In engineering mathematics, integral transform is a widely used tool for solving linear differential equations, In recent&nbsp; times &nbsp;the newly born &nbsp;fractional Hankel &nbsp;transform has been started for &nbsp;&nbsp;playing a very important &nbsp;&nbsp;&nbsp;role&nbsp; in various &nbsp;fields of applied &nbsp;mathematics and physics like fractional Fourier transform. This paper represent &nbsp;&nbsp;a formalization of differentiation property of a function invoving&nbsp; high order derivatives&nbsp; of&nbsp; newly introduced &nbsp;&nbsp;&nbsp;&nbsp;fractional Hankel &nbsp;transform. The differentiation property is &nbsp;proved for different higher differential equations

Iqbal, Saleem

Jafar, Sana

Sarwar, Farhana

English

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

  16  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/   Differentiation  Property of  Fractional  Hankel Transform of a Function Involving Higher Order Derivatives Sana Jafara, Saleem Iqbalb*, Farhana Sarwarc a,bDepartment of Mathematics, University of Balochistan, Quetta 87300, Pakistan cDepartment of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan aEmail: mssanajafar@yahoo.com bEmail: saleemiqbal81@yahoo.com cEmail: f_saleem10@yahoo.com   Abstract In engineering mathematics, integral transform is a widely used tool for solving linear differential equations, In recent  times  the newly born  fractional Hankel  transform has been started for   playing a very important    role  in various  fields of applied  mathematics and physics like fractional Fourier transform. This paper represent   a formalization of differentiation property of a function invoving  high order derivatives  of  newly introduced     fractional Hankel  transform. The differentiation property is  proved for different higher differential equations. Keywords: Hankel  Transform; Fractional Hankel  Transform;  Higher order Derivatives; Besssel’s Function. 1. Introduction  The fractional Fourier transform(FrFT) is undoubtedly one of the most valuable and powerful tools now a days in optics, signal communications and applied mathematics. The fractional Fourier transforms was properly introduced by Namias in 1980 and he established the mathematical formulation and identify the eigenvalues eigenfunctions and find its applications in Quantum mechanics and used the differential property to solve the Schrodinger differential equations [1].    ------------------------------------------------------------------------ * Corresponding author American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 17  After the introduction of  FrFT which the generalized form of classical Fourier transform,  McBride and Kerr  contributed interms of mathematical formulation [2] and Almaida applied the FrFT in time frequency plane [3]. Soon after the introduction of fractional order Fourier transform, it is then become a natural thought for the scientists and mathematicians whether other transforms could also be fractionalized or not. Keeping such thoughts the fractional Hankel transform was introduced by Namias and he   derive the fractional integral transforms corresponding to the Hankel transform  [4]. However, in case of fractional Hankel transform the range of the order is        which is unlike to FrFT [1]. The Hankel transformation arises in connection with the radial part of the Laplacian operator expressed in cylindrical polar co-ordinates. The operational calculus based on the conventional Hankel transform is somewhat limited in that most of its practical applications stem from the operational relation.  Fractionalization of the Hankel transforms gives rise to several useful operational relations and applications  [4-9 ]. 1.1 The  Fractional Hankel  transform (FRHT) The Hankel transform  and its inverse is defined by the following pair of Bessel order   [5]  ( )  ∫  ( )    (  )                                 ( )   ( )  ∫  ( )  (  )                                   ( ) Since eqns (1) and (2) are self -reciprocal., therefore the above pair of Hankel transforms can be in a single operator form    ( )      ( )   ∫  (  )  (   )                                  ( ) The classical Hankel transform pair is clearly correspond to angle say      or       In order to define the fractional Hankel transform Namias [4 ] consider the fractional order of  the fractional Hankel as                                                     (4) The classical Hankel transform have the order 1 or -1 and are self-reciprocal. The fractional Hankel transforms are not self-reciprocal and      is established as the inverse transform of  . The range of real fractional order is        unlike fractional Fourier transform whose range covered by        . Namias in 1980[4] and Kerr in 1991[5] worked on the integral representation of the fractional Hankel transform and its inverse transform the pair is give below   American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 18    [  ( )]( )  ∫  ( )    (   )                 ( ) The kernel of the transform is defined as   (   )  {          [    (      )      ] *         +     *         +       (   )                                                                       ( ) and                   *  (     ̂     ) (   )+          ̂                     ( ) 2. Results and Discussions In this section we are presenting the differential properties of higher derivative functions which will be very important to solve the high order differential equations by using the fractional Hankel transform.    2.1 Differentiation property of fractional Hankel  transform of    involing first and 2nd  order derivatives   By inserting      instead of  ( ) in eqn (5) which is the integral representation of fractional Hankel transform eqn  one can easily derive operational relation transform   *     +  given below   [     ]        (  )            (  )           ( )             ( ) Now for the second derivative we replace   by     As       so              Putting the values in eq (8 ) we get   [        ]        ( )            (   )           (  )   [      ]        ( )            (   )           (  )            (  ) Now replace   by    ,   as       so                     Putting the values in eq (8) we get American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 19    [           ]        (   )            ( )           (  )   [       ]       (   )            ( )           (  )                          (   ) Now replace   by      in equation (8) we get   [       ]        (       )            (     )           (    )                                    (  )        Now putting the values of         ,        and      from equation (10), (9) and (8) in eq (11) we get                  *       +      (     (   )            ( )           (  ))         (      ( )            (   )           (  ))          (      (  )            (  )           ( )   )   [       ]         (   )             ( )            (  )             ( )          (   )               (  )           (  )                (  )            ( )  Hence    [       ]         (   )           (  )                 (  )          (   ) (                 )  ( )                               (  )       Results eqns (8) and (12) are similar as in [8], we extended up to higher derivatives   2.2 Differentiation property of fractional Hankel  transform of f  involing higher  order  derivatives   For the third derivative replace   by      in equation  (12) we get American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 20    [       ]         (       )           (      )                 (     )          (      )  (                 )  (    )                               (  ) Now replace   by     in eq  (8)        so                     Hence    [           ]        (  )            (   )           (   )   [       ]        (  )     (  )            (   )           (   )   [       ]        (  )            (   )           (   )             (  )    Now replace   by     in eqn (8)                                                  Hence    [              ]        (   )            (  )           (   )                [        ]        (   )     (   )            (  )           (   )                [        ]       (   )            (  )           (   )                 (  )            Now putting the values from (8), (9), (10), (14) and (15) in eq  (13) we get American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 21    [       ]       *     (   )            (  )           (   )+        *     (   )            ( )           (  )+              *      ( )            (   )           (  )+       *      (  )            (   )           (   )+   (                 ) *      (  )            (  )           ( ) +   [       ]        (   )  *               +  (   )  *                                  +  (  ) *                                                 +  ( ) *                                               +  (  )  *                          +  (   )          (   )   [       ]        (   )  *        +  (   )  *                   +  (  )  *                      +  ( ) *                        +  (  )  *             +  (   )          (   )                    (  ) 2.3 Differentiaon property  involving fourth derivative For the fourth derivative we replace f by      in eq  (16)  American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 22    [       ]        (       )  *        +  (       )  *                   +  (      ) *                      +  (    )  *                        +  (     ) *             +  (      )           (      )            (  )               Now replace   by      and   by      in eq  (8)  respectively        so                        [            ]        (   )            (   )           (   )   [       ]       (   )     (   )            (   )           (   )   [       ]       (   )            (   )           (   )            (  )    For the case                                                         [              ]        (   )            (   )           (   )   [       ]        (   )     (   )            (   )           (   )   [       ]       (   )            (   )           (   )                         (  ) Now putting the values from(8), (9),(10),(14) , (15),(18) and (19) in eq (17) we get American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 23    [       ]      *     (   )            (   )           (   )+ *        + *     (   )            (  )           (   ) + *                   + *     (   )            ( )           (  )+ *                      + *      (  )            (  )           ( )+ *                        + *      ( )            (   )           (  )  + *             + *      (  )            (   )           (   )+        *     (   )            (   )           (   ) +                 [       ]          (   )  *                 +  (   ) *                                        +  (   ) *                                                                 +  (  ) *                                                                              +  ( ) *                                                                            +  (  ) *                                                    +  (   ) *                            +  (   )          (   )  American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 24    [       ]          (   )  *         +  (   )  *          +  (   )  *                        +  (  ) *                                     +  ( )  *                                +  (  ) *                          +  (   )  *              +  (   )         (   )             (  ) 2.4  Operational relation involving 5th derivatives will be established by replacing     by      in equation (20)  and    by     in eq (8)  we get    [       ]          (       )  *         +  (       )  *          +  (       ) *                        +  (      )  *                                     +  (    ) *                                +  (     )  *                          +  (      ) *              +  (      )          (      )         (  ) Now replace   by     in eq (8)        so                        [            ]        (   )            (   )           (   )   [       ]        (   )     (   )            (   )           (   )   [       ]        (   )            (   )           (   )                       (  ) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 25  Now replace   by     in eq (8)                                                   [             ]        (   )            (   )           (   )   [       ]        (   )     (   )            (   )           (   )   [       ]       (   )            (   )           (   )                     (  ) Now putting the values from(8), (9),(10),(14) , (15),(18) , (19),(22) and (23) in eq (21) we get   [       ]        *     (   )            (   )           (   ) + *         + *     (   )            (   )           (   )  + *          + *     (   )            (  )           (   ) + *                        + *     (   )            ( )           (  )  + *                                     + *      (  )            (  )           ( )  + *                                + *      ( )            (   )           (  )+ *                          + *      (  )            (   )           (   )   + *              + *     (   )            (   )           (   ) +       *      (   )            (   )           (   ) +                       American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 26    [       ]           (   )  *                    +  (   )  *                                 +  (   ) *                                                 +  (   ) *                                                                               +  (  ) *                                                                                        +  ( ) *                                                                                                      +  (  ) *                                                                                +  (   ) *                                                    +  (   ) *                             +   (   )            (   )         [       ]           (   )  *          +  (   )  *                       +  (   )  *          +  (   ) *                                      +  (  ) *                                         +  ( ) *                                            +  (  ) *                                +  (   )  *                            +  (   ) *                +   (   )            (   )             (  ) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 27  2.5  Differentiaon property   involving  6th derivative   Replace   by      in equation (24 ) we get and substitution of     and   by     in eq (8) One can established with the similar procedure    [       ]           (   )  *           +  (   )  *                          +  (   ) *                            +  (   )  *                     +  (   ) *                                            +  (  ) *                                                         +  ( ) *                                                   +  (  ) *                                              +  (   ) *                                   +   (   )  *                            +  (   ) *               +  (   )          (   )                             (  )  Thus the  differential property of fractional Hankel  transform for any higher order  derivatives of a function  can be established in a similar way  which will be helpful to solve differential equations of any order. 3. Conclusion  We have  contributed  the mathematical formulation of differential property of  Fractional Hankel transform in very easier manner and extended it  to higher order derivative of a function which will play a vital role for solving differential equations of higher order differential equations. References [1] V. Namias, “The fractional order Fourier transform and its application to quantum  mechanics”, IMA Journal   of Applied Mathematics, 25(3) 241-265(1980)  [2] A.C. McBride and F.H. Kerr. On Namias's fractional Fourier transforms. IMA J. Appl. Math., 39:159-  American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 65, No  1, pp 16-28 28  175, 1987 [3]  L.B. Almeida. The fractional Fourier transform and time-frequency representation.IEEE Trans. Sig. Proc.,  42:3084-3091, 1994 [4]. Namias, V: Fractionalization of Hankel transform. J. Inst. Math. Appl. 26, 187-197 (1980) [5]. Fiona H. Kerr, A Fractional Power Theory for Hankel Transform, in J. Mathematical Analysis and Application, 158, 114-123 (1991). [6]. Prasad, A, Mahato, KL: The fractional Hankel wavelet transformation. Asian-Eur. J. Math. 8(2), (2015) [7]. Sheppard, CJR, Larkin, KG: Similarity theorems for fractional Fourier transforms and fractional Hankel    transforms. Opt.Commun. 154, 173-178 (1998) [8] Taywade, RD, Gudadhe, AS, Mahalle, VN: Generalized operational relations and properties of fractional  Hankel transform. Sci. Rev. Chem. Commun. 2(3), 282-288 (2012) [9]. Taywade, RD, Gudadhe, AS, Mahalle, VN: Initial and final value theorem on fractional Hankel transform.  IOSR J. Math. 5, 36-39 (2013)   

Differentiation  Property of  Fractional  Hankel Transform of a Function Involving Higher Order Derivatives

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Differentiation Property of Fractional Hankel Transform of a Function Involving Higher Order Derivatives

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