Homotopy Perturbation and Elzaki Transform for Solving. Nonlinear Partial Differential Equations

Transcription

1 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 Homotopy Perturbatio ad Elzaki Trasform for Solig Noliear Partial Differetial Equatios Tarig M Elzaki 1* & Ema M A Hilal 1 Mathematics Departmet, Faculty of Scieces ad Arts-Alkamil, Kig Abdulaziz Uiersity, Jeddah-Saudi Arabia Mathematics Departmet, Faculty of Scieces, Suda Uiersity of Scieces ad Techology-Suda Mathematics Departmet, Faculty of Scieces for Girles Kig Abdulaziz Uiersity Jeddah-Saudi Arabia * of the correspodig author: Tarigalzaki@gmailcom Abstract I this work, we preset a reliable combiatio of homotopy perturbatio method ad Elzaki trasform to iestigate some oliear partial differetial equatios The oliear terms ca be hadled by the use of homotopy perturbatio method The proposed homotopy perturbatio method is applied to the reformulated first ad secod order iitial alue problem which leads the solutio i terms of trasformed ariables, ad the series solutio is obtaied by makig use of the ierse trasformatio The results show the efficiecy of this method Keywords: Homotopy perturbatio methods, Elzaki trasform oliear partial differetial equatios 1 Itroductio Liear ad oliear partial differetial equatios are of fudametal importace i sciece ad egieerig Some itegral trasform method such as Laplace ad Fourier ad Sumudu trasforms methods see [Kilicma ad EGadai (9), (1) lslam, Yasir Kha, Naeem Faraz ad Fracis Austi (1)], are used to sole liear partial differetial equatios ad use fulless of these itegral trasform lies i their ability to trasform differetial equatios ito algebraic equatios which allows simple ad systematic solutio procedures Howeer, usig itegral trasform i oliear differetial equatios may icrease its complexity I recet years, may research workers hae paid attetio to fid the solutios of oliear differetial equatios by usig arious methods Amog these are the Adomia decompositio method [Hashim, Noorai, Ahmed Bakar Ismail ad Zakaria, (6)], the tah method, the homotopy perturbatio method [ Sweilam, Khader (9), Sharma ad Giriraj Methi (11), Jafari, Amiataei (1), (11) ], the differetial trasform method [(8)], ad the ariatioal iteratio method Elzaki trasform [ Tarig ad Salih, (11), (1)] is totally icapable of hadlig the oliear equatios because of the difficulties that are caused by the oliear terms Various ways hae bee proposed recetly to deal with these oliearities, oe of these combiatios of homotopy perturbatio method ad Elzaki trasform which is studies i this paper The adatage of this method is its capability of combiig two powerful methods for obtaiig exact 33

2 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 solutios for oliear partial differetial equatios This article cosiders the effectieess of the homotopy perturbatio Elzaki trasform method i solig oliear partial differetial equatios both homogeeous ad o-homogeeous 11 Elzaki Trasform The basic defiitios of modified of Sumudu trasform or Elzaki trasform is defied as follows, Elzaki trasform of the fuctio f ( t ) is [ ] E f ( t ) = f ( t ) e dt, t > (1) t Tarig M Elzaki ad Sailh M Elzaki i [(11), (1)], showed the modified of Sumudu trasform [(7), (1)] or Elzaki trasform was applied to partial differetial equatios, ordiary differetial equatios, system of ordiary ad partial differetial equatios ad itegral equatios Elzaki trasform is a powerful tool for solig some differetial equatios which ca ot sole by Sumudu trasform i [(1)] I this paper, we combied Elzaki trasform ad homotopy perturbatio to sole oliear partial differetial equatios To obtai Elzaki trasform of partial deriatie we use itegratio by parts, ad the we hae: f x t f x t f x E = T x f x E = T x f x (, ) 1 (, ) 1 (,) (, ) (,) (, ) (,) t t t f ( x, t ) d f ( x, t ) d E = T x E T x = x dx x dx Proof: [ (, )] [ (, )] To obtai ELzaki trasform of partial deriaties we use itegratio by parts as follows: t p t t p p t f f f Ε ( x, t ) = e dt = lim e dt = lim e f ( x, t ) e f ( x, t ) dt t t p t p T ( x, ) = f ( x,) We assume that f is piecewise cotiuous ad it is of expoetial order Now ( x, t ) t t f f Ε = e dt = e f ( x, t ) dt, x x x usig the Leibitz rule to fid: f d Ε = T ( x, ) x dx f d, x dx By the same method we fid: Ε = T ( x ) 34

3 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 To fid: f ( x, t ) Ε t f Let = g, the t we hae: (, t ) g ( x, t ) f g x Ε ( x, t ) = Ε = Ε g x, t t ( ) f 1 Ε = f ( x, t ) T ( x, ) f ( x, ) ( x,) t t We ca easily exted this result to the th partial deriatie by usig mathematical iductio 1 Homotopy Perturbatio Method: Let X ad Y be the topological spaces If f ad g are cotiuous maps of the space X ito Y, it is said that f is homotopic to g, if there is cotiuous map F : X [,1] Y such that F ( x,) = f ( x ) ad F ( x,1) = g ( x ), for each x X, the the map is called homotopy betwee f ad g To explai the homotopy perturbatio method, we cosider a geeral equatio of the type, L ( u ) = () Where L is ay differetial operator, we defie a coex homotopy H ( u, p) by H ( u, p) = (1 p) F ( u ) + pl ( u ) (3) Where F ( u ) is a fuctioal operator with kow solutio which ca be obtaied easily It is clear that, for We hae: H ( u,) = F ( u ), H ( u,1) = L ( u ) H ( u, p ) = (4) I topology this show that H ( u, p) cotiuously traces a implicitly defied cares from a startig poit H (,) to a solutio fuctio H ( f,1) parameter ad write the solutio as a power series The HPM uses the embed lig parameter p as a small 35

4 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 u = u + pu + p u + p u (5) If p 1, the (5) correspods to (3) ad becomes the approximate solutio of the form, f limu u = = (6) p 1 i = We assume that (6) has a uique solutio The comparisos of like powers of p gie solutios of arious orders Homotopy Perturbatio Elzaki Trasform Method: Cosider a geeral oliear o-homogeous partial differetial equatio with iitial coditios of the form: i Du ( x, t ) + Ru ( x, t ) + Nu ( x, t ) = g ( x, t ) (7) u ( x,) = h( x ), u ( x,) = f ( x ) Where D is liear differetial operator of order two, R is liear differetial operator of less order tha D, N is the geeral oliear differetial operator ad g ( x, t ) is the source term Takig Elzaki trasform o both sides of equatio (7), to get: t E [ Du ( x, t )] + E [ Ru ( x, t )] + E [ Nu ( x, t )] = E [ g ( x, t )] (8) Usig the differetiatio property of Elzaki trasforms ad aboe iitial coditios, we hae: 3 E [ u ( x, t )] = E [ g ( x, t )] + h( x ) + f ( x ) E [ Ru ( x, t ) + Nu ( x, t )] (9) Applig the ierse Elzaki trasform o both sides of equatio (9), to fid: 1 u ( x, t ) = G ( x, t ) E E Ru ( x, t ) + Nu ( x, t ) (1) Where G ( x, t ) represets the term arisig from the source term ad the prescribed iitial coditios Now, we apply the homotopy perturbatio method Ad the oliear term ca be decomposed as Where H ( u ) are gie by: u ( x, t ) = p u ( x, t ) (11) = = N [ u ( x, t )] = p H ( u ) (1) 36

5 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 (13) 1 H ( u, u, u ) = N p u, =,1,, i 1 i! p i = p = Substitutig equatios (11) ad (1) i equatio (1), we get: 1 p u ( x, t ) = G ( x, t ) p E E R p u ( x, t ) + p H ( u ) = = = (14) This is the couplig of the Elzaki trasform ad the homotopy perturbatio method Comparig the coefficiet of like powers of p, the followig approximatios are obtaied p u x t = G x t : (, ) (, ) [ ] p : u ( x, t ) = E E Ru ( x, t ) + H ( u ) p : u ( x, t ) = E E Ru ( x, t ) + H ( u ) { } p : u ( x, t ) = E E Ru ( x, t ) + H ( u ) (15) The the solutio is u ( x, t ) = u ( x, t ), + u1( x, t ) + u ( x, t ) + 3 Applicatios: I this sectio we apply the homotopy perturbatio Elzaki trasform method for solig oliear partial differetial equatios Example 31: Cosider the followig homogeous oliear partial differetial equatios u + uu =, u ( x,) = x (16) Takig Elzaki trasform of equatio (16) subject to the iitial coditio, we hae: t x [ ] [ ] E u ( x, t ) = x E uu x (17) The ierse Elzaki trasform implies that: 1 u ( x, t ) = x E E uu x (18) Now applyig the homotopy perturbatio method, we get: = 1 p u ( x, t ) x p E E p H ( u ) = = (19) 37

6 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 Where H ( u ) are He's polyomials that represets the oliear terms Or [ ] p uu = p( u + pu + p u + )( u + pu + p u + ) x 1 x 1x x u = u + pu + p u + 1 The first few compoets of He's polyomials, are gie by H ( u ) = u u x H ( u ) = u u + u u 1 1x 1 x H ( u ) = u u + u u + u u x 1 1x x Comparig the coefficiets of the same powers of p, we get: p : u ( x, t ) = x, H ( u ) = x p : u ( x, t ) = E E H ( u ) = xt, H ( u ) = xt p : u ( x, t ) = E E H ( u ) = xt, H ( u ) = 3xt 1 1 p : u ( x, t ) = E E H ( u ) = xt Therefore the solutio u ( x, t ) is gie by: Example 3: 3 x u ( x, t ) = x (1 + t + t + t + ) = t 1 Cosider the first order oliear partial differetial equatio 3 ut uu x t x t xt, u ( x,) x + = = () To fid the solutio by homotopy perturbatio Elzaki trasform method, we applyig homotopy perturbatio method after takig Elzaki ad ierse Elzaki trasforms of equatio (), we get: Where t xt p u ( x, t ) = t + xt + + p E E p H ( u ) = = (1) 38

7 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 H ( u ) = u u x H ( u ) = u u + u u 1 1x 1 x H ( u ) = u u + u u + u u x 1 1x x Comparig the coefficiets of like powers of p, we hae: 3 4 xt t p : u ( x, t ) = t + xt t xt xt 7t xt t p : u1( x, t ) = The oise terms appear betwee the compoets u ( x, t ), u1( x, t ), therefore, the exact solutio is gie by: u ( x, t ) = t + xt Example 33: Let us cosider the secod order oliear partial differetial equatio u u u = + u, u ( x,) = x t x x Applyig Elzaki trasform of equatio (), ad makig use of the iitial coditio, to fid: () [ ] E u ( x, t ) = x + E u x + uu xx Take the ierse Elzaki trasform of equatio (3), we get: { } u x t x E E u uu 1 (, ) = + x + xx Apply the homotopy perturbatio method of (4), to get: = + 1 p u ( x, t ) x p E E p H ( u ) = = (3) (4) (5) Or p u x + uu xx =, u = u + pu1 + p u + Equatio (6), ca be writte i the form; (6) p( u + pu + p u + ) + p( u + pu + p u + )( u + pu + p u + ) = x 1x x 1 xx 1xx xx The first few compoets of He's polyomials are gie by: 39

8 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 H ( u ) = u + u u x xx Therefore H ( u ) = u u + u u + u u 1 x 1x 1xx 1 xx p : u ( x, t ) = x, H ( u ) = 6x p : u ( x, t ) = E E H ( u ) = 6 x t, H ( u ) = 7x t p : u ( x, t ) = E E H ( u ) = 36x t 1 1 The the solutio of equatio () is gie by: 4 Coclusio x u ( x, t ) = x (1 + 6t + 36 t + ) = 1 6t I this paper, we mixture Elzaki trasform ad homotopy perturbatio method to sole oliear partial differetial equatios The solutio by usig Adomia decompositio method is simple, but the calculatio of Adomia polyomials is complex The fact that the deeloped algorithm soles oliear partial differetial equatios without Adomia's polyomials ca be cosidered as a clear adatage of this techique oer the decompositio method Refereces [1] S lslam, Yasir Kha, Naeem Faraz ad Fracis Austi (1), Numerical Solutio of Logistic Differetial Equatios by usig the Laplace Decompositio Method, World Applied Scieces Joural 8 (9): [] Nura Guzel ad Muhammet Nurulay (8), Solutio of Shiff Systems By usig Differetial Trasform Method, Dulupiar uiersities Fe Bilimleri Estitusu Dergisi, ISSN , PP [3] Shi- Hsiag Chag, I-Lig Chag (8), A ew algorithm for calculatig oe dimesioal differetial trasform of oliear fuctios, Applied Mathematics ad Computatio 195, [4] Tarig M Elzaki (11), The New Itegral Trasform Elzaki Trasform Global Joural of Pure ad Applied Mathematics, ISSN ,Number 1, pp [5] Tarig M Elzaki & Salih M Elzaki (11), Applicatio of New Trasform Elzaki Trasform to Partial Differetial Equatios, Global Joural of Pure ad Applied Mathematics, ISSN ,Number 1, pp 65-7 [6] Tarig M Elzaki & Salih M Elzaki (11), O the Coectios betwee Laplace ad Elzaki trasforms, Adaces i Theoretical ad Applied Mathematics, ISSN Volume 6, Number 1, pp 1-11 [7] Tarig M Elzaki & Salih M Elzaki (11), O the Elzaki Trasform ad Ordiary Differetial Equatio With Variable Coefficiets, Adaces i Theoretical ad Applied Mathematics ISSN Volume 6, 4

9 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 Number 1, pp [8] Lokeath Debath ad D Bhatta (6) Itegral trasform ad their Applicatio secod Editio, Chapma & Hall /CRC [9] AKilicma ad HEGadai (9), A applicatio of double Laplace trasform ad Sumudu trasform, Lobacheskii J Math3 (3) pp14-3 [1] J Zhag, (7) A Sumudu based algorithm m for solig differetial equatios, Comp Sci J Moldoa 15(3), pp [11] Hassa Eltayeb ad Adem kilicma, (1), A Note o the Sumudu Trasforms ad differetial Equatios, Applied Mathematical Scieces, VOL, 4, o, [1] Kilicma A & H ELtayeb (1), A ote o Itegral trasform ad Partial Differetial Equatio, Applied Mathematical Scieces, 4(3), PP [13] Hassa ELtayeh ad Adem kilicma (1), o Some Applicatios of a ew Itegral Trasform, It Joural of Math Aalysis, Vol, 4, o3, [14] NH Sweilam, MM Khader (9) Exact Solutios of some Capled oliear partial differetial equatios usig the homotopy perturbatio method Computers ad Mathematics with Applicatios 58, [15] PR Sharma ad Giriraj Methi (11) Applicatios of Homotopy Perturbatio method to Partial differetial equatios Asia Joural of Mathematics ad Statistics 4 (3): [16] MA Jafari, A Amiataei (1) Improed Homotopy Perturbatio Method Iteratioal Mathematical Forum, 5, o, 3, [17] Jagde Sigh, Deedra, Sushila Homotopy Perturbatio Sumudu Trasform Method for Noliear Equatios Ad Theor Appl Mech, Vol 4, 11, o 4, [18] Nura Guzel ad Muhammet Nurulay (8), Solutio of Shiff Systems By usig Differetial Trasform Method, Dulupiar uiersities Fe Bilimleri Estitusu Dergisi, ISSN , PP [19] Shi- Hsiag Chag, I-Lig Chag (8), A ew algorithm for calculatig oe dimesioal differetial trasform of oliear fuctios, Applied Mathematics ad Computatio 195, []Hashim, I, MSMNoorai,RAhmedSABakarESIIsmailadAMZakaria, (6) Accuracy of the Adomia decompositio method applied to the Lorez system chaos8135 [1] Tarig M Elzaki, Salih M Elzaki, ad Ema M A Hilal, (1) Elzaki ad Sumudu Trasforms for Solig Some Differetial Equatios, Global Joural of Pure ad Applied Mathematics, ISSN , Volume 8, Number, pp

10 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol, No3, 1 Table 1 Elzaki trasform of some Fuctios f ( t) E [ f ( t) ] = T ( u) 1 t 3 t! + at e 1 a siat 3 a 1+ a cosat 1+ a Tarig M Elzaki Departmet of Mathematics, Faculty of Scieces ad Arts-Alkamil, Kig Abdulaziz Uiersity, Jeddah-Saudi Arabia tarigalzaki@gmailcom ad tfarah@kauedusa Ema M A Hilal Departmet of Mathematics, Faculty of Scieces for Girls Kig Abdulaziz Uiersity, Jeddah Saudi Arabia ehilal@kauedusa 4

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