioal works i compariso with the other methods Ackowledgemets The author would like to thak the referees for their commets ad discussios

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