An Efficient Technique in Finding the Exact Solutions for Cauchy Problems

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for Cauchy Problems Habibolla Laifizadeh, Shiraz Universiy of

1 Shiraz Universiy of Technology From he SelecedWorks of Habibolla Laifizadeh 1 An Efficien Technique in Finding he Eac Soluions for Cauchy Problems Habibolla Laifizadeh, Shiraz Universiy of Technology Available a: hps://works.bepress.com/habib_laifizadeh/155/

1. M. MIGHANI,. S. SAEEDI AN EFFICIENT TECHNIQUE IN FINDING THE EXACT SOLUTIONS FOR CAUCHY PROBLEMS 1. DEPARTMENT OF ARCHITECTURAL ENGINEERING, SHOMAL UNIVERSITY, AMOL, IRAN.

2 1. M. MIGHANI,. S. SAEEDI AN EFFICIENT TECHNIQUE IN FINDING THE EXACT SOLUTIONS FOR CAUCHY PROBLEMS 1. DEPARTMENT OF ARCHITECTURAL ENGINEERING, SHOMAL UNIVERSITY, AMOL, IRAN. DEPARTMENT OF ARCHITECTURAL ENGINEERING, SHAHID BEHESHTI UNIVERSITY, TEHRAN, IRAN ABSTRACT: Nonlinear phenomena play a crucial role in applied mahemaics and engineering. In his paper, he Reconsrucion of Variaional Ieraion Mehod (RVIM is used o solve he Cauchy problem. The new applied algorihm is a powerful and efficien echnique in finding soluions for he linear and nonlinear equaions using only few erms. The mehod used gives rapidly convergen successive approimaions. As saed before, we aim o obain analyical soluions o problems. We also aim o confirm ha he reconsrucion of variaional ieraion mehod is powerful, efficien, and promising in handling scienific and engineering problems. The Reconsrucion of Variaional Ieraion Mehod (RVIM echnique is independen of any small parameers a all. Besides, i provides us wih a simple way o ensure he convergence of soluion series, so ha we can always ge accurae enough approimaions. Some eamples are given o elucidae he soluion procedure and reliabiliy of he obained resuls. KEYWORDS: Reconsrucion of Variaional Ieraion Mehod, Cauchy problem, Inviscid Burgers' equaion, Transpor equaion INTRODUCTION Nonlinear phenomena play a crucial role in applied mahemaics and engineering. The heory of nonlinear problems has recenly undergone many sudies. Various approimaion mehods have been used for comple equaions. In recen years, several such echniques have drawn special aenion, such as Hiroa's bilinear mehod [1], he homogeneous balance mehod [-], inverse scaering mehod [4], Adomian's decomposiion mehod ADM [5,6], he variaional ieraion mehod [7,8] he - epansion mehod [9], as well as homoopy analysis mehod (HAM. Afer ha, many ypes of nonlinear problems were solved by he HAM by ohers [1]. One of he newes approimaion mehods is Reconsrucion of variaional ieraion mehod [11-15]. This new applied algorihm is a powerful and efficien echnique in finding he approimae soluions for he linear and nonlinear equaions. RVIM as a mehod ha based on Laplace ransform has high rapid convergence and reduces he size of calculaions using only few erms, so many linear and nonlinear equaions can be solved by his mehod. The mehod used gives rapidly convergen successive approimaions. As saed before, we aim o obain analyical soluions o problems. We also aim o confirm ha he reconsrucion of variaional ieraion mehod is powerful, efficien, and promising in handling scienific and engineering problems. The RVIM echnique is independen of any small parameers a all. Besides, i provides us wih a simple way o ensure he convergence of soluion series, so ha we can always ge accurae enough approimaions. The RVIM echnique has been successfully applied o many nonlinear problems in science and engineering. All of hese verify he grea poenial and validiy of he RVIM echnique for srongly nonlinear problems in science and engineering. In his paper, we use he mehod o discuss he firs-order parial differenial equaion in he form [16]: u (, + a(, u (, = ψ (, R, > (1 u(, = ϕ (, R ( When a (, = a is a consan and ψ ( = Eq. (1 is a linear equaion called he ranspor equaion which can describe many ineresing phenomena such as he spread of AIDS, he moving of wind. When a (, = u(, he equaion is called he inviscid Burgers' equaion arising in onedimensional sream of paricles or fluid having zero viscosiy. copyrigh FACULTY of ENGINEERING HUNEDOARA, ROMANIA 7

3 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inernaional Journal Of Engineering DESCRIPTION OF THE NEW METHOD To clarify he basic ideas of our proposed mehod in [19], we consider he following differenial equaion same as VIM based on Lagrange muliplier []: Lu( 1, L,k + Nu( 1, L,k = f ( 1, L,k ( By suppose ha Lu( L k = = 1,, k L i iu( i where L is a linear operaor, N a nonlinear operaor and f an inhomogeneous erm. We can rewrie equaion ( down a correcion funcional as follows: Lj u( j = f ( 1, L,k Nu ( 1, L,k i = Liu( i (5 i j h(( 1, L,k,u( 1, L,k herefore L j u( j = h(( 1, L, k,u( 1, L, k (6 Wih arificial iniial condiions being zero regarding he independen variable j. By aking Laplace ransform of boh sides of he equaion (6 in he usual way and using he arificial iniial condiions, we obain he resul as follows P( s. U( 1, L,i 1,s,i+ = H((1, L,i 1,s,i+,u (7 where P(s is a polynomial wih he degree of he highes derivaive in equaion (7, (he same as he highes order of he linear operaor L j. The following relaions are possible; l [h] = H (8-a k (4 1 B(s = (8-b P(s l [b( i ] = B(s (8-c Which ha in equaion (8-a he funcion H(( 1, L,i 1,s,i+ 1,k,u and h(( 1, L,i 1,i,i+ 1,k,u have been abbreviaed as respecively. Hence, rewrie he equaion (7 as; U(1, L,i 1,s,i+ = H((1, L, i 1,s,i+,u.B(s (9 Now, by applying he inverse Laplace Transform on boh sides of equaion (9 and by using he (8-a - (8-c, we have; i u(1, L,i-1, i,i + = h((1, L, i 1, τ,i +,u. b( i τ dτ (1 Now, we mus impose he acual iniial condiions o obain he soluion of he equaion (. Thus, we have he following ieraion formulaion: u n+ 1 ( 1, L, i-1, i, i+ 1, k = u ( 1, L, i-1, i, i+ 1, k i (11 + h(( 1, L, i 1, τ, i+ 1, k,u.b( i τ dτ where u is iniial soluion wih or wihou unknown parameers. Assuming u is he soluion of Lu wih iniial/boundary condiions of he main problem, In case of no unknown parameers, should saisfy iniial/ boundary condiions. When some unknown parameers are involved in u, he unknown parameers can be idenified by iniial/boundary condiions afer few ieraions, his echnology is very effecive in dealing wih boundary problems. I is worh menioning ha, in fac, he Lagrange muliplier in he He's variaional ieraion mehod is λ( τ = b( i τ as shown in [19]. The iniial values are usually used for selecing he zeroh approimaion. Wih u deermined, hen several approimaions u n n>, follow immediaely. Consequenly, he eac soluion may be obained by using u(1, L,i-1,i,i+ 1, k = limn un(1, L,i-1,i,i+ 1, k. (1 In wha follows, we will apply he RVIM mehod o homogeneous/non-homogeneous parabolic parial differenial equaions o illusrae he srengh of he mehod and o esablish eac soluions for hese problems. APPLICATION Since our focus is on he ideas and basic principles, we shall consider only he simples possible equaions o clearly illusrae he soluion procedure. In paricular, we will focus on pure Cauchy problems. These problems are iniial value problems. 8 Tome XI (Year 1. Fascicule. ISSN

4 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inernaional Journal Of Engineering Eample1. Consider he ranspor equaion [16-18]: u (, + au (, =, R, > (1 Subjec o he iniial condiions: u(, =, R (14 By selecion of auiliary linear operaor, so he Eq. (1 is rewrien as h(, ε,u 678 l{ u(, } = u = ( au (15 Now Laplace ransform is implemened wih respec o independen variable on boh sides of eq. (15 and by using he new arificial iniial condiion (which all of hem are zero we have s (, = l h(,,u (16 { } { h(,,u } l (, = s (17 and whereas Laplace inverse ransform of 1/s is as follows 1 1 l [ ] = 1 s (18 Therefore by using he Laplace inverse ransform and convoluion heorem i is concluded ha u(, = h(, ε,u dε (19 Therefore, by using he Eq. (1, (11 one can obain he following RVIM's ieraion formula in he -direcion, = u(, aun dε ( Now we sar wih an arbirary iniial approimaion u (, =, ha saisfies he iniial condiion and by using he RVIM ieraion formula (, we have he following successive approimaion, a a (, a a u (, a + a u = + (1 u = + ( = ( Whereas, he RVIM mehod admis he use of u(, = limn un(,, which gives he eac soluion u (, = a + a (4 Eample. Consider he nonlinear Cauchy problem [17] u (, + u (, =, R, > (5 Subjec o he iniial condiions: u(, =, R (6 Such as previous eamples, for implemenaion of he RVIM echnique, firs of all we need o choose he auiliary linear operaor as 678 l{ u(, } = u = ( u (7 Accordingly, afer aking Laplace Transform by using he arificial iniial condiion as in [19], on boh side of Eq. (5, he following RVIM ieraion formula in he -direcion can be obained h(, ε,u, = u(, un dε (8 By he RVIM'S recurren formula in Eq. (8, he erms of he sequence {u n } are consruced as follows, so ha we choose is iniial approimae soluion as u (, =. u, = (9 (, u = + ( 4 u (, = + + (1 Tome XI (Year 1. Fascicule. ISSN

5 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inernaional Journal Of Engineering The ne erms of {u n } can be deermined in a similar way and we can consruc he n h approimaion as u(, = e ( The approimaion obained by RVIM procedure converges o he eac soluion. Eample. Now we solve he following non-homogeneous Cauchy problem [17] u (, + u(, =, R, > ( u(, = e, R To apply RVIM o his equaion wih iniial and boundary condiions, according o (5 and (6, we have h(, ε,u l{ u(, } = u = ( u + (4 Therefore, as previous Eamples he RVIM ieraive formula can be epressed as:, = u(, un dε (5 Wih he aid of he iniial approimaion u (, = e and using he RVIM ieraion, we can obain direcly he res of he oher componens as follows u1 (, = e e + (6 (, e e + e u = + (7 6 u(, = e e + + e e 6 (8 The RVIM admis he use of u(, = limn un(,, which gives he eac soluion u(, = ( + e (9 Eample4. Consider he inviscid Burgers' equaion [17, 18] u (, + u(, u (, =, R, > (4 u(, =, R To implemenaion of he RVIM mehod o his differenial equaion wih iniial and boundary condiions, according o (5 and (6, he auiliary linear operaor is seleced as h(, ε,u 678 l{ u(, } = u = ( uu (41 Therefore, as previous Eamples he RVIM ieraive formula can be epressed as:, = u(, uun dε (4 We sar wih an iniial approimaion u (, = ; by he ieraion formula (4, we can obain he firs few componens as follows u 1 (, = (4 u(, = + ( u(, = (45 and he res of he componens are obained using he ieraion formula (41 and he soluion in closed form is given by. Recall ha u(, = limn un(,, Consequenly, he eac soluions u(, =. 1 (46 CONCLUSIONS There are wo main goals ha we aimed for his work. The firs is employing he powerful Reconsrucion of he variaional ieraion o invesigae Cauchy problems and he second one is showing he power of his mehod and is significan feaures. The wo goals are achieved. Tome XI (Year 1. Fascicule. ISSN

6 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inernaional Journal Of Engineering I is obvious ha he mehod gives rapidly convergen successive approimaions wihou any resricive assumpions or ransformaion ha may change he physical behavior of he problem. Reconsrucion of he variaional ieraion gives several successive approimaions hrough using he RVIM's ieraion relaion. Moreover, he RVIM reduces he size of calculaions by no requiring he edious Adomian polynomials, and hence he ieraion is direc and sraighforward. The RVIM uses he iniial values for selecing he zeroh approimaion, and boundary condiions, when given for bounded domains, can be used for jusificaion only. A clear conclusion can be drawn from he resuls ha Reconsrucion of he variaional ieraion mehod provides us an efficien ool o obaining eac soluions of parial differenial equaions REFERENCES [1.] X.B. Hu, Y.T. Wu, Applicaion of he Hiroa bilinear formalism o a new inegrable differenialdifference equaion, Phys. Le. A 46 ( [.] E. Fan, Two new applicaions of he homogeneous balance mehod, Phys. Le. A 56 ( [.] M. Wang, Y. Zhou, Z. Li, Applicaions of a homogeneous balance mehod o eac soluion of nonlinear equaions in mahemaical physics, Phys. Le. A 16 ( [4.] V.O. Vakhnenko, E.J. Parkes, A.J. Morrison, A Bäcklund ransformaion and he inverse scaering ransform mehod for he generalized Vakhnenko equaion, Chaos Solions Fracals 17 ( [5.] G. Adomian, Nonlinear dissipaive wave equaions, Appl. Mah. Le. 11 ( [6.] G. Adomian, R. Rach, Modified decomposiion soluion of linear and nonlinear boundary-value problems, Nonlinear Anal. ( [7.] M.T. Darvishi, F. Khani, A. A. Soliman, The numerical simulaion for siff sysems of ordinary differenial equaions, Compu. Mah. Appl. 54 ( [8.] A. Nikkar, M. Mighani, Applicaion of VIM for Solving Sevenh-Order Differenial Equaions American Journal ofcompuaional and Applied Mahemaics, (1 (1 7-4 [9.] A.V. Karmishin, A.I. Zhukov, V.G. Kolosov, Mehods of Dynamics Calculaion and Tesing for Thin-walled Srucures, Mashinosroyenie, Moscow, 199. [1.] Y. Khan, R. Taghipour, M. Fallahian, A. Nikkar, A new approach o modified regularized long wave equaion, Neural Compuing & Applicaions, (6 July 1, pp.1-7, doi. 1.17/s [11.] A. Nikkar, A new approach for solving Gas dynamics equaion. Aca Technica Corviniensis Bullein of Engineering, 1, 4: [1.] A. Nikkar, J. Vahidi, M. Jafarnejad Ghomi, M. Mighani, Reconsrucion of variaion Ieraion Mehod for Solving Fifh Order Caudrey-Dodd-Gibbon (CDG Equaion, Inernaional journal of Science and Engineering Invesigaions, 1, 6: [1.] A. Nikkar, Z. Mighani, S.M. Saghebian, S.B. Nojabaei and M. Daie, Developmen and validaion of an analyical mehod o he soluion of modelling he polluion of a sysem of lakes. Res. J. Appl. Sci. Eng. Technol., 5(1 ( [14.] S. Ghasempoor, J. Vahidi, A. Nikkar, M. Mighani, Analyical approach o some highly nonlinear equaions by means of he RVIM, Res. J. Appl. Sci. Eng. Technol., 5(1 ( [15.] A. A. Imani, D. D. Ganji, Houman B. Rokni, H. Laifizadeh, E. Hesameddini, M. Hadi Rafiee, Approimae raveling wave soluion for shallow waer wave equaion, Applied Mahemaical Modelling, 6(4 ( [16.] F. Khani, M. Ahmadzadeh Raji, H. Hamedi Nejad, Analyical soluions and efficiency of he nonlinear fin problem wih emperaure-dependen hermal conduciviy and hea ransfer coefficien, Commun. Nonlinear Sci. Numer. Simul. 14 ( [17.] A. Molabahrami, F. Khani, The homoopy analysis mehod o solve he Burgers-Huley equaion, Nonlinear Anal. RWA 1 ( ( [18.] M. Sajid, I. Ahmad, T. Haya, M. Ayub, Unseady flow and hea ransfer of a second grade fluid over a sreching shee, Commun. Nonlinear Sci. Numer. Simul. 14 (1 ( [19.] F. Khani, M. Ahmadzadeh Raji, S. Hamedi-Nezhad, A series soluion of he fin problem wih a emperaure-dependen hermal conduciviy, Commun. Nonlinear Sci. Numer. Simul. 14 (9 7_17. [.] F. Khani, A. Farmany, M. Ahmadzadeh Raji, A. Aziz, F. Samadi, Analyic soluion for hea ransfer of a hird grade viscoelasic fluid in non-darcy porous media wih hermophysical effecs, Commun. Nonlinear Sci. Numer. Simul. (9 doi:1.116/j.cnsns [1.] F. Khani, A. Aziz, Thermal analysis of a longiudinal rapezoidal fin wih emperaure dependen hermal conduciviy and hea ransfer coefficien, Commun. Nonlinear Sci. Numer. Simul (9, doi: Tome XI (Year 1. Fascicule. ISSN

ANNALS OF FACULTY ENGINEERING HUNEDOARA Inernaional Journal Of Engineering [.] R.C. McOwen, Parial Differenial Equaions: Mehods and Applicaions, Prenice Hall, Inc., 1996. [.] A. Tveio, R.

, 5. [5.] E. Hesameddini, H. Laifizadeh, Reconsrucion of Variaional Ieraion Algorihms using he Laplace Transform, In. J. Nonlinear Sci. Numer. Simul. 1 (9177-18. [6.] A.M.

7 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inernaional Journal Of Engineering [.] R.C. McOwen, Parial Differenial Equaions: Mehods and Applicaions, Prenice Hall, Inc., [.] A. Tveio, R. Winher, Inroducion o Parial Differenial Equaions, Springer-Verlag, Berlin, Heidelberg, 5. [4.] N.H. Asmar, Parial Differenial Equaions wih Fourier Series and Boundary Value Problems, Prenice Hall, Inc., 5. [5.] E. Hesameddini, H. Laifizadeh, Reconsrucion of Variaional Ieraion Algorihms using he Laplace Transform, In. J. Nonlinear Sci. Numer. Simul. 1 ( [6.] A.M. Wazwaz, The variaional ieraion mehod: A powerful scheme for handling Linear and nonlinear diffusion equaions, Compuers and Mahemaics wih Applicaions, ( ANNALS of Faculy Engineering Hunedoara Inernaional Journal of Engineering copyrigh UNIVERSITY POLITEHNICA TIMISOARA, FACULTY OF ENGINEERING HUNEDOARA, 5, REVOLUTIEI, 118, HUNEDOARA, ROMANIA hp://annals.fih.up.ro Tome XI (Year 1. Fascicule. ISSN

AN APPROXIMATION SOLUTION OF THE 3-D HEAT LIKE EQUATION

Shiraz Universiy of Technology From he SelecedWorks of Habibolla Laifizadeh 13 AN APPROXIMATION SOLUTION OF THE 3-D HEAT LIKE EQUATION Habibolla Laifizadeh, Shiraz Universiy of Technology Available a: