In this paper, modification of Adomian decomposition method is introduced for solving heat equation with derivative boundary conditions. Some examples and the obtained results demonstrate efficiency of the proposed method. Keywords: Modified decomposition method, Heat equation, Derivative boundary conditions

Gorial, Iman. I.

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Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  170  Approximation Method for the Heat Equation with Derivative  Boundary Conditions  Iman. I. Gorial  Department of Mathematics, College of Education for Pure Science / Ibn Al-Haitham, Baghdad University, Iraq * E-mail of the corresponding author: imanisho@yahoo.com Abstract In this paper, modification of Adomian decomposition method is introduced for solving heat equation with derivative boundary conditions. Some examples and the obtained results demonstrate efficiency of the proposed method. Keywords: Modified decomposition method, Heat equation, Derivative boundary conditions.  1. Introduction Many classes of linear and nonlinear differential equations can solve using Adomian decomposition method and in computation and faster in convergence it is much simpler than any other method available in the open literature.     Many Authors have proposed numerical methods for solving problems [1-10]. Later A. Cheniguel [11], present Adomian decomposition for solving non-homogeneous heat equation with derivative boundary conditions. In this work we propose a new technique based on the modification of Adomian decomposition method.   2. Solution Heat Equation by Modified Adomian’s Decomposition Method  In this section, we present modified decomposition method for solving the heat equation with derivative boundary conditions:                            ),(),(),( txqtxuDtxuD xxt            (1)    10),()0,(  xxfxu                                             (2)                  Tttgtu  0),(),0( 1                                             (3)                                                                                                     Tttgtu  0),(),1( 2                                                                               (4) Where 21,, ggf  and q are known functions, T is given constant.  We start with Adomian decomposition method                               0),(),(nn txutxu   Now, Equation (1) can be rewritten as Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  171                           ),(),(),( txqtxuLtxuL xxt                      (5)                  Where the differential operators )()( tLt  and 22xLxx  , the inverse 1L  is assumed an integral operator given by                             tdtL01 )(                                        (6) The operating with the operator 1L   on both sides of Equation (5) we have                         )),(())),((())),(((111 txqLtxuLLtxuLL xxt    Therefore, we can write,                             ),()0,(),( 101 txqLuLLxutxunnxxt         (7)  By Wazwaz [12], the modified decomposition method is based on the assumption that the function )(xK  can be divided into two parts, namely )(1 xK  and )(2 xK . Under this assumption we set                                   )()()( 21 xKxKxK    we suggest the following modification                    10 Ku             0121 uLLKu xxt            011nnxxtn uLLu , n 1 3. Numerical Illustration: Example 1: Consider the problem (1) with the following conditions, as taken in [11]                                            ),(),(2 txuDtxuD xt txe  2                                                                          10,)0,(  xexu x                                                                                                             Ttetu tx  0,),0(                                                Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  172                                        Ttetu t   0,),1( 1  Now after modified decomposition method, we obtain:  0),(0),(0),(),(3210 txutxutxuetxu tx Then the series form is given by: ),(),(),(),(),( 3210 txutxutxutxutxu               txe   This is the exact solution  ),( txutxe  .    The plot of the exact solution surface is shown in Figure 1 and the numerical solution surface is shown in Figure 2 for heat equation  050100150051015-6-4-20246x 10-5 050100150051015-6-4-20246x 10-5 Figure 1: Exact solution   Figure 2: Numerical solution  Example 2:  Consider the problem (1) with the following derivative boundary and initial conditions, as taken in [11]                    22 ),(),( xttxuDtxuD xt                                                                                                                                         Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  173                    10),sin()0,(  xxxu                                                                                                                Tttux  0,1),0(                              Ttttux  0),sin(),1(  We apply the above modified decomposition method; we obtain: 0),(0),(0),(31)sin(),(32130 txutxutxuxtxetxu t Then the series form is given by: ),(),(),(),(),( 3210 txutxutxutxutxu               xtxe t 331)sin(   Which gives the exact solution ),( txu xtxe t 331)sin(  .  Figure 3 and Figure 4 show the plot of the exact solution surface and the numerical solution surface for heat equation respectively. 010203005100200400600800 010203005100200400600800 Figure 3: Exact solution Figure 4: Numerical solution  Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  174  Example 3: Consider heat equation with derivative boundary conditions for the equation (1), as taken in [11]                       ),(),( 2 txuDtxuD xt                                                                               10),sin()0,(  xxxu                                          tx etu2),0(                                                                                tetu2),1(    Now we apply the above modified decomposition method, we obtain:  0),(0),(0),()sin(),(32102 txutxutxuxetxu t  Then the series form is given by: ),(),(),(),(),( 3210 txutxutxutxutxu               )sin(2xe t   This is the exact solution  ),( txu )sin(2xe t  .  Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  175    Figure 5 and Figure 6   show the plot of the exact and the numerical solution surface for heat equation respectively. 05101505100123456x 10-5 05101505100123456x 10-5 Figure 5: Exact solution Figure 6: Numerical solution 4. Conclusion In this paper, we have applied the modified decomposition method for the solution of the heat equation with derivative boundary conditions. This algorithm is simple and easy to implement. The obtained results confirmed a good accuracy of the method. On the other hand, the calculations are simpler and faster than in traditional techniques.  Mathematical Theory and Modeling                                                                                         www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.5, 2014  176  References [1] Wazwaz A. M. (1999), "The modified decomposition method and Pade approximations for solving     Thomas Fermi equations", App. Math.  And Computation, 105:11-19.   [2] Adomian G. and Rach R. (1986), "On linear and nonlinear integro-diff erential equations", J. Math. Anal. Appl. 113, no. 1: 199–201. [3] Iman. I. Gorial (2013), "A Semi-Analytic Method for Solving Two-Dimensional Fractional Dispersion   Equation", Advances in Physics Theories and Applications.22: p. 14-18. [4] Hossein J. and Varsha D. (2006), "Revised Adomian decomposition method for Solving a system of   nonlinear equations", App. Math. And Computation, 175: 1-7. [5] Hossein  J. and  Varsha  D. (2006),  "Revised  Adomian  decomposition  method  for  Solving  systems  of  Ordinary  and  Fractional Differential Equations" , App. Math. And Computation, 181: 598-608.   [6] Babolian E., Javadi S. H. and Sadeghi H. (2003), "Restarted Adomian Method for Integral Equations", App. Math.  And Computation, 167: 1150-1155. [7] Wazwaz A. M. (1999), "A Reliable Modification of Adomian Decomposition Method", App. Math.  And Computation, 102 :77-86. [8] Babolian E. and Javadi S. H. (2003), Restarted Adomian Method for Algebraic Equations, App. Math.  And Computation, 146: 533- 541. [9] Momani S., dibat Z.O (2006), ''Analytical solutions of a time-fractional Navier-Stokes equation by Adomiande composition method'', Appl. Math. Comput.Vol. 177, No.2: pp. 488-494 [10] Cheniguel A. and Ayadi A. (2011)," Solving Non Homogeneouse Heat Equation by the Adomian    Decomposition Method ", International Mathematical Forum, Vol.6, no.13: 639-649 [11] Cheniguel A.( 2011)," Numerical  Method  for  Solving Non-Homogeneous Heat  Equation  with  Derivative Boundary  Conditions ", International Mathematical  Forum,  Vol.6, , no.  14: 651 – 658 [12] Wazwaz A. (1999), ''A reliable modification of Adomian decomposition method'', Appl. Math. Comput., 102 (1): pp. 77–86.  The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   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Approximation Method for the Heat Equation with Derivative  Boundary Conditions

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Approximation Method for the Heat Equation with Derivative Boundary Conditions

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