Department of Mechanical Engineering, Salmas Branch, Islamic Azad University, Salmas, Iran

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1 Inernaional Parial Differenial Equaions Volume 4, Aricle ID 6759, 6 pages hp://dx.doi.org/.55/4/6759 Research Aricle Improvemen of he Modified Decomposiion Mehod for Handling Third-Order Singular Nonlinear Parial Differenial Equaions wih Applicaions in Physics Nema Dalir Deparmen of Mechanical Engineering, Salmas Branch, Islamic Azad Universiy, Salmas, Iran Correspondence should be addressed o Nema Dalir; dalir@au.ac.ir Received 4 June 4; Acceped 9 Ocober 4; Published 6 November 4 Academic Edior: Ahanasios N. Yannacopoulos Copyrigh 4 Nema Dalir. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. The modified decomposiion mehod (MDM) is improved by inroducing new inverse differenial operaors o adap he MDM for handling hird-order singular nonlinear parial differenial equaions (PDEs) arising in physics and mechanics. A few casesudy singular nonlinear iniial-value problems (IVPs) of hird-order PDEs are presened and solved by he improved modified decomposiion mehod (IMDM). The soluions are compared wih he exising exac analyical soluions. The comparisons show ha he IMDM is effecively capable of obaining he exac soluions of he hird-order singular nonlinear IVPs.. Inroducion Singular nonlinear PDEs appear in many cases in physics and mechanics. Examples of singular nonlinear PDEs in physics include cylindrical and spherical KdV equaions, Erns equaion, Clairau s equaion, Harree equaion, Yamabe problem, Zakharov-Schulman sysem, Cauchy momenum equaion, and reacion-diffusion equaions [ 3]. Examples of singular nonlinear PDEs in mechanics are equaion of moion of a poin mass in a cenral force field, generalized equaion of seady ransonic gas flow, cylindrical and spherical Navier- Sokes equaions, and cylindrical and spherical fluid hydrodynamic insabiliy equaions [4, 5]. However, despie such imporance in various fields of science and engineering, singular nonlinear PDEs are difficul o solve. In recen years, semianalyical mehods such as he Adomian decomposiion mehod (ADM) and he modified decomposiion mehod (MDM) have gained significance in solving many problems in physics and mahemaics. ODEs and PDEs of various ypes have been solved by he MDM, such ha singular and nonsingular nonlinear ODEs and also nonsingular nonlinear PDEs have been solved by he MDM [6 ]. Neverheless, lieraure survey makes i clear ha he MDM is never aemped on solving he singular nonlinear PDEs. Wazwaz [] and Wazwaz and Mehanna [] used he combined Laplace ransform-adm for handling nonlinear Volerra inegrodifferenial equaions and singular inegral equaion of hea ransfer, respecively. Sivakumar and Baiju [3] used a shooing ype Laplace-Adomian decomposiion algorihm for solving nonlinear differenial equaions wih boundary condiions a infiniy. Noghrehabadi e al. [4] combined Pade approximans and he ADM o sudy he deflecion and pull-in insabiliy of nanocanilever elecromechanical swiches. Duan and Rach [5] developednew numerical modified Adomian decomposiion algorihms by using he Wazwaz-El-Sayed modified decomposiion recursion scheme and invesigaed heir pracicaliy and efficiency for several nonlinear examples. Lin e al. [6], based on he new definiion of he Adomian polynomials, he ADM, and he Pade approximans echnique, proposed a new algorihm o consruc analyical approximae soluions for nonlinear fracional differenial equaions wih iniial or boundary condiions. Kermani and Dehesani [7] applied headmosolvehenonlinearequaionsfromhedmodel for a nanosized oscillaor. Song and Wang [8] inroduced a convergence-conrol parameer ino sandard ADM and esablished a new ieraive formula. In he presen sudy, he goal is o develop new inverse differenial operaors incorporaed ino he MDM o solve he hird-order singular nonlinear PDEs. Thus he improved

2 Inernaional Parial Differenial Equaions modified decomposiion mehod (IMDM), which is he MDM in conjuncion wih he new developed inverse differenial operaors, is used o solve he singular nonlinear IVPs in he hird-order PDEs. The resuls of he IMDM soluions agree wih he exising exac soluions of IVPs.. IMDM for Third-Order Singular Nonlinear PDEs.. Third-Order Singular Nonlinear PDEs. We consider he hird-order, in, singular nonlinear PDE as follows: 3 n =F(x,u, x, ), () where and x are independen variables, u is he dependen variable, F is a nonlinear funcion of x, u, u x, u xx,andu,and n is a real consan, n>. The iniial condiions are as follows: u (x, ) =f(x), u (x, ) =g(x), u (x, ) =h(x). By defining he linear differenial operaor L ( ) = 3 ( ) / 3 (n/) ( ( )/ ),helefhandsideof() becomes as L u= / 3 (n/) (/ ).Then,() canberewrienas () L u=f(x,u, x, ). (3) The inverse of operaor L,hais,, is defined such ha {L (u)} = u(x, ) u(x, ) u (x, ) in he following form: ( ) n n ( ) d d d. (4) I can be shown in he following manner ha he applicaion of on / 3 (n/) (/ ) gives u(x, ) u(x, ) u (x, ): ( 3 u 3 n ) n n n n ( 3 u 3 n )ddd ( n 3 nn )ddd (n )ddd u n (n ) d d ( ( = (u) n (n )dd ( )dd= )d )d (x,) ( ) ( )d (x,) =u(x, ) u(x, ) u (x, ). (x,) (5) The inverse differenial operaor of (4), proposed in he presen work, inroduces he improved modified decomposiion mehod (IMDM) for solving hird-order singular nonlinear PDEs. Applying he inverse differenial operaor of (4) o (3) and use of he boundary condiions of () gives: u (x, ) =f(x) g(x) (F (x, u, x, )). (6) The ADM and MDM sae ha he dependen variable u(x, ) and he nonlinear erms F in (6) should be subsiued wih he following infinie series [7]: u (x, ) = u m (x, ), m= d F(x,u, x, )= A m (x, ), m= where A m s,calledheadomianpolynomials,aredefinedas [8] A m = [ d m m m! dλ m F ( λ i u i )] i= λ= Subsiuionof infinieseries of (7) in (6) resuls in (7). (8) u m (x, ) =f(x) g(x) ( A m (x, )). (9) m= m= Due o he ADM, all erms of u(x, ) excep u (x, ) are deermined by a recursive relaion, as follows [9]: u (x, ) =f(x) g(x), u (x, ) = (A (x, )), u m (x, ) = (A m (x, ), m. () In MDM, a sligh modificaion is applied o he ADM o enhance he convergence behavior of decomposiion mehod, such ha f(x) g(x) is spli ino wo pars; he firs par,

3 Inernaional Parial Differenial Equaions 3 f(x), iswrienwihu (x, ), andhesecondpar,g(x), is wrien wih u (x, ) as follows []: u (x, ) =f(x), u (x, ) =g(x) (A (x, )), u m (x, ) = (A m (x, ), m. ().. General Third-Order Singular Nonlinear PDEs. We ake ino accoun he general hird-order (in )singular nonlinear PDE as follows: 3 n n (n ) =F(x,u, x, ), () where and x are independen variables, u is he dependen variable, F is a nonlinear funcion of x, u, u x, u xx,andu,and n is a real consan, n>. The iniial condiions are as follows: u (x, ) =f(x), u (x, ) =g(x), u (x, ) =h(x). (3) By defining he linear differenial operaor L ( ) = 3 ( ) / 3 (n/) ( ( ) / )(n(n )/ ) ( ( )/),helefhandsideof () is rewrien as L u=/ 3 (n/) (/ ) (n(n )/ ) (/).Thus,() becomes as L u=f(x,u, x, ). (4) The is defined in he following form such ha {L (u)} = u(x, ) u(x, ): ( ) n n ( ) d d d. (5) I can be shown in he following manner ha {L (u)} = u(x, ) u(x, ): ( 3 u 3 n n n n (n ) n ( 3 u 3 n ) ( n 3 nn n (n ) n(n ) n )ddd n (n nn )ddd n ( n nn ) d d )ddd n n ( n nn )dd (n )dd n (n ) d n (n )d ( )d=(u) =u(x, ) u(x, ). (6) The inverse differenial operaor of (5), defined in he presen work, can be used in conjuncion wih he MDM o solve he general hird-order singular nonlinear PDEs. Applying he inverse operaor of (5)o he differenial equaion (4) resuls in u (x, ) =f(x) (F (x, u, x, )). (7) Subsiuing he infinie series of (7) in (7),dueoheMDM, gives he following form [9]: u m (x, ) =f(x) ( A m (x, )), (8) m= m= where he Adomian polynomials A m s are defined in (8).The MDM form of (8) can be wrien as [] u (x, ) =f(x), u (x, ) = (A (x, )), u m (x, ) = (A m (x, ), m. (9).3. General Complee Third-Order Singular Nonlinear PDEs. We consider he general complee hird-order, in, singular nonlinear PDE as follows: 3 3n 3n (n ) =F(x,u, x, ), n (n )(n ) 3 u () where and x are independen variables, u is he dependen variable, F is a nonlinear funcion of x, u, u x, u xx,andu,and n is a real consan, n>.theiniialcondiionsareas u (x, ) =f(x), u (x, ) =g(x), () u (x, ) =h(x). By defining he linear operaor L ( ) = 3 ( ) / 3 (3n/) ( ( )/ ) (3n(n )/ ) ( ( )/) (n(n )(n )/ 3 )( ), he lef hand side of () is rewrien as L u = / 3 (3n/) (/ ) (3n(n )/ ) (/) (n(n )(n )/ 3 )u,suchha() becomes as L u=f(x,u, x, ). ()

4 4 Inernaional Parial Differenial Equaions The inverse operaor of L,hais,,isdefinedinhe following form such ha {L (u)} = u: ( ) = n n ( ) d d d. (3) The credibiliy of definiion of he inverse operaor, (3), is shown in he following manner: ( 3 u 3 3n 3n (n ) n (n )(n ) 3 u) = n n ( 3 u 3 3n = n 3n (n ) n (n )(n ) 3 u) d d d ( n 3 3nn 3n(n ) n n(n )(n ) n 3 u) d d d = n (n nn = n n(n ) n u) d d d ( n nn n(n ) n u) d d = n (n nn u) d d = n ( n nn u) d = n ( n nn u) d = n (n u)d = n (n u) = n (n u)= u (x, ). (4) The inverse operaor of (3), which is developed in he presen work, improves he MDM for solving he hird-order singular nonlinear PDEs. Applicaion of he inverse operaor of (3) on () resuls in u (x, ) = (F (x, u, x, )). (5) Subsiuion of he infinie series of (7) in (5) gives he following form [9]: u m (x, ) = ( A m (x, )), (6) m= m= where A m s are defined in (8). TheMDMfor(6) can be wrien as [] u (x, ) =, u (x, ) = (A (x, )), u m (x, ) = (A m (x, ), m. 3. Case Sudies of Third-Order Singular Nonlinear PDEs Solved by IMDM (7) 3.. Case Sudy. We consider he following singular nonlinear iniial-value problem (IVP) of PDE: 3 4 = 3 u 3 x x, u (x, ) =, u (x, ) = x, u (x, ) =. (8) Applicaion of inverse operaor developed in (5) wih n =, hais, ( ) (/ ) ( ) d d, on he hird-order singular nonlinear PDE of (8) gives u (x, ) = x (u u x ). (9) Now, wih he subsiuion of he dependen variable u(x, ) and he nonlinear erm u u x wih he infinie series of (7), (6) becomes u m (x, ) = x m= ( m= A m (x, )) u { (x, ) = x, u { (x, ) = (A (x, )), { u m (x, ) = (A m (x, ), m. The Adomian polynomials, A m s, are obained as [8] A (x, ) =u (x, ) u x (x, ) =, A (x, ) =u (x, ) u x (x, ) (3) u (x, ) u x (x, ) =, (3) A (x, ) =u (x, ) u x (x, ) u (x, ) u x (x, ) u (x, ) u x (x, ) =, A m (x, ) =, m 3.

5 Inernaional Parial Differenial Equaions 5 Also, he expressions for various erms of he soluion u(x, ), ha is, u m s, become u (x, ) = x, u (x, ) =, u (x, ) = (A (x, )) =, u 3 (x, ) = (A (x, )) =, u m (x, ) =, m 4. (3) In conclusion, he soluion of he hird-order singular nonlinear iniial-value problem of (8) is u (x, ) =u (x, ) u (x, ) = x (33) which is he exac soluion of (8). I can be seen ha he exac soluion of (8), (33), is obained only by he firs-order approximaion using he new developed inverse differenial operaor, (5). This reveals ha he new developed inverse differenial operaor is an effecive ool in handling he general singular nonlinear IVPs of hird-order PDEs. 3.. Case Sudy. We ake ino consideraion he following singular nonlinear IVP of hird-order PDE: u=( x ) x, (34) u (x, ) =, u (x, ) =, u (x, ) =. The inverse operaor ( ) = (/ 3 ) 3 ( ) d d d, defined in (3) wih n=3,isappliedohepdeof(34),whichgives u (x, ) = x3 4 L ((u x ) ). (35) Due o he MDM, wih he use of he infinie series of (7), (35) canberewrienas u m (x, ) = x3 m= 4 L ( A m (x, )) m= u (x, ) = x3 {, u (x, ) = 4 { L (A (x, )), { u m (x, ) = (A m (x, ), m. (36) By doing compuaions in he symbolic sofware Mahemaica, he A m s are obained as [8] A (x, ) =(u x (x, )) 3 = 44, A (x, ) =u x (x, ) u x (x, ) =, A (x, ) =u x (x, ) u x (x, ) (u x (x, )) =, A m (x, ) =, m 3. (37) The u m sarealsoobainedasfollows: u (x, ) = x3, u (x, ) = , u (x, ) = (A (x, )) =, u 3 (x, ) = (A (x, )) =, u m (x, ) =, m 4. (38) Therefore, he soluion of he singular iniial-value problem of (34) is obained as follows: u (x, ) =u (x, ) u (x, ) = x (39) which is he exac soluion of (34). Here, he exac soluion, (39), is obained using he new inverse operaor developed in Secion.3, (3). This fac indicaes ha he newly developed inverse differenial operaor is very efficien in solving he general complee singular nonlinear IVPs of hird-order PDEs. 4. Conclusions The modified decomposiion mehod (MDM) in conjuncion wih a few proposed inverse differenial operaors is used o solve he singular nonlinear iniial-value problems (IVPs) of hird-order PDEs. The new MDM, called he improved MDM (IMDM), is applied on wo case sudies for hird-order singular nonlinear IVPs, for which he IMDM analyical soluions are obained and seen o be he same as exac analyical soluions. The fas rae of convergence of he IMDM resuls owards he exac soluions indicaes ha he IMDM is a very efficien mehod for handling he hird-order singular nonlinear PDEs. Conflic of Ineress The auhor declares ha here is no conflic of ineress regarding he publicaion of his paper. References [] W. F. Ames, Nonlinear Parial Differenial Equaions in Engineering, Academic Press, New York, NY, USA, 97. [] L. Debnah, Nonlinear Parial Differenial Equaions for Scieniss and Engineers, Birkhäuser, Berlin, Germany, 997. [3] A.D.PolyaninandV.F.Zaisev,Handbook of Nonlinear Parial Differenial Equaions, Chapman and Hall, New York, NY, USA, 3. [4] F. C. Cirsea, Nonlinear mehods in he sudy of singular parial differenial equaions [Ph.D. hesis], Vicoria Universiy of Technology, Vicoria, Ausralia, 4. [5] A. M. Wazwaz, Parial Differenial Equaions and Soliary Wave Theory, Springer, New York, NY, USA, 9.

6 6 Inernaional Parial Differenial Equaions [6] A. M. Wazwaz, Consrucion of solion soluions and periodic soluions of he Boussinesq equaion by he modified decomposiion mehod, Chaos, Solions and Fracals,vol.,no.8,pp ,. [7] A.-M. Wazwaz, The modified decomposiion mehod and Padé approximans for a boundary layer equaion in unbounded domain, Applied Mahemaics and Compuaion, vol. 77, no., pp , 6. [8] A.-M. Wazwaz, The modified decomposiion mehod for analyic reamen of differenial equaions, Applied Mahemaics and Compuaion,vol.73,no.,pp.65 76,6. [9] S. A. Kechil and I. Hashim, Non-perurbaive soluion of freeconvecive boundary-layer equaion by Adomian decomposiion mehod, Physics Leers A: General, Aomic and Solid Sae Physics,vol.363,no.-,pp. 4,7. [] Y. Q. Hasan and L. M. Zhu, Modified adomian decomposiion mehod for singular iniial value problems in he second order ordinary differenial equaions, Surveys in Mahemaics and is Applicaions,vol.3,pp.83 93,8. [] A.-M. Wazwaz, The combined Laplace ransform-adomian decomposiion mehod for handling nonlinear Volerra inegro-differenial equaions, Applied Mahemaics and Compuaion,vol.6,no.4,pp.34 39,. [] A.-M. Wazwaz and M. S. Mehanna, The combined Laplace- Adomian mehod for handling singular inegral equaion of hea ransfer, Inernaional Nonlinear Science, vol., no., pp. 48 5,. [3] T. R. Sivakumar and S. Baiju, Shooing ype Laplace-Adomian decomposiion algorihm for nonlinear differenial equaions wih boundary condiions a infiniy, Applied Mahemaics Leers,vol.4,no.,pp.7 78,. [4] A. Noghrehabadi, M. Ghalambaz, and A. Ghanbarzadeh, A new approach o he elecrosaic pull-in insabiliy of nanocanilever acuaors using he ADM Padé echnique, Compuers & Mahemaics wih Applicaions,vol.64,no.9,pp.86 85,. [5] J.-S. Duan and R. Rach, Higher-order numeric Wazwaz-El- Sayed modified Adomian decomposiion algorihms, Compuers & Mahemaics wih Applicaions, vol.63,no.,pp ,. [6] Y. Lin, Y. Liu, and Z. Li, Symbolic compuaion of analyic approximae soluions for nonlinear fracional differenial equaions, Compuer Physics Communicaions, vol.84,no., pp. 3 4, 3. [7] M. M. Kermani and M. Dehesani, Solving he nonlinear equaions for one-dimensional nano-sized model including Rydberg and Varshni poenials and Casimir force using he decomposiion mehod, Applied Mahemaical Modelling, vol. 37,no.5,pp ,3. [8] L. Song and W. Wang, A new improved Adomian decomposiion mehod and is applicaion o fracional differenial equaions, Applied Mahemaical Modelling, vol.37,no.3,pp , 3.

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