The aim of this study is to solve some linear and nonlinear partial differential equations using the new integral transform "Elzaki transform" and projected differential transform method. The nonlinear terms can be handled by using of projected differential transform method; this method is more efficient and easy to handle such partial differential equations in comparison to other methods. The results show the efficiency and validation of this method. Keywords: Elzaki transform, projected differential transform method, nonlinear partial differential equations

Elzaki, Tarig. M.

Hilal, Eman M. A.

English

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Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 50 Solution of Linear and Nonlinear Partial Differential Equations Using Mixture of Elzaki Transform and the Projected Differential Transform Method  Tarig. M. Elzaki1* & Eman M. A. Hilal2  1. Mathematics Department, Faculty of Sciences and Arts-Alkamil, King Abdulaziz University, Jeddah-Saudi Arabia. Mathematics Department, Faculty of Sciences, Sudan University of Sciences and Technology-Sudan. 2. Mathematics Department, Faculty of Sciences for Girles King Abdulaziz University Jeddah-Saudi Arabia * E-mail of the corresponding author: Tarig.alzaki@gmail.com  Abstract The aim of this study is to solve some linear and nonlinear partial differential equations using the new integral transform "Elzaki transform" and projected differential transform method. The nonlinear terms can be handled by using of projected differential transform method; this method is more efficient and easy to handle such partial differential equations in comparison to other methods. The results show the efficiency and validation of this method. Keywords: Elzaki transform, projected differential transform method, nonlinear partial differential equations.  1. Introduction Many problems of physical interest are described by linear and nonlinear partial differential equations with initial or boundary conditions, these problems are fundamental importance in science and technology especially in engineering. Some valuable contributions have already been made to showing differential equations such as Laplace transform method [lslam, Yasir Khan, Naeem Faraz and Francis Austin (2010), Lokenath Debnath and Bhatta (2006)], Sumudu transform [A.Kilicman and H.E.Gadain (2009), (2010)], differential transform method etc. Elzaki transform method is very effective tool for solve linear partial differential equations [Tarig   Elzaki & Salih Elzaki (2011)]. In this study we use the projected differential transform method [Nuran  Guzel and Muhammet Nurulay (2008), Shin- Hsiang Chang , Ling Chang (2008)  ] to decompose the nonlinear terms, this means that we can use both Elzaki transform and projected differential transform methods to solve many nonlinear partial differential equations. 1.1. Elzaki Transform: The basic definitions of modified of Sumudu transform or Elzaki transform is defined as follows, Elzaki transform of the function  ( )f t  is  Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 51 [ ]0( ) ( ) , 0tvE f t v f t e dt t∞−= >∫  Tarig M. Elzaki and Sailh M. Elzaki in [2011], showed the modified of  Sumudu transform [Kilicman  & ELtayeb (2010)] or Elzaki transform was applied to partial differential equations, ordinary differential equations, system of ordinary and partial differential equations and integral equations. In this paper, we combined Elzaki transform and projected differential transform methods to solve linear and nonlinear partial differential equations. To obtain Elzaki transform of partial derivative we use integration by parts, and then we have: [ ] [ ]22 22 22 2( , ) 1 ( , ) 1 ( ,0)( , ) ( ,0) ( , ) ( ,0)( , ) ( , )( , ) ( , )f x t f x t f xE T x v vf x E T x v f x vt v t v tf x t d f x t dE T x v E T x vx dx x dx ∂ ∂ ∂ = − = − −  ∂ ∂ ∂    ∂ ∂ = =  ∂ ∂   Proof:  To obtain ELzaki transform of partial derivatives we use integration by parts as follows:                     ( )( ) ( )00 00, lim ( , ) ( , ),,0limpp pt t t tv v v vppf f fx t v e dt ve dt ve f x t e f x t dtt t tT x vvf xv− − − −∞→∞→∞  ∂ ∂ ∂   Ε = = = −   ∂ ∂ ∂     = −∫ ∫ ∫ We assume that f is piecewise continuous and it is of exponential order.  Now   ( ) ( )0 0,, ,t tv vf x tfve dt ve f x t dtx x x∞− −∞ ∂∂ ∂ Ε = = ∂ ∂ ∂  ∫ ∫ using the Leibnitz rule to find:                       ( ),f d T x vx dx∂  Ε =   ∂                                                By the same method we find:   ( )2 22 2 ,f d T x vx dx ∂ Ε =   ∂                                              To find:       22 ( , )fx tt ∂Ε  ∂  Let       ,f g thent∂=∂ we have:  ( ) ( ) ( ) ( )22,,, ,0g x tg x tfx t vg xt t v  ∂ ∂  Ε = Ε = Ε −  ∂ ∂    Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 52 ( ) ( ) ( ) ( )22 21, , , 0 ,0f fx t T x v f x v xt v t ∂ ∂Ε = − − ∂ ∂  We can easily extend this result to the nth partial derivative by using mathematical induction.   1.2. Projected Differential Transform Methods:  In this section we introduce the projected differential transform method which is modified of the differential transform method. Definition: The basic definition of projected differential transform method of function 1 2( , , , )nf x x xLL  is defined as              1 21 2 10( , , , )1( , , , , )!nknn knxf x x xf x x x kk x−= ∂ = ∂  LLLL                (1)                  Such that   1 2( , , , )nf x x xLL is the original function and  1 2 1( , , , , )nf x x x k−LL  is projected transform function. And the differential inverse transform of 1 2 1( , , , , )nf x x x k−LL  is defined as                    1 2 1 2 0( , , , ) ( , , , )( )kn nk of x x x f x x x x x∞== −∑LL LL               (2)                               The fundamental theorems of the projected differential transform are      Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 53 Theorems:     ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 2 1 2 1 21 2 1 1 2 1 1 2 11 2 1 21 2 1 1 2 11 , ,......, , ,......, , ,......,, ,......, , , ,......, , , ,......, ,2 , ,......, , ,......,, ,......, , , ,......,n n nn n nn nn nIf z x x x u x x x v x x xThen z x x x k u x x x k v x x x kIfz x x x cu x x xThen z x x x k cu x x x− − −− −= ±= ±== ( )( ) ( ) ( )( ) ( ) ( )1 21 21 2 1 1 2 1,, ,......,3 , ,......,, ,......, , 1 , ,....., , 1nnnn nkdu x x xIfz x x xdxThen z x x x k k u x x x k− −== + + ( ) ( ) ( )( ) ( ) ( )1 21 21 2 1 1 2 1, ,......,4 , ,......,!, ,......, , , ,....., ,!nnn nnn nd u x x xIfz x x xdxk nThen z x x x k u x x x k nk− −=+= + ( ) ( ) ( ) ( )( ) ( ) ( )1 2 1 2 1 21 2 1 1 2 1 1 2 105 , ,......, , ,......, , ,......,, ,......, , , ,......, , , ,......, ,n n nkn n nmIf z x x x u x x x v x x xThen z x x x k u x x x m v x x x k m− − −=== −∑          ( ) ( ) ( ) ( ) ( )( ) ( ) ( )1 3 21 2 2 11 2 1 1 2 2 1 2 1 21 2 1 1 1 2 1 1 2 1 2 1 2 10 0 0 01 16 , ,......, , ,......, , ,......, ..... , ,......,, ,......, , ...... , ,......, , , ,......, ,..... ,nn nn n n n nk k kkn n nk k k knIf z x x x u x x x u x x x u x x x Thenz x x x k u x x x k u x x x k ku x−− −− − −= = = =−== −×∑ ∑ ∑∑( ) ( )2 1 1 2 1 2 1 1,......, , , ,......, ,n n n n n nx x k k u x x x k k− − − − −− − ( ) ( )( ) ( )1 21 2 1 21 2 1 1 2 17 , ,......, ........1, ,......, , , ,......, ,0nqq qn nnn n nnIf z x x x x x xk qThen z x x x k x x x q kk qδ− −=== − = ≠ Note that c  is a constant and n  is a nonnegative integer. 2. Applications:  Consider a general nonlinear non-homogenous partial differential equation with initial conditions of the form:                ( , ) ( , ) ( , ) ( , )Du x t Ru x t Nu x t g x t+ + =                            (3)                                                             ( ,0) ( ) , ( ,0) ( )tu x h x u x f x= =  Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 54 Where D is linear differential operator of order two, R  is linear differential operator of less order than D , N is the general nonlinear differential operator and ( , )g x t is the source term. Taking Elzaki transform on both sides of equation (3), to get:                 [ ( , )] [ ( , )] [ ( , )] [ ( , )]E Du x t E Ru x t E Nu x t E g x t+ + =               (4)                               Using the differentiation property of Elzaki transforms and above initial conditions, we have:    2 2 3 2[ ( , )] [ ( , )] ( ) ( ) [ ( , ) ( , )]E u x t v E g x t v h x v f x v E Ru x t Nu x t= + + − +           (5)            Appling the inverse Elzaki transform on both sides of equation (5), to find:                  [ ]{ }1 2( , ) ( , ) ( , ) ( , )u x t G x t E v E Ru x t Nu x t−= − +                 (6)                           Where ( , )G x t represents the term arising from the source term and the prescribed initial conditions. Now, we apply the projected differential transform method.              [ ]{ }1 2( , 1) , ( ,0) ( , )m mu x m E v E A B u x G x t−+ = − + =                (7)                    Where ,m mA B are the projected differential transform of ( , ) , ( , )Ru x t Nu x t . From equation (7), we have: [ ]{ }[ ]{ } [ ]{ }1 20 01 2 1 21 1 2 2( ,0) ( , ) , ( ,1)( , 2) , ( ,3).....u x G x t u x E v E A Bu x E v E A B u x E v E A B−− −= = − += − + = − +  Then the solution of equation (3) is ( , ) ( ,0) ( ,1) ( , 2) ........u x t u x u x u x= + + +  To illustrate the capability and simplicity of the method, some examples for different linear and nonlinear partial differential equations will be discussed.  Example 2.1: Consider the simple first order partial differential equation                      32 , ( ,0) 6 xy y y y x ex t−∂ ∂= + =∂ ∂                           (8)                                             Taking Elzaki transform of (8), leads to [ ] [ ]2 3( , ) 62xm mvE y x t v e E A B−= + −  Take the inverse Elzaki transform to find,                { [ ] }1 3( , 1) , ( ,0) 62xm mvy x m E E A B y x e− −+ = − =                                       (9) Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 55 Where ( , ), ( , )m my x mA B y x mx∂= =∂are the projected differential transform of       ( , ), ( , )y x t y x tx∂∂. The standard Elzaki transform defines the solution ( , )y x t by the series           0( , ) ( , )my x t y x m∞== ∑  From equation (9) we find that:   3( ,0) 6 xy x e −=   3 3 1 3 3 30 03 3 1 4 3 2 31 13 318 , 6 , ( ,1) 12 1236 , 12 , ( , 2) 24 12.................................................................. ( ,3) 8x x x xx x x xxA e B e y x E v e teA te B e y x E v e t ey x t e− − = − −− − = − −− = − = = − = −  = = − = = = − The solution in a series form is given by 3 3 2 3 3 2 (3 2 )( , ) 6 12 12 ........ 6 6x x x x t x ty x t e te t e e e e− − − − − − += − + + = =  Example 2.2: Consider the following linear second order partial differential equation                                    0 , ( ,0) 0, ( ,0) cosxx tt tu u u x u x x+ = = =                                            (10) To find the solution we take Elzaki transform of equation (10) and making use of the conditions to find, [ ] [ ]23 22( , )( , ) cos ,m mu x mE u x t v x v E A Ax∂= − =∂ Take the inverse Elzaki transform we get:                   [ ]{ }1 2( , 1) , ( ,0) 0 , ( ,1) cosmu x m E v E A u x u x t x−+ = − = =                        (11) By using equation (11), we find that: [ ]{ } 31 213 3 51 22cos ( , 2) cos cos3!cos ( ,3) cos cos3! 3! 5!...tA t x u x E v E t x xt t tA x u x E v E x x−−= − ⇒ = =   = − ⇒ = =      Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 56 Then the solution is            3 5( , ) cos ..... cos sinh3! 5!t tu x t x t x t = + + + =   Example 2.3: Consider the following second order nonlinear partial differential equation                    2 222 , ( .0)u u uu u x xt x x∂ ∂ ∂ = + = ∂ ∂ ∂                         (12) To find the solution take Elzaki transform of (12) and using the condition we get: [ ] [ ]2 2( , ) m mE u x t v x vE A B= + +  Where220 0( , ) ( , ) ( ), ( , )h hm mm mu x m u x h m u h mA B u x mx x x= =∂ ∂ − ∂ −= =∂ ∂ ∂∑ ∑, are projected differential transform of   2 22,u uux x∂ ∂  ∂ ∂  Take the inverse Elzaki transform to get:                                   [ ]{ }1 2( , 1) , ( ,0)m mu x m E vE A B u x x−+ = + =                                (13) From equation (13) we find that: 2 2 1 2 3 20 02 2 1 2 4 2 21 14 , 2 ( ,1) 6 648 , 24 ( ,2) 72 36...A x B x u x E x v x tA x t B x t u x E x v x t−− = = ⇒ = =  = = ⇒ = =  Then the solution of equation (12) is 22 2 2 2 2 1( , ) 6 36 ..... (1 6 )1 6xu x t x x t x t x tt−= + + + = − =− Which is an exactly the same solution obtained by the Adomian decomposition method. Example 2.4: Let us consider the nonlinear partial differential equation                             2 22212 , ( ,0)2u u u xu u u xt x x∂ ∂ ∂ + = + = ∂ ∂ ∂                                          (14) Taking Elzaki transform of equation (14) and using the condition leads to: Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 57 [ ] [ ]2 1( , ) 22 m mxE u x t v vE A B+ = + +   Taking the inverse Elzaki transform to find:                              [ ]{ }1 1( , 1) 2 , ( ,0) 2m mxu x m E vE A B u x− ++ = + =                                 (15) Where                 0 0220 0( , ) ( , )( , )( , )( , ) ( , )h kmk mh kmk mu x h m u x h kA u x mx xu x h kB u x m u x h mx= == =∂ − ∂ −=∂ ∂∂ −= −∂∑∑∑∑  From equation (15) we have: 21 1 3( 1)( ,0) , ( ,1) , ( , 2) ,...........2 2 8x x xu x u x t u x t+ + += = =  Then the solution of equation (14) is ( ) 121 1( , ) 12 2 1x xu x t tt−+ += − =− 3. Conclusion The solution of linear and nonlinear partial differential equations can be obtained using Elzaki transform and projected differential transform method without any discretization of the variables. The results for all examples can be obtained in series form, and all calculations in the method are very easy. This technique is useful to solve linear and nonlinear partial differential equations.  References [1] S. lslam, Yasir Khan, Naeem Faraz and Francis Austin (2010), Numerical Solution of Logistic Differential Equations by using the Laplace Decomposition Method, World Applied Sciences Journal 8 (9):1100-1105.  [2] Nuran  Guzel and Muhammet Nurulay (2008), Solution of Shiff Systems By using Differential Transform Method, Dunlupinar universities Fen Bilimleri Enstitusu Dergisi, ISSN  1302-3055, PP. 49-59. [3] Shin- Hsiang Chang , I. Ling Chang (2008), A new algorithm for  calculating one – dimensional differential transform of nonlinear functions, Applied Mathematics and Computation 195, 799-808. [4] Tarig M. Elzaki (2011), The New Integral Transform “Elzaki Transform” Global Journal of Pure and Applied Mathematics, ISSN 0973-1768, Number 1, pp. 57-64. [5] Tarig M. Elzaki & Salih M. Elzaki (2011), Application of New  Transform “Elzaki Transform” to Partial Differential Equations, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768,Number 1, pp. 65-70. [6] Tarig M. Elzaki & Salih M. Elzaki (2011), On the Connections Between Laplace and Elzaki transforms, Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 Volume 6, Number 1, pp. 1-11. [7] Tarig M. Elzaki & Salih M. Elzaki (2011), On the Elzaki Transform and Ordinary Differential Equation With Variable Coefficients,  Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 6, Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 58 Number 1, pp. 13-18. [8] Lokenath Debnath and D. Bhatta (2006). Integral transform and their Application second Edition, Chapman & Hall /CRC.  [9] A.Kilicman and H.E.Gadain (2009). An application of double Laplace transform and Sumudu transform, Lobachevskii J. Math.30 (3), pp.214-223. [10] J. Zhang (2007), A Sumudu based algorithm m for solving differential equations, Comp.  Sci. J.  Moldova 15(3), pp – 303-313. [11] Hassan Eltayeb and Adem  kilicman (2010), A Note on  the  Sumudu  Transforms  and  differential   Equations, Applied  Mathematical  Sciences, VOL, 4, no.22,1089-1098    [12]  Kilicman A. & H. ELtayeb (2010). A note on Integral transform and Partial Differential Equation, Applied Mathematical Sciences, 4(3), PP.109-118. [13] Hassan ELtayeh and Adem kilicman (2010), on Some Applications of a new Integral Transform, Int. Journal of Math. Analysis, Vol, 4, no.3, 123-132. [14] N.H. Sweilam, M.M. Khader (2009). Exact Solutions of some capled nonlinear partial differential equations using the homotopy perturbation method. Computers and Mathematics with Applications 58 2134-2141. [15] P.R. Sharma and Giriraj Methi (2011). Applications of Homotopy Perturbation method to Partial differential equations. Asian Journal of Mathematics and Statistics 4 (3): 140-150. [16] M.A. Jafari, A. Aminataei (2010). Improved Homotopy Perturbation Method. International Mathematical Forum, 5, no, 32, 1567-1579. [17] Jagdev Singh, Devendra, Sushila (2011). Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations. Adv. Theor. Appl. Mech., Vol. 4, no. 4, 165-175. Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.4, 2012 59  Table 1. Elzaki transform of some Functions                               ( )tf  ( )[ ] ( )uTtfE =  1 2v  t  3v  nt  2! nn v +  ate  21vav− sinat  32 21ava v+ cosat  22 21va v+  This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  More information about the publisher can be found in the IISTE’s homepage:  http://www.iiste.org  The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world.   Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/Journals/ The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. 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Solution of Linear and Nonlinear Partial Differential Equations Using Mixture of Elzaki Transform and the Projected Differential Transform Method

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Solution of Linear and Nonlinear Partial Differential Equations Using Mixture of Elzaki Transform and the Projected Differential Transform Method

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