In this paper we have discussed fractional Sumudu transforms and its relationship with fractional Fourier transform and obtained the mathematical expression of kernel of fractional Sumudu transform. Such findings &nbsp;will play a significant role for fractional Sumudu transform to recognize its importance in the&nbsp; fields of engineering and applied mathematics &nbsp;like other fractional transforms

Iqbal , Saleem

Rehman, Abdul

Saeed , Sadaf

Sarwar , Farhana

English

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

  41   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 Relationship Between the Fractional Sumudu Transform and Fractional Fourier Transform Sadaf Saeeda, Saleem Iqbalb*, Farhana Sarwarc, Abdul Rehmand a,b,dDepartment of Mathematics, University of Balochistan, Quetta 87300, Pakistan cDepartment of Mathematics F.G.Girls Degree College, Madrissa Road, Quetta, Cantt, 87300, Pakistan aEmail: sadafimran6zg@gmail.com bEmail: saleemiqbal81@yahoo.com cEmail: f_saleem10@yahoo.com cEmail: rehman_maths@hotmail.com   Abstract In this paper we have discussed the kernel of fractional Sumudu transforms and its relationship with fractional Fourier transform which will play a significant role for fractional Sumudu transform to recognize its importance like other fractional transforms. Keywords:  Sumudu Transform; Fractional Sumudu transform; Fractional Laplace Transform; Fractional Mellin Transform; Fractional Fourier Transform. 1.  Introduction   The Sumudu transform which is introduced in 1993[1] is not as well-known as other transforms like Laplace, Fourier, Mellin, Hilbert and other transforms. Sumudu transform carries some advantages as compare to other transform and it is found capable in solving complicated problems. The fundamental properties of Sumudu transform   were established in [2] and subsequently applied to partial differential equations [3-5]. The Sumudu transform which is equivalent to the Laplace transform also leads to a series of properties in a similar way like Laplace transform. The Sumudu transform may be used to solve problems without resorting to a new frequency domain [6-7].   ------------------------------------------------------------------------ * Corresponding author. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 74, No  1, pp 41-45   42  In 2003, Belgacem and his colleagues [8] have shown it to be the theoretical dual to the Laplace transform, the relationship of Sumudu transform with Laplace transform is discussed in [9]. The inverse Sumudu transform is also exist and discussed in [10]. The idea of fractional transforms attracted the researchers after the introduction of fractional Fourier transform by Namias in 1980 [11]. Now a days  the mathematicians, physicists are paying their attention not only on fractional Fourier transforms but also working on  many other fractional integral transforms and   provided well-established and valuable methods for solving problems in many areas of applied mathematics, physics, optics, signal processing, quantum mechanics through integral transforms techniques. Fractional Sumudu transform were also introduced which is the generalization of Sumudu transform [12]. Still lot of properties of fractional Sumudu transform needs to find out, so researchers are also paying their attention to this newly introduced transform. In this paper we have established the relationship of fractional Sumudu transform with fractional Fourier transforms and find out the formula for kernel of fractional Sumudu transform. 2. Results and Discussions In this section we are presenting the kernel  of fractional Sumudu transform  and its relationship with  fractional Fourier transform,  which can play a significant role in solving  complicated problems  arising in signal processing and other fields of applied mathematics and will also helpful to obtained the solutions of  fractional differential equation. 2.1. The Kernel of fractional Sumudu transform (FRST) The fractional Laplace transform are discussed in [13]. And defined as    * ( )+   ̅ ( )  ∫  ( )  (   )                                      ( )            Where   (   ) is called kernel and it is defined as   (   )    √                (              )                                                       ( ) The classical ST is defined as   ( )  , ( )-    ∫   .  /   ( )                               ( ) provided the integral exists for some u. where  is the Sumudu transform operator. Since there is a close relation between Laplace and Sumudu transform the relationship can be written as  (  )   , ( )-   ∫        ( )     ( )          ( ) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 74, No  1, pp 41-45   43  Where  ( ) is the Laplace transform of  ( ) The fractional Sumudu transform discussed and defined in the following manner in [12]     (  )      ( )               ( ) or                                                       ( )       .  /                       ( ) where    is the fractional Laplace transform and defined as     ( )  √         ∫  ( )       (     )                 ( ) From equations (6) and (7) the integral formula of fractional Sumudu transform is defined as    ( )  √            ∫  ( )       .   (  ) /                ( )       (    )      √                .   (  ) /                 ( ) The kernel of fractional Sumudu transform is defined in equation (8) Since there is a close relation between fractional Laplace and fractional Sumudu transform [13] and the relationship between FRST with FRFT discussed in this article the kernel properties will be followed as discussed in [14].  2.2. The relationship between fractional Sumudu transform with fractional Fourier Transform          Since fractional Fourier transform of  ( ) is defined as [14]   * ( )+   ( )̅̅ ̅̅ ̅̅  ( ) √         ∫  ( )       (     ) (  )           whe    is   t   u ti  e  f                        (  ) Replace    by       in eqn (8) we get,    .   /     ̅ ( )    √         ∫  ( )       (     )                  since    is complex therefore substituting         in equation (6) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 74, No  1, pp 41-45   44  we get   (      )  (    ) √         ∫  ( )       (   (    ) ) (    )           √         ∫  ( )       (             ) (    )                    (       )√         (    )∫  ( )               (     ) (  )                    (       )       (    ) [√         ∫ ( ( )        )       (     ) (  )          ]              (  ) using (8) and (10) we get the result from eqn (10)                                  * ( )+         (       )       (    )   * ( )        +                                                      (  ) Eq (12) shows the relationship of fractional Sumudu transform with fractional Fourier transform  3. Conclusion We have established relationship between  fractional Sumudu transform and  fractional Fourier  transform and find the of Kernel of fractional Sumudu which will play a significant role  to recognized the importance of fractional Sumudu transform as compare  to  fractional Fourier transform.    Reference [1]. G. K. Watugala, “Su udu tr  sf r :    ew i tegr   tr  sf r  t  solve differential equations and c  tr   e gi eeri g  r b e s” I t. J. Math. Educ. Sci. Technol., Vol. 24, No. 1, 35-43 , 1993 [2]. F.B.M Belgacem, A.A Karaballi “Sumudu transform fundamental properties investigations and applications.  International Journal of Stochastic Analysis. Vol. 2006; 1-23, 2006. [3]. M. A. Asiru, “Further  r  erties  f the Su udu tr  sf r    d its     ic ti  s,” I ter  ti     J ur     of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441–449, 2002. [4]. A. Kı ıc¸      dH. E t yeb, “A   te    i tegr   tr  sf r s   d   rti   differe ti   equ ti  s,” Applied Mathematical Sciences, vol. 4, no. 1–4, pp. 109–118, 2010. [5]. A. K,¸   , H. E t yeb,   d P. R. Ag rw  , “O  Su udu tr  sf r    d syste   f differe ti   equ ti  s,” Abstr ct   d A   ied A   ysis, v  . 2010,  2010. [6]. Keji Li, Y. X. (). A brief introduction of Sumudu transform and comparison with other integral  American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 74, No  1, pp 41-45   45  transforms. IEEE, 285-287. 2011, [7]. V. G. Gu t    d B. Sh r  , “A   ic ti    f Su udu tr  sf r  i  reaction-diffusion systems and     i e r w ves,” A   ied M the  tic   Scie ces, v  . 4,   . 9–12, pp. 435–446, 2010. [8]. F. B. M. Be g ce , A. A. K r b   i,   d S. L. K    , “A   ytic   i vestig ti  s  f the Su udu transform    and applications to integral producti   equ ti  s,” M the  tic   Pr b e s i  E gi eeri g, no. 3-4, pp.  103–118, 2003 .  [9]. M. Baloch., S. Iqbal,  The Relationship of Fractional Laplace Transform with Fractional Fourier, Mellin and Sumudu Transforms. American Scientific Research Journal for Engineering, Technology, and  Sciences (ASRJETS), 55(1), pp.11-16., 2019 [10]. R. Silambarasan, K.S Nisar, et al “Discrete inverse Sumudu transform application to Whittaker and Zettl  equations. AIP Conference Proceedings ,Vol. 2046, No. 1, p. 020091, 22018 [11]. V. N  i s, “The fr cti      rder Fourier transform and its application to quantum  ech  ics”, IMA Journal of Applied Mathematics, 25(3), 241-265, 1980. [12]. Gupta VG, Shrama B, Kiliçman A. A note on fractional Sumudu transform. Journal of Applied Mathematics, 2010 [13]. K.K Sharma. Fractional Laplace transform, Springer-Verlag London, 377-379, 2010 [14]. L.B. A  eid , “The fr cti     F urier tr  sf r    d ti e-freque cy re rese t ti  s”, IEEE Transactions  on Signal Processing, 42(11) ,  3084–3091, 1994   

On the Relationship Between the Fractional Sumudu Transform and Fractional Fourier Transform

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On the Relationship Between the Fractional Sumudu Transform and Fractional Fourier Transform

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