11 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ The Relationship of Fractional Laplace Transform with Fractional Fourier, Mellin and Sumudu Transforms Maria Balocha, 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: maria.umbreen@yahoo.com bEmail: saleemiqbal81@yahoo.com cEmail: f_saleem10@yahoo.com eEmail: abdul_maths@yahoo.com Abstract We have developed in this research paper, some of the fundamental relationship of fractional Laplace transform with fractional Fourier, fractional Mellin and fractional Sumudu transforms. These results are expressed mathematically, and such relationships should be very useful in applications to signal processing and optics. Keywords: Fractional Laplace Transform; Fractional Mellin Transform; Fractional Fourier Transform; Fractional Sumudu transform. 1. Introduction Fractional integral transforms provide a well-established and valuable method for solving problems in many areas of applied mathematics, physics like optics, signal processing, quantum mechanics. Since the introduction of fractional Fourier transform by Namias in 1980 [1], the applied mathematicians, physicists are paying their attention not only on fractional Fourier transforms but also working on many other transforms like fractional Hilbert transform fractional Mellin transform fractional Laplace transform and fractional Sumudu transform. In recent times the fractional integral transforms have become a very important tool and are playing a key role in various branches of applied mathematics and physics. Fractional Laplace transform was introduced by many researchers in their research articles in different ways, in 2003 it was first defined by A. Torre as a special case of canonical transform with characteristic matrix and its relation to canonical transform, and parabolic differential equations are discussed in [2], the properties of fractional Laplace transform are also developed [3]. ----------------------------------------------------------------------- * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 11-16 12 In 2009 Guy Jumarie developed a new form of fractional Laplace transform with the Mittage Leffler function for the entire class of functions that are fractional differentiable [4]. In 2010 in K. K. Sharma defined fractional Laplace Transform as a special case of linear canonical transform with representative matrix and used it in problems [5]. The convolution structure for the two versions of fractional Laplace transform was developed [6]. Various of fractional Laplace transform properties are discussed which are useful in application to differential and integral equations or problems in non-extensive statistical mechanics [7]. In this paper we have established the relationship of fractional Laplace transform with other transforms. The fractional Laplace transform is defined in [ 5] ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = πœ™πœ™οΏ½π›Όπ›Ό(𝑠𝑠) = οΏ½ πœ™πœ™(𝑑𝑑)π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒) ∞ βˆ’βˆž 𝑑𝑑𝑑𝑑 (1) Where π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒) is called kernel and it is defined as π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒) = οΏ½ 1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ 𝑒𝑒𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2βˆ’π‘ π‘ 2+2𝑖𝑖𝑠𝑠𝑑𝑑 sec𝛼𝛼� (2) This reduces into classical Laplace transform if we put 𝛼𝛼 = πœ‹πœ‹ 2 . The inversion formula for fractional Laplace transform can be obtained by replacing 𝛼𝛼 by βˆ’π›Όπ›Ό in equations (1) and (2) Properties of kernel function of Laplace transform stated in [3] by Gudadhe are given as follows ⎩ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎧ π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒) = π‘˜π‘˜π›Όπ›Ό(𝑒𝑒, 𝑑𝑑) π‘˜π‘˜βˆ’π›Όπ›Ό(𝑑𝑑,𝑒𝑒) = π‘˜π‘˜π›Όπ›Ό βˆ—(𝑑𝑑,𝑒𝑒) π‘˜π‘˜π›Όπ›Ό(βˆ’π‘‘π‘‘,𝑒𝑒) = π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,βˆ’π‘’π‘’) οΏ½ π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒)π‘˜π‘˜π›½π›½(𝑒𝑒, 𝑧𝑧)𝑑𝑑𝑒𝑒 ∞ βˆ’βˆž = π‘˜π‘˜π›Όπ›Ό+𝛽𝛽(𝑑𝑑, 𝑧𝑧) (3) οΏ½ π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒) ∞ βˆ’βˆž π‘˜π‘˜π›Όπ›Ό βˆ—(𝑑𝑑,𝑒𝑒)𝑑𝑑𝑑𝑑 = 𝛿𝛿(𝑒𝑒 βˆ’ 𝑒𝑒′) The kernel which is defined in eq (3) has the similar expressions as Almeida defined [8]. 2. Results and Discussions In this section we are presenting some important relationship of fractional Laplace transform with other transform like fractional Fourier transform, fractional Mellin transform and fractional Sumudu transform which can play a significant role in signal processing and other fields of applied mathematics 2.1. The relationship between fractional Laplace transform with fractional Fourier transform Since fractional Fourier transform of πœ™πœ™(𝑑𝑑) is defined as [8] American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 11-16 13 ℱ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = πœ™πœ™οΏ½π›Όπ›Ό(πœ”πœ”) = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2+πœ”πœ”2οΏ½βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑, ∞ βˆ’βˆž when 𝛼𝛼 is not a multiple of πœ‹πœ‹ (5) The fractional Laplace transform is defined as after combining equations (1) and (2) ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = πœ™πœ™οΏ½π›Όπ›Ό(𝑠𝑠) = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2βˆ’π‘ π‘ 2+2𝑖𝑖𝑠𝑠𝑑𝑑 sec 𝛼𝛼� ∞ βˆ’βˆž 𝑑𝑑𝑑𝑑, when 𝛼𝛼 is not a multiple of πœ‹πœ‹ (6) since 𝑠𝑠 is complex therefore substituting 𝑠𝑠 = 𝜎𝜎 + π‘–π‘–πœ”πœ” in in equation (6) we get ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = πœ™πœ™οΏ½π›Όπ›Ό(𝜎𝜎 + π‘–π‘–πœ”πœ”) = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒 𝑖𝑖 cot𝛼𝛼 2 �𝑑𝑑2βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)2οΏ½βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2βˆ’πœŽπœŽ2+πœ”πœ”2βˆ’2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž = π‘’π‘’βˆ’ 𝑖𝑖 cot𝛼𝛼 2 �𝜎𝜎2+2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ csc 𝛼𝛼𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2+πœ”πœ”2οΏ½βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž = π‘’π‘’βˆ’ 𝑖𝑖 cot 𝛼𝛼 2 �𝜎𝜎2+2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½ οΏ½οΏ½ 1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ (πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ csc 𝛼𝛼)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2+πœ”πœ”2οΏ½βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž οΏ½ (7) From (5) and (7) we get the result ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = π‘’π‘’βˆ’ 𝑖𝑖 cot𝛼𝛼 2 �𝜎𝜎2+2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½β„±π›Όπ›Ό{πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ csc 𝛼𝛼} (8) Eq (8) shows the relationship of fractional Laplace transform with fractional Fourier transform and this relation will reduces to classical relation of Laplace transform and Fourier transform if we put 𝛼𝛼 = πœ‹πœ‹ 2 . 2.2. Relation between Fractional Laplace Transform and Fractional Mellin Transform In order to develop a relation of FRLT with FRMT let us consider the change of variable which is defined by π‘’π‘’βˆ’π‘‘π‘‘ = 𝑦𝑦 Replacing 𝑑𝑑 by π‘’π‘’βˆ’π‘‘π‘‘ in equation (6) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 11-16 14 we get ℒ𝛼𝛼{πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)}(𝑠𝑠) = οΏ½1βˆ’π‘–π‘– cot 𝛼𝛼 2πœ‹πœ‹ ∫ πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)𝑒𝑒 𝑖𝑖cot𝛼𝛼 2 �𝑑𝑑2βˆ’π‘ π‘ 2οΏ½βˆ’π‘ π‘ π‘‘π‘‘ cscπ›Όπ›Όπ‘‘π‘‘π‘‘π‘‘βˆž βˆ’βˆž (9) Substituting π‘’π‘’βˆ’π‘‘π‘‘ = 𝑦𝑦 ⟹ 𝑑𝑑𝑑𝑑 = βˆ’π‘‘π‘‘π‘‘π‘‘ 𝑑𝑑 When 𝑑𝑑 β†’ βˆ’βˆž then 𝑦𝑦 β†’ ∞ , when 𝑑𝑑 β†’ ∞ then 𝑦𝑦 β†’ 0 in equation (6) we get = βˆ’οΏ½ 1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑦𝑦)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 οΏ½ln𝑑𝑑2βˆ’π‘ π‘ 2οΏ½+𝑠𝑠 𝑐𝑐𝑠𝑠𝑐𝑐 𝛼𝛼 ln 𝑑𝑑 𝑑𝑑𝑦𝑦 𝑦𝑦 0 ∞ = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑦𝑦)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 οΏ½ln𝑑𝑑2βˆ’π‘ π‘ 2οΏ½+ln 𝑑𝑑𝑠𝑠csc𝛼𝛼 𝑑𝑑𝑦𝑦 𝑦𝑦 ∞ 0 = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑦𝑦)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 οΏ½ln𝑑𝑑2βˆ’π‘ π‘ 2�𝑦𝑦𝑠𝑠 csc π›Όπ›Όβˆ’1𝑑𝑑𝑦𝑦 (10) ∞ 0 In [3] fractional Mellin transform is defined as ℳ𝛼𝛼{πœ™πœ™(𝑦𝑦)} = οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑦𝑦)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 οΏ½ln 𝑑𝑑2βˆ’π‘ π‘ 2�𝑦𝑦𝑠𝑠 csc π›Όπ›Όβˆ’1𝑑𝑑𝑦𝑦 ∞ 0 (11) From (10) and (11) we get the relation between fractional Laplace transform and fractional Mellin transform ℒ𝛼𝛼{πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)} = ℳ𝛼𝛼{πœ™πœ™(𝑦𝑦)} This reduces to classical relation between Laplace transform Mellin transform for 𝛼𝛼 = πœ‹πœ‹ 2 . 2.3. Relation between Fractional Laplace Transform and Fractional Sumudu Transform In [10] the duality relation between two sided Laplace transform and two sided Sumudu transform is given by 𝐺𝐺(𝑒𝑒) = 1 𝑒𝑒 𝐹𝐹(𝑠𝑠)οΏ½ 𝑠𝑠=1𝑒𝑒 (12) Or 𝐹𝐹(𝑠𝑠) = 1 𝑠𝑠 𝐺𝐺(𝑒𝑒)| 𝑒𝑒=1𝑠𝑠 (13) Where 𝐺𝐺(𝑒𝑒) = 𝑆𝑆{𝑓𝑓(𝑑𝑑)} and 𝐹𝐹(𝑠𝑠) = β„’{𝑓𝑓(𝑑𝑑)} are the Sumudu and Laplace transform respectively American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 11-16 15 The generalized fractional Sumudu transform can be established by using the duality relation between Laplace transform and Sumudu transform as defined in a similar way as defined in equations (12) and (13)[11] Let πœ™πœ™(𝑑𝑑) be of exponential order and ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = πœ™πœ™οΏ½π›Όπ›Ό(𝑠𝑠) and 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑)} = 𝐺𝐺𝛼𝛼(𝑒𝑒) then 𝐺𝐺𝛼𝛼(𝑒𝑒) = 1 𝑒𝑒 πœ™πœ™οΏ½π›Όπ›Ό οΏ½ 1 𝑒𝑒 οΏ½ = 1 𝑒𝑒 οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2βˆ’(1𝑒𝑒)2οΏ½βˆ’π‘‘π‘‘π‘’π‘’ csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž (14) and 𝐺𝐺𝛼𝛼(𝑒𝑒) = 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑)} = 1 𝑒𝑒 οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2βˆ’(1𝑒𝑒)2οΏ½βˆ’π‘‘π‘‘π‘’π‘’ csc𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž (15) Substituting 𝑒𝑒 = 1 𝑠𝑠 we get 𝐺𝐺𝛼𝛼 οΏ½ 1 𝑠𝑠 οΏ½ = 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑)} = 𝑠𝑠� 1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)𝑒𝑒 𝑖𝑖 cot𝛼𝛼 2 �𝑑𝑑2βˆ’(𝑠𝑠)2οΏ½βˆ’π‘ π‘ π‘‘π‘‘ csc𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž β†’ (16) From (16) and (1) we get the relation of fractional Laplace transform with fractional Sumudu transform 1 𝑠𝑠 𝐺𝐺𝛼𝛼 οΏ½ 1 𝑠𝑠 οΏ½ = πœ™πœ™οΏ½π›Όπ›Ό(𝑠𝑠) (17) 3. Conclusion We have established mathematically the relationship of the fractional Laplace transform with fractional Mellin transform and Fractional Fourier transform which will play a significant role in signal processing, optics and other field of applied mathematics, physics and engineering References [1] V. Namias, β€œThe fractional order Fourier transform and its application to quantum mechanics”, IMA Journal of Applied Mathematics, 25(3) (1980) 241-265 [2] A. Torre. (2003). Linear and Radial Canonical Transforms of Fractional Order. Computational and Applied Mathematics, 477-486. [3] P. R.Gudadhe, Analytical study of a special case of complex canonical transform. Global Journal of Mathematical Sciences, (2010) 293-302 [4] G.Jumarie, Fractional Laplace transform with Mittage Leffler function. Applied Mathematics, (2009). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 11-16 16 [5] K.K Sharma. Fractional Laplace transform, Springer-Verlag London Limited, 4, (2010) 377-379 [6] R.Prabhakar, Deshmukhi, S.D. Alka, Convolution structure for two version of fractional Laplace Transform, Journal of Science and Arts, 2(15),(2011) 143-150. [7] R. A. Treumann, W. Baumjohann, Fractional Laplace transform-A Perspective, Frontier Physics, 2,(2014). 1-4. Institute for Nonlinear Science. Springer, New York, NY (2003) [8] L.B. Almeida, β€œThe fractional Fourier transform and time-frequency representations”, IEEE Transactions on Signal Processing, 42(11) (1994) 3084–3091 [9] A. Peng, Y. Wang, X. Zuo, L. Gong, Properties of fractional Mellin transform, Advances in Information Sciences and Services Sciences 5(11) (2013) 90-96. [10] Keji Li, Y. X. (2011). A brief introduction of Sumudu transform and comparison with other integral transforms. IEEE, 285-287. [11] Gupta, V.J et el. (2010). A note of fractional Sumudu Transform, journal of Applied Mathematics Research Article (9 pages), Article ID 154189, Volume 2010. https://arxiv.org/find/physics/1/au:+Treumann_R/0/1/0/all/0/1 https://arxiv.org/find/physics/1/au:+Baumjohann_W/0/1/0/all/0/1