Assessment of an Analytical Approach in Solving Two Strongly Boundary Value Problems

Transcription

1 Iteratioal Joural of Sciece ad Egieerig Ivestigatios vol., issue, Jauar ISSN: Assessmet of a Aaltical Approach i Solvig Two Strogl Boudar Value Problems Pema Nikaee, D.D.Gaji, S.E.Ghasemi, Hadi Hasapour 4, Gh.R.Mehdizade Ahagar 5,, Departmet of Mechaical Egieerig, Noshirvai Uiversit of Techolog, Babol, Ira. 4 Departmet of Mechaical Egieerig, Sharif Uiversit of Techolog, Tehra, Ira. 5 Departmet of Mechaical Egieerig, Yazd Uiversit, Yazd, Ira. ( pema.ikaee@gmail.com, 4 hhadi68@ahoo.com) Abstract- I this research, a powerful aaltical method called Recostructio of Variatioal Method (RVIM) is itroduced to hadle two boudar value problems. Oe is a parameterized sith order boudar value problem ad the other is a oliear boudar value problem arisig i the stud of thi film flow of a third grade fluid dow a iclied plae. With similarit method, the goverig equatios ca be reduced to a sstem of oliear ordiar differetial equatios. The effectiveess of the method, which is idepedet of the small parameter, is ivestigated b comparig the results obtaied with the umerical oes (4th order Rug-kutta method) ad the eact oes. For the secod problem the velocit profile is plotted ad the effect of varig the material costat o the velocit profile is studied. Kewords- Rotatig disk; Codesatio film; Aaltical approach; Recostructio of Variatioal Method, RVIM I. INTRODUCTION Noliear problems ad pheomea pla a importat role i applied mathematics, phsics, egieerig ad other braches of sciece. Ecept for a limited umber of these problems, most of them do ot have precise aaltical solutios; therefore, these oliear equatios should be solved usig approimatio methods. The perturbatio method is oe well-kow method to solve oliear equatios. But sice, usig the commo perturbatio method is based o the eistece of a small parameter, developig the method for differet applicatio is difficult. Therefore, ma differet ew techiques have bee recetl itroduced to elimiate the small parameter, such as the Adomia decompositio method [-], the Homotop aalsis method [4-6], the Homotop perturbatio method [7- ]. I this letter we emplo a ew ad effective aaltical method amed Recostructio of Variatioal Iteratio method (RVIM) to solve two strogl boudar value problems. B applig Laplace Trasform, RVIM overcomes the difficult of the perturbatio techiques ad other variatioal methods i case of usig small parameters ad Lagrage multipliers, respectivel. Reducig the size of calculatios ad omittig the difficult arisig i calculatio of oliear itricatel terms are other advatages of this method. Besides, it provides us with a simple wa to esure the covergece of solutio series, so that we ca alwas get accurate eough approimatios eve i first orders of the result iteratio. II. RVIM METHOD AND CONVOLUTION THEOREM I this sectio, a alterative method for fidig the optimal value of the Lagrage multiplier b the use of the Laplace trasform will be itroduced [ -]. Suppose is the idepedet variable; applig Laplace trasform to u(, t) with respect to as variable, ( ) ( ); st u(, t ); s e u (, t ) dt U s u s () We ofte come across fuctios which are ot the trasform of kow fuctios. But, b meas of the covolutio theorem, we ca take the iverse Laplace trasform. The covolutio of u ( ) ad v ( ) is writte as u( )* v ( ). It is defied as the itegral of the fuctios after oe is reversed ad shifted. If U( s) ad V ( s ) are the Laplace trasforms of u ( ) ad v( ), respectivel. The U ( s)* V ( s) is the Laplace Trasform of iverse Laplace Trasform as below, u ( ) v ( ) d U ( s ) V ( s ) u( ) v ( ) d () so b takig To illustrate the cocept of the RVIM, we cosider the followig geeral differetial equatio L( u( )) N ( u( )) f ( ) () 6

2 Where L ad N are liear ad oliear operators respectivel ad f ( ) is the forcig term. To facilitate our discussio of RVIM, with itroducig the ew fuctio h( u( )) f ( ) N ( u( )) ad cosiderig the ew equatio, Eq. ca be rewritte as, L( u( )) h( u, ) (5) Now, for implemetatio the RVIM techique based o ew idea of Laplace trasform, we appl Laplace Trasform o both sides of the Eq. (5). Itroducig all artificial iitial coditios as zero for the mai problem, the left side of the equatio after trasformatio will be L( u( )) U ( s) P( s) (6) Where Ps ( ) is polomial with the highest order derivative of the selected liear operator L( u( )) U ( s) P( s) h( u, ) (7) h( u, ) U( s) (8) Ps ( ) Suppose that Ds ( ) Ps ( ) ad h( u, ) H ( s ). Usig the covolutio theorem ; U ( s) D( s) H ( s) d ( )* h( u, ) (9) Takig the iverse Laplace trasform o both sides of Eq. (9) u( ) d ( ) h( u, ) d () Thus the followig recostructed method of variatioal iteratio formula ca be obtaied u ( ) u( ) d ( ) h( u, ) d () III. A PARAMETERIZED SIXTH ORDER BOUNDARY VALUE A. Goverig Equatio Cosider the followig problem (6) u ( ) ( c) u ( ) cu( ) c,, () With boudar coditios: u(), u(), u(), 7 u() sih(), u() cosh(), u() sih(). 6 () The eact solutio of this problem is: u( ) sih( ). 6 We see the eact solutio of this problem does ot deped o the parameter c but the problem itself does. This ca be viewed b rewritig Eq.() as (6) { u ( ) u ( )} c{ u ( ) u( ) }, (5) Which shows that, o matter what the value of c is, a solutio of fourth order problem is also a solutio of the sith-order problem. B. Implemetatio of RVIM Cosiderig the iitial approimatios for u as follow 5 4 u ( ), a b d (6) Rewritig equatios (), based o selective liear operator (6) u ( ) ( c) u ( ) cu( ) c, (7) Now Laplace trasform is implemeted with respect to idepedet variable o both sides of Eqs. (7). Usig the ew artificial boudar coditios (which all of them are zero) 6 s U( s) L ( c) u ( ) cu( ) c, (8) B usig the Laplace iverse Trasform ad covolutio theorem, it is cocluded that 5 ( ) u( ) ( c) u ( ) cu( ) c d (9) Hece, we arrive at the followig iterative formula for the approimate solutio of (), subject to the boudar coditios (), u ( ) 5 ( ) u( ) ( c) u ( ) ( ) cu c d () Accordig to above equatios, for first order approimatio : ( ) u 5 ( ) ( ) ( ) ( ) ( ) u c u cu c d () With substitutig boudar coditios () i the iterative formula () the ukow costats of the iitial approimatio (6) will be determied ad with pittig them i the result formula the fial aswer is approached. More the order of the iteratio growth more the accurac of the solutio icreases. We calculated the first order of approimatio for c : u ( ) () I Table, results obtaied from the first order of RVIM are compared with the eact results (4 th order Rug-Kutta method) ad the Error value is compared with the Error of two other approimate methods called HAM ad OHAM. It is revealed that a good accurac to the eact results is achieved ad RVIM is more accurate ad rapid tha two other methods, i covergig to the eact results. Iteratioal Joural of Sciece ad Egieerig Ivestigatios, Volume, Issue, Jauar 7 ISSN: Paper ID: -

3 TABLE. COMPARISON BETWEEN RVIM RESULTS AND NUMERICAL RESULTS AND HAM AND OHAM RESULTS FOR C= Numerical Results th order RVIM Errors ( Num. Obtaied Results ) HAM 4 [] OHAM [4] RVIM E-6.E-5.4E E-4.E-6 5.6E E-4.4E-5.E E-4 5.E-6.4E E-4 4.E-5.E E-4 5.7E-5 5.4E E-4 4.9E-5 5.7E E-5 4.5E-5.9E E-6.4E E E-6.E-5.4E-5 IV. A. Goverig Equatios THIN FILM FLOW PROBLEM The thi film flow of a third grade fluid dow a iclied plae of icliatio is govered b the followig oliear boudar value problem [5] d u 6 du d u g si () d d d du u(), at. d Itroducig the parameters * g si *, u u, (5) * g si The problem i Equatios () ad, after omittig asterisks, takes the followig form d u du d u 6, (6) d d d du u(), at, (7) d where is the damic viscosit, g is the gravit, is the fluid desit ad is the material costat of a third grade fluid. We ote that Equatio (6) is a secod order oliear ad ihomogeeous differetial equatio with two boudar coditios; therefore, it is a well-posed problem. Through itegratio of Equatio (6) C, (8) d d where C is a costat of itegratio. Emploig the secod coditio of (7) i Equatio (8), we obtai C. Thus, the sstem (6)-(7) ca be writte as, d d u() B. Implemetatio of RVIM (9) () I this sectio, we emplo RVIM to solve Equatio (9). The iitial guess is i the followig form: u ( ) a, () Rewritig equatios (9), based o selective liear operator, () d d Now Laplace trasform is implemeted with respect to idepedet variable o both sides of Eqs. (). Usig the ew artificial boudar coditios (which all of them are zero) du su s L, () d B usig the Laplace iverse Trasform ad covolutio theorem, it is cocluded that du u( ) d d Hece, we arrive at the followig iterative formula for the approimate solutio of (9), subject to the boudar coditio (), du u ( ) u( ) d d (5) Accordig to above equatios, for first order approimatio : du u( ) u( ) d d (6) With substitutig boudar coditio () i the iterative formula (5) the ukow costat of the iitial approimatio, a, will be determied ad with pittig it i the result formula the fial aswer is approached. We calculated the secod ad third order of approimatios whe.: u ( ).5 u ( ) (7) u ( ) (8) (9) The third ad fourth orders are calculated but ot metioed for brevit. All the results preseted i figures ad Iteratioal Joural of Sciece ad Egieerig Ivestigatios, Volume, Issue, Jauar 8 ISSN: Paper ID: -

4 table i the et sectio are obtaied from the fourth order of RVIM solutio. C. Results ad Discussio The effectiveess of Recostructio of Variatioal Method (RVIM) is depicted i Table ad Figs ad. The results are well matched with the results carried out b umerical solutio (Ruge Kutta). I Table, error is itroduced as follow: Error u( ) u( ) NM RVIM Errors of first four orders of the approimatio are preseted i Table. It is obvious that the solutio rapidl coverges to the eact results ad just i the 4 th order of the iteratio we got efficiet accurac. Figs. ad describes the compariso betwee the RVIM ad umerical results for β =.5 ad β =.. A ecellet agreemet is observed. This accurac gives high cofidece i the validit of this problem, ad reveals a ecellet agreemet i egieerig accurac. Fig..Compariso betwee RIVM ad Numerical results for. Fig..Compariso betwee RIVM ad Numerical results for.5 Fig..Cotour of u() for differet values of material costat from to TABLE. COMPARISON BETWEEN NUMERICAL RESULTS AND FIRST 4 ORDERS OF RVIM SOLUTION Result Error Numerical 4 th order RVIM th order d order rd order 4th Order E-5.857E E E E E E E E E E-5 Iteratioal Joural of Sciece ad Egieerig Ivestigatios, Volume, Issue, Jauar 9 ISSN: Paper ID: -

5 I figure the effects of varig β have bee ivestigated. As it is show b cotour i this figure the cocetratio of u() occurs i = for small amout of material costat. It is illustrated that with icreasig β the amout of u() decreases geerall. I low values of these chages are ot sesible, while i greater values, the chages of u() are more oticeable. V. CONCLUSION I this paper, a strog aaltical method called Recostructio of Variatioal Iteratio Method (RVIM) has bee successfull applied to fid eplicit solutios of two boudar value problems, which ma occur i differet fields of sciece ad egieerig. Both of illustratig eamples cofirm the coveiece, reliabilit ad efficiec of this method. RVIM ca be itroduced to overcome the limitatios ad difficulties eistig i other approimate method, e.g. large computatio eed, use of small parameters, covergece i high orders of approimatios. It is predicted that RVIM ca be used widel i mathematical, phsical ad egieerig problems, due to its simplicit ad efficiec. [] E.Hesameddii, H.Latifizadeh, Recostructio of Variatioal Iteratio Algorithms usig the Laplace Trasform, It. J. Noliear Sci. Numerical Simulatio (9)77-8 [] Siraj-Ul-Islam, Sirajul-Haq ad Javed Ali, Numerical solutio of special th-order boudar value problems usig differetial trasform method, Comm.Nol.Sc.Nu.Sim., 4, 4, pp. -8, 9. [4] Javed ALI, Saeed ISLAM, Hamid KHAN, Gul ZAMAN, The solutio of a parameterized sith order boudar value problem b the optimal asmptotic method, Proceedigs of the Romaia Academ S.A, Vol., pp [5] A. M. Siddiqui, R.Mahmood ad Q. K. Ghori, Homotop Perturbatio Method for Thi Film Flow of a Third Grade Fluid Dow a Iclied Plae, Chaos, Solitos& Fractals, Vol. 5, No., 8, pp Pema Nikaee received his B.S. degree from the Departmet of Mechaical Egieerig at Babol Uiversit of Techolog, Ira, i, where he is ow a researcher. His research iterests are heat ad mass trasfer i fluid flows. He is also workig o applicatios of Noliear Sciece i Mechaical Egieerig. REFERENCES [] G. Adomia, Noliear dissipative wave equatios, Appl. Math. Lett. (998) 5_6. [] G. Adomia, R. Rach, Modified decompositio solutio of liear ad oliear boudar-value problems. Noliear Aalsis: Theor, Methods & Applicatios Volume Issue 5, Sept. 994 Pages [] Abdul-Majid Wazwaz, A compariso betwee the Adomia decompositio method ad the Talor series method i the series solutios, Applied Mathematics ad Computatio, 97(998), [4] Abdul-Majid Wazwaz, The variatioal iteratio method (VIM) for solvig liear ad oliear Volterra itegral ad itegro differetial equatios, Iteratioal Joural of Com. Math., 87(5) () - 4. [5] D.D.Gaji, Hafez Tari, H.Babazadeh, The applicatio of He s variatioal iteratio method to oliear equatios arisig i heat trasfer.phsics Letters A 6 ()(7). [6] M.Rafei, D.D.Gaji, H.Daiali,H.Pashaei,The variatioal iteratio method for oliear oscillators with discotiuities.joural of Soudad Vibratio 5 (7) 64. [7] A. Sadighi, D.D. Gaji, Aaltic treatmet of liear ad oliear Schrödiger equatios: A stud with Homotop perturbatio ad Adomia decompositio methods. Phs. Lett. A 7 (8) 465. [8] S.H. HoseiNia, H. Soltai, J. Ghasemi, A.N. Rajbar, D.D. Gaji, maitaiig the stabilit of oliear differetial equatios b the ehacemet of HPM. Phs.Lett. A 7 (8) [9] M. Esmaeilpour ad D. D. Gaji, Applicatio of He s Homotop perturbatio method to boudar laer flow ad covectio heat trasfer over a flat plate, Phsics Letters A, vol. 7, o., pp. 8,7. [] D.D. Gaji, G.A. Afrouzi, H. Hosseizadeh, R.A. Talarposhti, Fourth order Volterraitegro-differetial equatios usig modified Homotop perturbatio method. Phs. Lett. A 7 (7). [] A.A. Imai, D.D. Gaji, Houma B. Roki, H. Latifizadeh, E.Hesameddii, M. HadiRafiee. Approimate travelig wave solutio for shallow water wave equatio. Applied Mathematical Modelig 6 () Davood Domiri Gaji (D.D.Gaji) received his Ph.D. degree i Mechaical Egieerig from Tarbiat Modares Uiversit, Ira, ad is curretl Associate Professor i the Mechaical Egieerig Departmet of Babol Noshirvai Uiversit of Techolog, Ira. His research iterests are oliear sciece i egieerig, CFD, FDM, FVM, iverse problems ad spra modelig. He is Editor-i-chief of Iteratioal Joural of Noliear Damics i Egieerig ad Scieces (IJNDES). Iteratioal Joural of Sciece ad Egieerig Ivestigatios, Volume, Issue, Jauar ISSN: Paper ID: -