The main objective of this paper has to investigate the uniqueness of the solution of fractional differential equation by using different integral transforms, we applied Laplace transform, Elzaki transform and Sumudu transform on a linear ordinary fractional differential equation. The uniqueness of the solution is achieved in fractional differential equations by applying different integral transform methods

Abdul Rehman

Iqbal, Saleem

Rehman, Mariam

Sarwar, Farhana

English

American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS)

  57  American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers  http://asrjetsjournal.org/   On the  Uniquness of  Solutions of  Linear  Ordinary Fractional  Differential  Equations   by Using Different Integral Transform Methods Mariam Rehmana, Saleem Iqbalb*, Farhana Sarwarc, Abdul Rehmand a,b,c,dDepartment of Mathematics, University of Balochistan, Quetta 87300, Pakistan cDepartment of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan aEmail: mariamrehman27@yahoo.com bEmail: saleemiqbal81@yahoo.com cEmail: f_saleem10@yahoo.com dEmail: abdul_maths@yahoo.com   Abstract The main objective of this paper has to investigate the uniqueness of the solution of fractional differential equation by using different integral transforms, we applied Laplace transform, Elzaki transform and Sumudu transform on a linear ordinary fractional differential equation. The uniqueness of the solution is achieved in fractional differential equations by applying different integral transform methods. Keywords: fractional differential equation; Laplace transform; Elzaki transform; Sumudu transform. 1. Introduction  The Fractional Calculus which is the branch of mathematics remained inactive from the 17th century to early 20th century. Fractional Calculus deals with derivative and integrals of arbitrary order, since last three decades the fractional calculus found applications in various areas of studies in applied mathematics and science like fluid flow, rheology, diffusion, oscillation, anomalous diffusion, reaction-diffusion, turbulence, electric network, physics, chemistry, waves, dynamical problems and statistical distribution theory.  ------------------------------------------------------------------------ * Corresponding author American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 58  Now a day’s mathematicians and researcher working with fractional differential equations and discover numerous applications in the areas of applied mathematics, engineering and physics [1]. Various important phenomena in many fields of science such as electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are discussed by fractional differential equations (FDE) [2–4]. Also, fractional differential equations have been found to be effective to describe some. The physical phenomena such as damping laws, rheology, diffusion processes are also well described by fractional differential equations.  There are many methods have been used to solve the fractional differential equations, such as Adomian’s decomposition method (ADM) [5-7], Fourier transform method [8], Laplace transform method [2,3,9], and so on.  2. Results and Discussions               Let us consider a linear fractional differential equation      ( )   (    ) (    )       (    ) (    )     (    ) (  )                                            ( ) where                                        Let us discuss the solution of equation (1) by using Laplace, Sumudu and Elzaki transforms to solve equation (1)  2.1 Solution by using Laplace transform method Laplace transform is an extremely useful method for solving linear ODEs and related initial value problems, as well as systems of linear ODEs, much easier. Fundamental formulas and definitions are given in [10].  In order to get solution of equation (1), we first consider        and    , the equation ( ) takes the form       ⁄  ( )   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )    ⁄                                          ( )    with subject to initial condition                               ( )    Applying the Laplace transform on equation ( )  we get   ( )   ( )     ⁄  ( )   ⁄  ( )   ⁄  ( )   ⁄                                                                           ( ) using  ( )   , we obtain    ( )    ⁄  ( )   ⁄  ( )   ⁄  ( )   ⁄             American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 59     ⁄  ( )   ( )   ⁄  ( )   ⁄  ( )   ⁄             Or         ( )   ( )    ( )    ( )       ⁄     ⁄     ⁄                                                                               ( ) Now using inverse Laplace transform we get the solution   ( )             ⁄ (  ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                                                         ( ) Now For     and    equation ( )  takes the form as follows    ⁄  ( )   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )   ⁄                                   ( ) Subject to initial condition                               ( )    ( )    Applying the Laplace transform on equation ( )  ( ) is obtained as follows    ( )    ( )    ( )     ⁄  ( )   ⁄  ( )   ⁄  ( )   ⁄             Applying the initial condition    ( )   ⁄  ( )   ⁄  ( )   ⁄  ( )   ⁄                ⁄  ( )   ( )   ⁄  ( )   ⁄  ( )   ⁄              ( )   ( )    ( )    ( )        ⁄     ⁄     ⁄ Using inverse Laplace Transform  ( ) is obtained as follows  ( )               ⁄ (  ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )  ( )               ⁄ (  ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                                                 (7)  For     and     equation ( ) takes the form as follows    ⁄  ( )   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )   ⁄                                                          ( ) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 60  subject to initial condition                               ( )    ( )     ( )    Applying the Laplace transform on equation ( )  ( ) is obtained as follows    ( )     ( )    ( )   ( )     ⁄  ( )   ⁄  ( )   ⁄  ( )   ⁄             applying the initial condition    ( )   ⁄  ( )   ⁄  ( )   ⁄  ( )   ⁄                ⁄  ( )   ( )   ⁄  ( )   ⁄  ( )   ⁄              ( )   ( )    ( )    ( )         ⁄     ⁄     ⁄ using inverse Laplace Transform  ( ) is obtained as follows  ( )               ⁄ (   ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                                                                (9) For      and     equation ( ) takes the form     ⁄  ( )   ( ) (   ⁄ )   ⁄   ( ) (  ⁄ )   ⁄   ( ) (  ⁄ )   ⁄                                      (  ) subject to initial condition                              ( )    ( )     ( )      ( )    using the Laplace transform on equation (  )  ( ) is obtained as follows    ( )     ( )     ( )     ( )    ( )   ( )     ⁄  ( )    ⁄  ( )   ⁄  ( )   ⁄             using the initial conditions, we obtain    ( )   ⁄  ( )    ⁄  ( )   ⁄  ( )   ⁄                ⁄  ( )   ( )    ⁄  ( )   ⁄  ( )   ⁄              ( )   ( )    ( )    ( )         ⁄      ⁄     ⁄ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 61  Using inverse Laplace transform  ( ) is obtained as follows  ( )                ⁄ (   ⁄ )    ⁄ (   ⁄ )    ⁄ (  ⁄ )                                       (11) Similarly, for     and    , equation ( ) takes the form    ⁄  ( )   (  ) (   ⁄ )    ⁄   (  ) (   ⁄ )   ⁄   ( ) (  ⁄ )   ⁄                                 (  ) applying the Laplace transform on  (  )  the obtained results is as follows  ( )                 ⁄ (   ⁄ )     ⁄ (   ⁄ )    ⁄ (   ⁄ )                                                     (13)  similarly, for          and   , equation ( )  takes the form       ⁄  ( )   (    ) (     ⁄   )      ⁄     (    ) (     ⁄   )      ⁄   (    ) (     ⁄ )      ⁄                                                                                                                                             (  ) combining results of (5), (7). (9) and (11) we can generalize the solution for all cases  ( )                        (   )      (   )    (   )                                   (15) 2.2 Solution by using Sumudu transform method The Sumudu transform method (STM) was introduced  in 1993, by Watugala  [11-,12] Taking Sumudu transform on equation ( )  ( ) is obtained as follows  ( )   ⁄     ⁄  ( )       ( )   ⁄   ( )   ⁄   ( )    ⁄          Using  ( )   , we get  ( )   ⁄   ( )   ⁄   ( )   ⁄   ( )    ⁄           ( )   ( )    ( )   ( )      ⁄     ⁄     ⁄  using inverse Sumudu transform  ( ) is obtained as follows American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 62   ( )             ⁄ (  ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                                           (  ) using the Sumudu transform on equation ( )  ( ) is obtained as follows  ( )   ⁄    ⁄  ( )       ( )   ⁄   ( )   ⁄   ( )   ⁄          using the initial conditions  ( )   ⁄   ( )   ⁄   ( )   ⁄   ( )   ⁄           ( )   ( )    ( )    ( )       ⁄     ⁄     ⁄  using inverse Sumudu transform on equation  ( ) is obtained as follows  ( )               ⁄ (  ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ ) using the Sumudu transform on equation ( )  ( ) is obtained as follows  ( )   ⁄    ⁄  ( )       ( )   ⁄   ( )   ⁄   ( )   ⁄          Applying the initial conditions  ( )   ⁄   ( )   ⁄   ( )   ⁄   ( )   ⁄              ( )   ( )    ( )    ( )       ⁄     ⁄     ⁄       using inverse Sumudu transform  ( ) is obtained as follows  ( )               ⁄ (   ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                   (  )                    Using the Sumudu Transform on equation (  )  ( ) is obtained as follows  ( )   ⁄    ⁄  ( )       ( )   ⁄   ( )   ⁄   ( )   ⁄                                  Applying the initial conditions American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 63   ( )   ⁄   ( )   ⁄   ( )   ⁄   ( )   ⁄           ( )   ( )    ( )    ( )        ⁄     ⁄     ⁄  using inverse Sumudu transform  ( ) is obtained as follows  ( )                ⁄ (   ⁄ )    ⁄ (   ⁄ )    ⁄ (  ⁄ )                                         (  ) In a similar way as for Laplace transform case, the solution of equation (14) also found as  ( )                        (   )      (   )    (   )                                (  ) 2.3    Solution by using Elzaki transform The Elzaki transform was introduced by Elzaki [13], using Elzaki transform on equation ( )   ( ) is obtained as follows  ( )   ⁄    ⁄       ( )   ⁄   ( )   ⁄   ( )   ⁄            Applying the initial conditions  ( )   ⁄   ( )   ⁄   ( )   ⁄   ( )   ⁄             ( )   ( )    ( )    ( )        ⁄     ⁄     ⁄  Using inverse Elzaki Transform  ( ) is obtained as follows  ( )               ⁄ (  ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                               (  ) applying Elzaki transform on equation ( )  ( ) is obtained as follows  ( )   ⁄    ⁄       ( )    ⁄   ( )   ⁄   ( )   ⁄            applying the initial conditions  ( )   ⁄   ( )    ⁄   ( )   ⁄   ( )   ⁄            American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 64   ( )   ( )    ( )    ( )        ⁄      ⁄     ⁄  Using the inverse Elzaki Transform  ( ) is obtained as follows  ( )               ⁄ (   ⁄ )    ⁄ (  ⁄ )    ⁄ (  ⁄ )                  (  ) applying Elzaki transform on equation ( )  ( ) is obtained as follows  ( )   ⁄    ⁄       ( )    ⁄   ( )    ⁄   ( )   ⁄            using the initial conditions  ( )   ⁄   ( )    ⁄   ( )    ⁄   ( )   ⁄             ( )   ( )     ( )    ( )        ⁄      ⁄      ⁄  Using the inverse Elzaki transform  ( ) is obtained as follows  ( )                ⁄ (   ⁄ )    ⁄ (   ⁄ )    ⁄ (  ⁄ )                            (  ) In a similar way the equation (14) also gives the following solution after applying the Elzaki transform  ( )                        (   )      (   )    (   )                                            (  ) Equations (15), (19) and (23) are same, which shows that the fractional differential equations have found  the unique solutions after applying different transforms methods             3. Conclusion Fractional differential also have the unique solution if the same fractional differential equation solved by different integral transforms methods in a similar manner as in case of classical differential equations.  References [1] B. J. West, M. Bolognab, and P. Grigolini, Physics of  fractal operators, Springer, New York 2003. [2] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York 1993. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 57-65 65  [3] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential  equations, to methods of their solution and some of their applications, Academic Press, New York 1999. [4] S.G. Samko, A.A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: theory and applications,   Gordon and Breach, Amsterdam 1993. [5] S. Abbasbandy, J. Comput. Appl. Math. 207, 53 (2007). [6] G. Adomian, J. Math. Anal. Appl. 55, 441 (1976). [7] S.Momani and N. T. Shawagfeh, Appl.Math. Comput.182, 1083 (2006). [8] S. Kemple and H. Beyer, Global and causal solutions of fractional differential equations, Transform methods    and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP), Singapore, 96, 210   (1997). [9] L. Debnath and D. Bhatta, Integral transforms and their applications (second edition), Chapman and  Hall/CRC, Boca Raton 2007. [10] B. Eugene, Mathematical Physics, Addison Wesley, 1968. [11] G. K. Watugala, ―Sumudu transform: a new integral transform to solve differential equations and control  engineering problems,‖ International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, pp. 35–43, 1993. [12] G. K.Watugala, ―Sumudu transform—a new integral transform to solve differential equations and control     engineering problems,‖ Mathematical Engineering in Industry, vol. 6, no. 4, pp.319–329, 1998. [13] T. M. Elzaki and S. M. Elzaki.  ―The new transform Elzaki Transform‖, Global Journal of Pure and Applied Mathematics, 1: 57-64, 2011.  

On the  Uniquness of  Solutions of  Linear  Ordinary Fractional  Differential  Equations   by Using Different Integral Transform Methods

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On the Uniquness of Solutions of Linear Ordinary Fractional Differential Equations by Using Different Integral Transform Methods

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