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 Jafar a , Saleem Iqbal b *, Farhana Sarwar c a,b Department of Mathematics, University of Balochistan, Quetta 87300, Pakistan c Department of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan a Email: mssanajafar@yahoo.com b Email: saleemiqbal81@yahoo.com c Email: 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 2 nd 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 5 th 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 6 th 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)