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Explicit Solution of Hyperbolic Partial Differential Equations by an Iterative Decomposition Method

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					                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                         Vol. 9, No. 6, June 2011




                  Explicit Solution of Hyperbolic Partial
                  Differential Equations by an Iterative
                         Decomposition Method
                                          Adekunle, Y.A.1*, Kadiri, K.O.2, Odetunde, O.S.3
    1
        Department of Computer Science and Mathematics, Babcock University, Ilisan-Remo Ogun State, Nigeria
             adekunleya@gmail.com
    2
        . Department Electrical/Electronics Engineering, Federal Polytechnic, Offa, Kwara State, Nigeria
             kadiritoyin2007@yahoo.com
    3
        . Department of Mathematical Science, Olabisi Onabanjo University, Ago-Iwoye, Ogun State,Nigeria
             todetunde@yahoo.co.uk

Abstract- In this paper, an iterative decomposition method              such as the wave equations, much attention is given
is applied to solve partial differential equations. The                 to the study of this class of partial differential
solution of a partial differential equation of the                      equation by mathematicians.
hyperbolic form is obtained by the stated method in the                 Various numerical methods have been applied to
form of an infinite series of easily computable terms.
Some examples are given and the solutions obtained by
                                                                        solve hyperbolic partial differential equations. See
the method are found to compare favourably with the                     Ref[4 ]. However, many of these methods, commonly
known exact solutions.                                                  used as characteristic methods require large size of
                                                                        computational work, and round-off error causes loss
Keywords: Hyperbolic partial differential equation,                     of accuracy [2]. The present paper attempts to present
iterative decomposition method, analytical solution.                    a method for the expilict solution of hyperbolic
                                                                        partial differential equation. The results obtained are
                                                                        compared with known analytical solution. For
                          I.    INTRODUCTION                            problems whose analytical solutions are unknown,
   An equation involving two or more partial                            the method will be appropriate.
derivatives of an unknown function of two or more                            The paper is organised as follows: section 2
independent variables is called a Partial Differential                  presents the iterative decomposition method, section
equation.                                                               3 considers the method of section 2 for hyperbolic
The most general second order linear partial                            equations. In section 4, we present examples to
differential equation in two independent variables x                    illustrate the simplicity, efficiency and accuracy of
and y in this paper can be expressed generally in the                   the method. A conclusion is drawn in section 5
form
         +            +         +     +e                  (1)               II THE ITERATIVE DECOMPOSITION
                                                                            METHOD
                                                                           The idea of the iterative decomposition method
             Or                                                         can be conveyed considering equation (1) as an
                                                                        equation in the form
        +         +            = H (x, y, u,      )       (2)
                                                                                                                       (4)
                                                                        Where k is a constant and N(u) is the nonlinear term.
where a, b and c are function of x and y and H is a                     We may find the solution of (4) in a series form such
linear function of u,            ,    .                                 as

Thus equation (2) is called linear, and is a form of                                                              (5)
(1). The second order derivatives occur only to the                      We can decompose the nonlinear operator N as
first degree .If in equation (2), b2_ 4ac >0, the
equation is called Hyperbolic.

    Since the governing equation in many
phenomena     of    engineering,   science,   and                                                                             (6)
mathematical physics lead to hyperbolic equations,



                                                                  307                               http://sites.google.com/site/ijcsis/
                                                                                                    ISSN 1947-5500
                                                             (IJCSIS) International Journal of Computer Science and Information Security,
                                                             Vol. 9, No. 6, June 2011




From equations (5) and (6), equation (4) is then
equivalent to                                                                       =        {-                          }                (15)

                                                      (7)

    From equation (7), we define the following
                                                                              Then, the solution of equation two will be given as
iterative scheme


                (x, y) = N (                         (8)

                                                                              We can determine the component un as far as our
                                                                              desired accuracy of the approximation. The n-terms
                                                                                      can be used to approximate the
m=1, 2, 3...                                                                  solution.
Thus,
                                          (9)
III Application of the iterative Decomposition
Method.                                                                       III       Numerical Experiments.
    Consider the hyperbolic equation (2) with the
following initial conditions;                                                 We shall now consider the application of the iterative
                                                                              decomposition method.
                                         (10)

           (x,0) =g(x)                               (11)                     Example 4. [ ] Consider the hyperbolic partial
                                                                              differential equation
   Using                       , we can put
                                                                                                                                   (16)

                                      ,                                       with the initial condition

                                                     (12)
                                                                                                                                   (17)
in operation form as


            u = H(x,y,u,          ,       )-
                                                                              The exact solution of this problem is
                                                     (13)

The inverse operator of                   denoted           is
                                                                              Putting equation (16) in operator form, we have
defined as

            (.) =                                                                                            -                         (18)

Then                                                                          L=


            u(x,y) = u(x,0) +              (x,0) +           H                Applying the inverse operator L1 to both sides of
                                                                              (18)

(x,y,u,     ,     )-                             }               (14)         where

and



                                                                        308                                http://sites.google.com/site/ijcsis/
                                                                                                           ISSN 1947-5500
                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                     Vol. 9, No. 6, June 2011




                                                                    Then                               .

                                                                    Then we have the exact solution
-(                }                                (19)

 By the iterative decomposition method,

                                                                    after only two terms of the solution series.

                                          -(                        IV     Conclusion

     } dy dy =0                                     (20)                 Our aim in this work has been to derive an
                                                                    approximation algorithm for the explicit solution of
                                                                    hyperbolic partial differential equations. This we
                                                                    have been able to do via a method which does not
                                                                    require elaborate mathematical computation and
    Then the solution is                       which is             assumptions. This method is a significant
the same as the given exact solution.                               improvement over some other decomposition
                                                                    methods, notably the well-known Adomian
 Example 4.2 [ ] consider the equation                              Decomposition Method.
                                                                                               REFERENCES

                                                   (21)
                                                                    [1] G. Adomian, Nonlinear Stochastic Systems and Application to
                                                                    Physics, Kluwer Academic Press. 1989
With the associated initial condition
                                                                    [2] G. Adomian, Solving Frontier Problems of Physics: The
                                                                    Decomposition Method, Kluwer Academic Press. 1994.
      u(x,0) =x                                    (22)
                                                                    [3] J. Biazar, and H. Ebrahimi, An approximation to the solution of
                                                                    hyperbolic equations by Adomian decomposition method and
                                                                    comparison with characteristic method, Appl. Math. Comput. 163,
                                                                    633-638. 2005.

 The exact solution of the problem is                               [4] G. Evens, J. Blackledge, and P. Yardley, Numerical Method for
                                                                    Partial Differential Equations, Springer. 2000.
     The equation may be written in operator form as
                                                                    [5] O. Taiwo, S. Odetunde, and .Y Adekunle, Numerical
                                                                    Approximation of One Dimensional Biharmonic Equations by an
      LU=                                                           Iterative Decomposition Method, J. Math. Sc. 20 (1) 37-44. 2008



Applying the inverse operator           ,we have




  From (14) and (15)




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                                                                                                   ISSN 1947-5500