International Journal of Modern Mathematical Sciences, 2012, 3(2): International Journal of Modern Mathematical Sciences

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1 Article International Journal of Modern Mathematical Sciences (2): International Journal of Modern Mathematical Sciences Journal homepage:wwwmodernscientificpresscom/journals/ijmmsaspx On Goursat Problems ISSN: X Florida USA Muhammad Usman Tamour Zubair Umair Ali Syed Tauseef Mohyud-Din * Department of Mathematics HITEC University Taxila Cantt Pakistan * Author to whom correspondence should be addressed; syedtauseefs@hotmailcom Article history: Received 29 May 2012 Received in revised form 1 August 2012 Accepted 8 August 2012 Published 13 August 2012 Abstract: In this paper we apply Homotopy Analysis Method (HAM) to find appropriate solutions of linear and nonlinear Goursat problems which are of utmost importance in applied and engineering sciences The proposed modification is the elegant coupling of Homotopy Analysis Method (HAM) Numerical results coupled with graphical representation explicitly reveal the complete reliability of the proposed algorithm Keywords: Homotopy analysis method linear and nonlinear Goursat problems exact solution Maple Mathematics Subject Classification (2000): 35Q79 1 Introduction The rapid development of nonlinear sciences [1-17] witnesses a wide range of analytical and numerical techniques by various scientists Most of the developed schemes have their limitations like limited convergence divergent results linearization discretization unrealistic assumptions and noncompatibility with the versatility of physical problems [1-11] In the similar context Liao [7-9] developed Homotopy Analysis Method (HAM) which is being applied on a wide range of nonlinear problems of physical nature see [1-17] and the references therein The basic motivation of present study is modification of traditional Homotopy Analysis Method (HAM) to tackle nonlinear partial differential equations It is observed that proposed technique is highly effective Moreover this method (HAM) is more user-friendly and it overcomes the complexities of selection of initial value Several examples are given which reveal the efficiency and reliability of the proposed algorithm

2 64 2 Analysis of Homotopy Analysis Method (HAM) We consider the following differential equation = 0 (1) where is a nonlinear operator denotes independent variables is an unknown function respectively For simplicity we ignore all boundary or initial conditions which can be treated in the similar way By means of generalizing the traditional Homotopy method Liao constructs the so called zero - order deformation equation 1 ; 0 = h; (2) where 01 is the embedding parameter h 0 is a nonzero parameter 0 is an auxiliary function is an auxiliary linear operator 0 is an initial guess of ; is a unknown function respectively It is important that one has great freedom to choose auxiliary things in HAM Obviously when and = 0 and = 1 it holds ;0 = ;1 = Respectively Thus as increases from 0 to 1 the solution ; varies from the initial guesses 0 to the solution Expanding in Taylor series with respect to we have ; = + where = 1 ;! at = 0 If the auxiliary linear operator the initial guess the auxiliary h and the auxiliary function are so properly chosen the above series converges at = 1 then we have Define the vector = + = { } Differentiating equation (2) Times with respect to the embedding parameter and then setting = 0 and finally dividing them by! we obtain the %h-order deformation equation where and & ' = h( ' (3) ( 1 = 1 1 ; at = 0 1! 1 & = 0 1 = 1 > 1 Applying 1 both sides of (3) we get = & 1 + h 1 ( 1

3 65 In this way it is easily to obtain for m 1 at %h- order we have = when - we get an accurate approximation of the original equation(1) For the convergence of the above method we refer the reader Liao s work If equation (1) admits unique solution then this method will produce the unique solution If equation (1) does not possess unique solution the HAM will give a solution among many other (possible) solutions 3 Goursat Problems The Goursat problems arise in linear and nonlinear partial differential equations which mixed derivatives The slandered form of the Goursat problems is given by /% = 0/% / % 0 / 20 % 3 /0 = 4/ 0% = h% 40 = h0 = 00 The slandered form of the homogenous linear Goursat problems is given by /% = 0 0 / 20 % 3 /0 = 4/ 0% = h% 40 = h0 = 00 The slandered form of the inhomogeneous linear Goursat problems is given by /% = 0 + 5/% 0 / 20 % 3 /0 = 4/ 0% = h% 40 = h0 = 00 4 Numerical Application In this section we apply Homotopy analysis method (HAM) for linear and nonlinear Goursat problems Numerical results are very encouraging Example 41 We first consider the following homogenous linear Goursat problem /% = with initial conditions /0 = 6 7 0% = = 1 To solve the given Equation by HAM we choose the linear operator with the property /%;9 = :; :7:8 </%;9= > + />? = 0

4 66 > + %>? = 0 where > 1 and > 2 are the integral constants The inverse operator 1 is given by / % 0 0 A/A% we now define a nonlinear operator as /%;9 = /%;9 78 /%;9 using the above definition we construct the zeroth-order deformation equation 1 9/%;9 0 /% = 9h/%;9/%;9 for 9 = 0 and 9 = 1 we can write /%;0 = /% /%;1 = /% Thus we obtain the %h order deformation equation /% & ' /% = h/%( ' with initial condition /0 = 0 00 = 1 where ( 1 = B 2 /% 1 1 C Now the solutions of the %h order deformation equation are /% = & 1 /% + 1 h/%( 1 1 we start with an initial approximation /% = by means of the iteration formula as discuss above if h = 1 = 1 we can obtain directly the others components as the series solution is given by /% = /% = 1 2 % 2 % + 268? /% = 1 2 %? % %E 1 24 %G 26 8 /% = % 2 % %? % %E 1 24 %G and the close form solution is /% = 6 7I8

5 67 Figure 41 Depicts approximate and exact solutions Example 42 We first consider the following homogenous linear Goursat problem /% = 2 with initial conditions / % 6 '? To solve the given Equation by HAM we choose the linear operator with the property /%;9 :; :7:8 </%;9= > />? 0 > %>? 0 where > 1 and > 2 are the integral constants The inverse operator 1 is given by / % 0 A/A% we now define a nonlinear operator as /%;9 /%;9 78 2/%;9 using the above definition we construct the zeroth-order deformation equation 1 9/%;9 0 /% 9/%;9/%;9 for 9 0 and 9 1 we can write /%;0 /% /%;1 /% Thus we obtain the % order deformation equation /% & ' /% /%( ' with initial condition / where ( 1 B 2 /% C

6 68 Now the solutions of the % order deformation equation are /% & 1 /% 1 /%( we start with an initial approximation /% '?8 1 by means of the iteration formula as discuss above if 1 1 we can obtain directly the others components as /% '?8 1 /% 3 2 % %? 2% '?8? /% 7 2% 2%? 26 8 % '?8 2 3 %E the series solution is given by /% '? % %? 2% '?8 7 2% 2%? 26 8 % '?8 2 3 %E and the close form solution is /%6 7'?8 Figure 42 Depicts approximate and exact solutions

7 69 Example 43 We first consider the following inhomogeneous linear Goursat problem /% % with initial conditions /0 = 6 7 0% = % = 1 To solve the given Equation by HAM we choose the linear operator with the property /%;9 = :; :7:8 </%;9= > + />? = 0 > + %>? = 0 where > 1 and > 2 are the integral constants The inverse operator 1 is given by / % 0 0 A/A% we now define a nonlinear operator as /%;9 = /%;9 78 /%;9 + % using the above definition we construct the zeroth-order deformation equation 1 9/%;9 0 /% = 9h/%;9/%;9 for 9 = 0 and 9 = 1 we can write /%;0 = /% /%;1 = /% Thus we obtain the %h order deformation equation /% & ' /% = h/%( ' with initial condition /0 = 0 00 = 1 where ( 1 = B 2 /% %C Now the solutions of the %h order deformation equation are /% = & 1 /% + 1 h/%( 1 1 we start with an initial approximation /% = % by means of the iteration formula as discuss above if h = 1 = 1 we can obtain directly the others components as 2 /% = 1 2 %2 + 2%6 % % %4 26 % 0 /% = 6 / + % + 6 % 1 1 /% = 1 2 % 2 % + 26%

8 70 the series solution is given by /% 6 7 % %2 % %? 2% %E 1 24 %G 26 8 and the close form solution is /%%6 7I8 Figure 43 Depicts approximate and exact solutions Example 44 We first consider the following inhomogeneous linear Goursat problem /% 4/%% 2 / 2 with initial conditions /06 7 0% To solve the given Equation by HAM we choose the linear operator with the property /%;9 :; :7:8 </%;9= > />? 0 > %>? 0 where > 1 and > 2 are the integral constants The inverse operator 1 is given by / % 0 we now define a nonlinear operator as /%;9/%;9 78 /%;94/%/? %? using the above definition we construct the zeroth-order deformation equation

9 71 1 9/%;9 0 /% = 9h/%;9/%;9 for 9 = 0 and 9 = 1 we can write /%;0 = /% /%;1 = /% Thus we obtain the %h order deformation equation /% & ' /% = h/%( ' with initial condition /0 = 0 00 = 1 where ( 1 = B 2 /% /% + / 2 % 2 C Now the solutions of the %h order deformation equation are /% = & 1 /% + 1 h/%( 1 1 we start with an initial approximation 0 /% = 6 / + 6 % 1 by means of the iteration formula as discuss above if h = 1 = 1 we can obtain directly the others components as 0 /% = 6 / + 6 % 1 1 /% = 1 18 % 18 9% + 186% + 9% 3 % 5 2 /% = 1 2 %2 + 2%6 % % % %3 26 % the series solution is given by /% = % 18 9% % E % O 1 2 %? + 2% %G %P %E and the close form solution is /% = %? /? + 6 7I8

10 72 Figure 44 Depicts approximate and exact solutions Example 45 We first consider the following inhomogeneous linear Goursat problem /% 3 / 3 3/ 2 % 3/% 2 % 3 with initial conditions /0 / 0% % 00 0 To solve the given Equation by HAM we choose the linear operator with the property /%;9 :; :7:8 </%;9= > />? 0 > %>? 0 where > 1 and > 2 are the integral constants The inverse operator 1 is given by / % 0 A/A% we now define a nonlinear operator as /%;9 /%;9 78 E /%;9 / E 3/? % 3/%? % E using the above definition we construct the zeroth-order deformation equation 1 9/%;9 0 /% 9/%;9/%;9 for 9 0 and 9 1 we can write /%;0 /% /%;1 /% Thus we obtain the % order deformation equation /% & ' /% /%( ' with initial condition /

11 73 Q where ( 1 B 2 /% 1 1 S0 R0 Q QR 1S / 3 3/ 2 % 3/% 2 % 3 C Now the solutions of the % order deformation equation are /% & 1 /% 1 /%( we start with an initial approximation /% / % by means of the iteration formula as discuss above if 1 1 we can obtain directly the others components as 0 /% / % 1 /% 0 2 /% 0 the series solution is given by /% / % and the close form solution is /% / % Figure 45 Depicts approximate and exact solutions Example 46 We first consider the following inhomogeneous linear Goursat problem /% 2 6 2/ 6 2% 26 /%

12 74 with initial conditions /0 = % = = 2 To solve the given Equation by HAM we choose the linear operator with the property /%;9 = :; :7:8 </%;9= > + />? = 0 > + %>? = 0 where > 1 and > 2 are the integral constants The inverse operator 1 is given by / % 0 0 A/A% we now define a nonlinear operator as /%;9 = /%;9 78 +? /%;9 6?7 + 6? I8 using the above definition we construct the zeroth-order deformation equation 1 9/%;9 0 /% = 9h/%;9/%;9 for 9 = 0 and 9 = 1 we can write /%;0 = /% /%;1 = /% Thus we obtain the %h order deformation equation /% & ' /% = h/%( ' with initial condition /0 = 0 00 = 1 where ( 1 = B 2 /% U S=0 T=0 T U T 6 2/ + 6 2% + 26 /+% C Now the solutions of the %h order deformation equation are /% = & 1 /% + 1 h/%( 1 1 we start with an initial approximation 0 /% = 6 % + 6 / by means of the iteration formula as discuss above if h = 1 = 1 we can obtain directly the others components as 2 /% = 0 the series solution is given by 0 /% = 6 % + 6 / 1 /% = 0 /% =

13 75 and the close form solution is /% Figure 46 Depicts approximate and exact solutions 5 Conclusion In this paper we apply Homotopy analysis method on linear and nonlinear Goursat problems The result which obtained is very good and reliable as compared to other methods The advantages of HAM are illustrated It is clear that HAM is very powerful and efficient method to find the exact solution of wide range of linear and nonlinear problems References [1] S Abbasbandy Homotopy analysis method for generalized Benjamin-Bona-Mahony equation Z Angew MathPhys 59 (2008): [2] S Abbasbandy and F S Zakaria Soliton solutions for the fifth-order K-dVequation with the homotopy analysis method Nonlinear Dyn 51 (2008): [3] S Abbasbandy Homotopy analysis method for the Kawahara equation Nonlinear Anal (B) 11 (2010):

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