An Iterative Method for Solving Two Special Cases of Nonlinear PDEs

Transcription

1 Conemporary Engineering Sciences, Vol. 10, 2017, no. 11, HIKARI Ld, hps://doi.org/ /ces An Ieraive Mehod for Solving Two Special Cases of Nonlinear PDEs Carlos Albero Abello Muñoz Universidad del Quindío Grupo de Modelación Maemáica en Epidemiología Armenia, Colombia Pedro Pablo Cárdenas Alzae Deparmen of Mahemaics and GEDNOL Universidad Tecnológica de Pereira Pereira, Colombia Anibal Muñoz Loaiza Universidad del Quindío Grupo de Modelación Maemáica en Epidemiología Armenia, Colombia Copyrigh c 2017 Carlos Albero Abello Muñoz e al. This aricle is disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac In his work we apply an ieraive mehod Adomian Decomposiion Mehod or ADM for solving some classes of nonlinear PDEs. The efficiency of his mehod is illusraed by invesigaing he convergence resuls for his ype of models. The numerical resuls show he reliabiliy and accuracy of he ADM. Keywords: Ieraive mehod, Adomian Decomposiion Mehod, Nonlinear iniial value problems

2 56 Carlos Albero Abello Muñoz e al. 1 Inroducion Many mahemaical models can be expressed using nonlinear parial differenial equaions and hose can be found in wide variey engineering applicaions. We know ha he mos general form of nonlinear PDEs is given by F u, u, u x 1 Therefore, he Adomian Decomposiion Mehod or ADM is applied here. ADM is a semi-numerical-analyic mehod for solving ordinary and parial differenial equaions. Adomian firs inroduced he concep of ADM in [1,]. This echnique consrucs an analyical soluion in he form of a polynomial [2]. This mehod is an alernaive procedure for obaining analyical series soluion of he differenial equaions. The series ofen coincides wih he Taylor expansion of he rue soluion a poin x 0 = 0, in he value case, alhough he series can be rapidly convergen in a small region. 2 Descripion of ADM The Adomian decomposiion mehod is applied o a general nonlinear equaion in he form [3] Lu + Ru + Nu = g 2 Here, he linear erms are decomposed ino L + R and he nonlinear erms are represened by Nu. Here, L is he operaor of he highes-ordered derivaives wih respec o and R is he remainder of he linear operaor. Thus we ge Lu = Ru Nu + g 3 is regarded as he inverse operaor of L and is defined by = 0 0 d d If L is a second order operaor, hen is defined by a wo-fold indefinie inegral ux, 0 Lu = ux, ux, 0 Now, operaing on boh sides of Eq.2 by using we obain 5 Lu = g Ru Nu 6

3 An ieraive mehod for solving wo special cases of nonlinear PDEs 57 Therefore we have ux, 0 ux, = ux, g Ru Nu 7 The ADM represens he soluion of Eq.7 as a series ux, = + Here, he operaor N u nonlinear is decomposed as Nu = + Now, subsiuing 8 and 9 ino 7 we obain u n x, 8 A n 9 where + u n x, = u 0 R + + u n x, A n 10 ux, 0 u 0 = ux, g 11 Then, consequenly we can obain u 1 = Ru 0 A 0 u 2 = Ru 1 A u n+1 = Ru n A n where u n x, will be deermined recurrenly, and A n are he so-called polynomials Adomian of u 0, u 1,..., u n defined by [ A n = 1 d n ] F λ n! dλ n i u i, n = 0, 1, 2, In his case we obain i=0 A 0 = fu 0 A 1 = u 1 f u 0 A 2 = u 2 f u ! u2 1f u 0 1.

4 58 Carlos Albero Abello Muñoz e al. Now, if we inroduce he parameer λ convenienly, we can obain ha where uλ = λ n u n 15 N uλ = λ n A n 16 Therefore, expanding by Taylor s series a λ = 0 we have N uλ = = [ ] 1 d n n! dλ Nuλ λ n n [ ] 17 λ i u i λ n 1 n! d n dλ n N The Adomian s polynomials A n can be calculaed using he recurrence equaion [ A n = 1 d n ] n! dλ N λ i u n i 18 If we are working wih sysems of differenial equaions or algebraic ype alike, he nonlinear erms N can be of he form N = Nu 1, u 2,..., u k,... λ=0 where u k = u ki 3 Applicaion and resuls For an illusraion of his mehod, we use wo es problems in his secion. We will show ha how he Adomian Decomposiion Mehod is compuaionally efficien and we shall also consider he performance of ADM wih heoreical soluion [5]. Tes problem 1: BBM model. We consider he nonlinear problem { u = u xx u x uu x ux, 0 = sech 2 x 19

5 An ieraive mehod for solving wo special cases of nonlinear PDEs 59 The Eq.19 can be wrien in erms of he differenial operaor as L u = L xx u L x u ul x u 20 As before, assuming he exisence of he inverse operaor, we apply i o Eq.20 and we ge L u = L xx u L x u ul x u 21 where by lineariy we obain an equivalen expression L u = L xx u L x u ul x u 22 Now, as in he previous example, he expression on he lef side of his las expression is equal o herefore L u = ux, ux, 0 L u = ux, ux, 0 = ux, = sech 2 x L xx u L x u L xx u + ul x u L x u ul x u L xx u L x u ul x u The erm non-linear corresponds o Nu = uu x, which is equivalen in erms of he Adomian s polynomials o A n = Nu = uu x. Thus, assuming a soluion in he form ux, = u n we obain recurrence relaion x u n = sech 2 + L xx u n L x Now, idenifying he firs erm as u 0 = sech 2 x, we ge u n x u 0 = sech 2 u 1 = L xx u 0 L x u 0 A 0. u n+1 = L xx u n L x u n A n A n 23

6 550 Carlos Albero Abello Muñoz e al. o for all n 1. Therefore, we have ha A 0 = Nu 0 = u 0 u 0 x, which leads A 0 = u 0 u 0 x = = sech 2 x sech 2 x = 1 2 sech x Then, he erm u 1 is deermined by x sech sech2 x anh x anh x x u 1 = = = 1 2 L xx u 0 L x u 0 A 0 x L xx sech 2 0 = 1 2 sech2 x L x sech 2 x x x x x sech 2 anh + sech anh d x x anh 1 + sech 2 In he same way, we can compue u 2, ha is or wha is he same u 2 = L xx u 1 L x u 1 A 1 u 2 = L xx u 1 L x u 1 A 1 In his case, he Adomian s polynomial A 1 is deermined by A 1 = u 0 x u 1 + u 0 u 1 x Therefore, afer performing he respecive calculaions we obain 1 x x 2 sech anh u 2 = L xx u 1 L x u 1 A 1 = 1 x [ x 3x 256 sech cosh + 16 coshx + cosh x ] 3x 159 sinh 2 sinhx 3 sinh 2 2 Finally, he soluion is of he form ux, = u n. So ha,

7 An ieraive mehod for solving wo special cases of nonlinear PDEs 551 ux, = u n = u 0 + u 1 + u 2 + x = sech x x x 2 sech2 anh 1 + sech x [ x 3x 256 sech cosh + 16 coshx + cosh 2 2 x ] 3x 159 sinh 2 sinhx 3 sinh Tes problem 2: Burgers s dimensional model. We consider he nonlinear problem [6] { u + 2uu x + uu y u xx + u yy ux, y, 0 = x + y 2 The exac soluion of his model is ux, y, = x + y. Now, wriing his + 1 model in he form of he differenial operaor we have L u = L xx u + L yy u 2uL x u + ul y u 25 Assuming he exisence of he inverse operaor Eq.25 we ge and applying i o L u = L xx u + L yy u 2 ul x u + ul y u knowing ha L u = ux, y, ux, y, 0 and nex o he iniial condiion we obain he expression ux, y, = x + y + L xx u + L yy u 2 ul x u + ul y u 26 Now, assuming a soluion of he form ux, y, = u n and knowing ha he non-linear erm is Nu = ul x u + ul y u we ge u n = x+y+ [ L xx u n + L yy u n 2 ] A n 27 Idenifying he firs erm u 0 = x+y, we can compue he oher erms as follows

8 552 Carlos Albero Abello Muñoz e al. [ u 1 = Lxx u 0 + L yy u 0 2 A 0 ] [ u 2 = Lxx u 1 + L yy u 1 2 A 1 ]. u n+1 = [ Lxx u n + L yy u n 2 A n ] So ha, knowing ha u 0 = x + y, we can compue he firs polynomial, A 0 = u 0 L x u 0 + u 0 L y u 0 = 2x + 2y, where u 1 is deermined as u 1 = [L xx x + y + L yy x + y] 2 2x + 2y = 0 2 = x + y 0 2x + 2y d Knowing he value of A 0, we can compue he following Adomian s polynomial A 1 using he expression A 1 = u 1 u 0 x + u 0 u 1 x + u 1 u 0 y + u 0 u 1 y. Therefore, we have ha A 1 = 16x + y. Then, o compue u 2 we ge u 2 = [L xx x + y + L yy x + y] 2 16x + y [ ] 2 x + y 2 = = 16 2 x + y And so on we can obain A 2 = 96 2 x+y and herefore u 3 = 6 3 x+y. Finally, we obain he general soluion ux, y, = x + y x + y x + y 6 3 x + y + = x + y [ n n] = x + y + 1 Conclusion In his paper, we calculaed he exac soluion of he BBM equaion and Burgers s dimensional model by using ADM. We demonsraed ha his ieraive mehod is quie efficien o deermine soluion in closed form. The new

9 An ieraive mehod for solving wo special cases of nonlinear PDEs 553 scheme obained using he ADM yields an analyical soluion in he form of a rapidly convergen series. This is why he ADM makes he soluion procedure much more aracive. Acknowledgemens. We would like o hank he referee for his valuable suggesions ha improved he presenaion of his paper and our graiude o Grupo de Modelación Maemáica en Epidemiología of he Universidad del Quindío and Deparmen of Mahemaics of he Universidad Tecnológica de Pereira Colombia and he group GEDNOL. References [1] A. Parra, Resolución de Ecuaciones Diferenciales Parciales de Segundo Orden no Lineales Mediane el méodo de Adomian, Tesis de Maesría en Maemáicas, UTP, [2] A. Boumenir, M. Gordon, The rae of convergence for he decomposiion mehod, Numerical Funcional Analysis and Opimizaion, , hps://doi.org/ /nfa [3] A. Wazwaz, A new approach o he nonlinear advecion problem: An applicaion of he decomposiion mehod, Applied Mahemaics and Compuaion, , hps://doi.org/ / [] G. Adomian, R. Rach, Noise erm in decomposiion soluion series, Comp. Mah. Appl., , hps://doi.org/ / c [5] H. Ismail, K. Raslan, Aziza A. Abd Rabboh, Adomian decomposiion mehod for Burgers-Huxley and Burgers-Fisher equaions, Applied Mahemaics and Compuaion, , hps://doi.org/ /j.amc [6] O. González, R. Bernal, Applying Adomian decomposiion mehod o solve Burgers equaion wih a nonlinear source, Inernaional Journal of Applied and Compuaional Mahemaics, , hps://doi.org/ /s Received: July 8, 2017; Published: July 20, 2017