Homotopy Perturbation Method for Solving Linear Boundary Value Problems

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1 International Jornal of Crrent Engineering and Technolog E-ISSN , P-ISSN INPRESSCO, All Rights Reserved Available at Research Article Homotop Pertrbation Method for Solving Linear Bondar Vale Problems Nada. F. Alshehri * Math Department, Faclt of science, King Khaled Universit Abha, KSA Accepted 10 Sept 2016, Available online 26 Sept 2016, Vol.6, No.5 (Oct 2016) Abstract A homotop pertrbation method (HPM) is introdced for obtaining soltions of sstems of linear partial differential eqations. The non-newtonian flid nder consideration is obeing the Casson model. Heat and mass transfer for nstead flow with heat sorces and the effect of chemical reaction are stdied. The main advantage of this method is the flexibilit to give approximate and exact soltions to both linear and nonlinear problems. The reslts show that the methods are simple and effective. The distribtions of the velocit, temperatre and concentration fnctions are discssed and illstrated graphicall for different vales of the phsical s of the problem. Kewords: HPM, Casson model etc. Introdction 1 The theor and application of optimal control are relevant to fields sch as biomedicine, phsics, econom, aerospace, chemical engineering, and robotics. However, the optimal control of nonlinear sstems is a difficlt task, which has been stdied for decades. Identification of the thermal condctivit coefficient and heat sorce is important in engineering practice. Both problems belong to the class of inverse problems and can be solved b means of homotop pertrbation method. The homotop pertrbation method (HPM) has been worked ot over a nmber of ears b nmeros athors, mainl b He J. In (HPM), a complicated problem nder std is continosl deformed into a simple problem which is eas to solve to obtain an analtic or approximate soltion investigated b He J. Demir A et al. explained that a new homotop pertrbation techniqe is proposed to find a soltion based on the decomposition the fnction f (x, ) which leads to the constrction of new homotopies and clarif that the proposed method (HPM) is ver effective and simple. Khati R. et al. Investigate the exact soltion of linear and nonlinear diffsion eqations are obtained b Homotop Pertrbation Method. a hbrid method that combines the homotop pertrbation method (HPM) and Padétechniqe to obtain the approximate analtic soltion of the Hamilton Jacobi Bellman eqation is Investigated b Ganjefar S. and Rezaei S. Babolian E.et. al. se the homotop pertrbation method for solving time-dependent differential eqations. Patra A. and Saha S. enhance the heat transfer between primar *Corresponding athor: Nada. F. Alshehri srface and the environment, radiating extended srfaces are commonl tilised. Temperatre distribtion and effectiveness of convective radial fins with constant and temperatre-dependent thermal condctivit are solved b appling homotop pertrbation smd transform method (HPSTM). The homotop pertrbation method (HPM) combined with Trefftz method is emploed to find the soltion of two kinds of nonlinear inverse problems for heat condction investigated b Grsa K. and Maciag A. Tsai C. investigated the homotop analsis method (HAM) is combined with the method of fndamental soltions (MFS) and the agmented polharmonic spline (APS) to solve certain nonlinear partial differential eqations (PDE). The prpose of the present investigation is to std the homotop pertrbation method for solving linear deferential eqation which flow of cople-stress. We restrict orselves to the case of linear differential eqations and sggest a qite simple techniqe for sing HPM. Comparativel speaking, even thogh we don t claim that the sggested techniqe is the best one, this is a reliable techniqe which one can simpl se withot having mch experience and nderstanding of HPM. Mathematical formlation and soltion Channel flow between two oscillating poros plates and the flid is being injected b one plate with constant velocit and scked off b the other plate with the same velocit. Then the continit eqation redces to so that is the fnction of and onl. A niform magnetic field 1734 International Jornal of Crrent Engineering and Technolog, Vol.6, No.5 (Oct 2016)

2 Homotop Pertrbation Method for Solving Linear Bondar Vale Problems with magnetic flx densit vector ( ) is applied. Also the heat and mass transfer in the channel are taken into accont b giving temperatre and concentration to the lower and pper plate as and respectivel. A two dimensional a Cartesian coordinates ( ) are characterized and considered, where is in the direction of flid motion and is perpendiclar to the mean position of the channel. Fig.1: Geometr of the problem he constittive eqation of the Bi-viscosit model can be written in the following tensor notation as: ( ), ( ) Where is the plastic viscosit, is the ielding stress,, which is the component of the deformation rate and the vale of denotes the pper limit of apparent viscosit coefficient. For Newtonian flid. Where are the for nstead twodimensional flows, the velocit, temperatre and concentration can be written as a fnction of. The viscoelastic flid in plsatile pressre gradient -direction is driven b a The temperatre eqation with radiation * ( ) + (3) The concentration eqation with diffsion and chemical reaction * ( ) + (4) Where is the velocit, is the extra stress tensor obeing Casson model, is the liqid densit, is the pressre, is the viscosit of the flid. permeabilit of magnetic field, is crrent densit. temperatre and concentration of the flid, is the coefficient of mass diffsivit and is the specific heat capacit at constant pressre, is the mean absorption coefficient, is the mean temperatre, is heat flx, is constants of chemical reaction. Ths, for the effect of cople stress to be present mst be nonzero. For nstead two-dimensional flow, the velocit, temperatre and concentration can be written as a fnction of and onl. ( ( ) ) ( ) ( ) where be the longitdinal and transverse velocit components of the flid, respectivel. Eqations (1-4) redce to ( ) ( ) (5) ( ) (6) ( ) (7) Where are known qantities represents the stead-state of the pressre gradient and oscillator parts respectivel where is the freqenc of plsatile flow. Basic eqations The eqations governing the flow of an incompressible viscoelastic flid are given b: The continit eqation ( ) (8) In mathematical form, the corresponding bondar conditions can be pt as ( ) ( ) } (9) Let s introdce the following dimensionless qantities as follow: (1) The momentm eqation (10) * ( ) + ( ) (2) 1735 International Jornal of Crrent Engineering and Technolog, Vol.6, No.5 (Oct 2016) }

3 Homotop Pertrbation Method for Solving Linear Bondar Vale Problems Where is the magnetic field, is the Schmidt nmber, and the Eckhart nmber, is the Renolds nmber, is the Prandtle nmber. The sstem of eqations (5-8) with the condition (10) can be written in dimensionless form after dropping star mark as follows: ( ) ( ) ( ) ( ) (11) ( ) (12) ( ) ( ) (13) Where is the cople stress and is the radiation. The corresponding dimensionless bondar conditions are: Soltion of the problem } (14) To solve eqations (11-13) sbjected to the bondar conditions (14), we write the velocit, temperatre and concentration in the form: ( ) ( ) (15) ( ) ( ) (16) ( ) ( ) (17) Eqations ((11)-(13)) take the following form: Stead case Unstead case (18) (19) (20) ( ) ( ) ( ) ( ) (25) Collecting the coefficients of like power of ( ), we get the following set of eqations of motion: (26) (27) ( ) (28) ( ) (29) While the bondar conditions will take the following form: } (30) Hence, the soltions of eqations ((18)-(23)) sbject to bondar conditions (30) can be obtained as follows respectivel: Stead case (31) ( ) ( ) (32) Unstead case (33) (34) (35) ( ) ( ) (36) Reslts and discssion (37) (38) ( ) (21) ( ) (22) (23) Eqations ((18)-(23)) can be solved b sing the following pertrbation method for small homotop pertrbation ( ), ( ) ( ) ( ) ( ) (24) In the present work we applied the homotop pertrbation method to solve linear differential eqations. The problem of an incompressible nstead (MHD) plsatile flow of cople-stress non-newtonian flid obeing Casson model with heat and mass transfer in two-dimensional channel has been stdied. We discssed the inflences of the magnetic field and porosit of the medim on the flid. The formla of the velocit, temperatre and concentration have been obtained. Different s of the problem have been showed graphicall sch as the magnetic, the cople stress,the 1736 International Jornal of Crrent Engineering and Technolog, Vol.6, No.5 (Oct 2016)

4 Nada. F. Alshehri Homotop Pertrbation Method for Solving Linear Bondar Vale Problems of non-newtonian flid, Prandtl nmber, and Shmidt nmber and the chemical reaction. We compare the reslts with other researches solved with other was. Figre (2) illstrates the change of the velocit distribtion verss the coordinate with several vale of, In this figre the velocit increases when the Renolds nmber increases. The relation between velocit distribtion and cople stress has been presented in figre (3) where the velocit distribtions decreases with the increasing of Figre (4) illstrates the change of the velocit distribtion verss the coordinate with several vale of. in this figre velocit distribtion increases with the increasing of the magnetic field We can observe from figre (5) that the velocit increases with the increases of according to the time variation. It is fond that the velocit distribtion increases (decreases) with the increases of according to the time variation in figre (6). in figres (7) the velocit distribtion increases with the increases of according to the time variation. 0.2 M 4 M 5 M Figre 4: The velocit distribtion is plotted with the coordinates for different vales of magnetic field 1.5 R e 1 R e 2 R e R e 0.1 R e R e Figre 5: The velocit distribtion is plotted with the time t for different vales of Renolds nmber t Figre 2: The velocit distribtion is plotted with the coordinates for different vales of Renolds nmber R c R c 0.7 R c t R c 10 R c 20 R c 30 Figre 6: The velocit distribtion is plotted with the coordinates for different vales of cople stress 0.2 M 1 M 3 M Figre 3: The velocit distribtion is plotted with the coordinates for different vales of cople stress Figre 7: The velocit distribtion is plotted with the time t for different vales of magnetic field 1737 International Jornal of Crrent Engineering and Technolog, Vol.6, No.5 (Oct 2016)

5 Homotop Pertrbation Method for Solving Linear Bondar Vale Problems Conclsion In this work, we sed homotop pertrbation method for solving linear differential eqations. The reslts have been approved the efficienc of this method for solving this problem. Frthermore, accrate soltions were derived from first-order approximations in the examples presented in this paper. To clarif the inflences of the varios s of the problem at hand, a nmerical calclating is made. The conclding remarks ma be listed as follows. 1) The velocit distribtion of the flid has the same behavior with and Renolds nmber. It increases with the increase of and. bt the opposite with. 2) The reslts which obtained in the stead state for bondar conditions agreed with both of Das U. and Adhikar S. and Misra J. where the velocit distribtion of the flid in the stead state has the same behavior with magnetic field. Reference He J. (2006) Homotop pertrbation method for solving bondar vale problems Phsics Letters A. Vol. 350, pp He J. (1999) Homotop pertrbation techniqe Compt. Math. Appl. Mech. Eng. Vol. 178, pp He J. (2004) Comparison of homotop method and homotop analsis method. Appl. Math. Compt. Vol.156, pp Demir1 A., Erman1 S., Özgür1 B. and Korkmaz E. (2013) Analsis of the new homotop pertrbation method for linear and nonlinear Bondar Vale Problems, Vol.61, pp Khati R. Desai1 V., Pradhan H. (2013) Soltion b Homotop Pertrbation Method of Linear and Nonlinear Diffsion Eqation Vol.3, pp Ganjefar S. and Rezaei S. (2016) Modified homotop pertrbation method for optimal control problems sing the Padé approximant Applied Mathematical Vol. 40, pp Babolian E., Azizi A., Saeidian J. (2009) Some notes on sing the homotop pertrbation method for solving timedependent differential eqations Mathematical and Compter, Vol. 50, pp Patra A. and Saha S. (2014) Homotop pertrbation smd transform method for solving convective radial fins with temperatre-dependent thermal condctivit of fractional order energ balance eqation International Jornal of Heat and Mass Transfer, Vol. 76, pp Grsa K. and Maciag A. (2016) Temperatre dependent thermal condctivit determination and sorce identification for nonlinear heat condction b means of the Trefftz and homotop pertrbation methods International Jornal of Heat and Mass Transfer, Vol. 100, pp Tsai C. (2012) Homotop method of fndamental soltions for solving certain nonlinear partial differential eqations Engineering Analsis with Bondar Elements, Vol. 36, pp Das U. (2013 Viscoelastic Effects on Unstead Two- Dimensional and Mass Transfer of a Viscoelastic Flid in a Poros Channel with Radiative Heat Transfer. Jornal of Engineering, Vol. 5, pp Adhikar S. and Misra J. (2011) nstead two-dimensional hdromagnetic flow and heat transfer of a flid Appl. Math. and Mech. Vol. 7, pp Int. J. of 1738 International Jornal of Crrent Engineering and Technolog, Vol.6, No.5 (Oct 2016)