International Journal of Trend in Scientific Research and Development (IJTSRD) International Open Access Journal ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 2 | Issue – 5 Fractional Quadruple Laplace Transform and its Properties 1 Monisha. G1, Savitha. S2 Research Scholar, 2Faculty of Mathematics Department of Mathematics, Vivekanandha College of Arts and S Sciences ciences for Women [Autonomous], Trichengode, Namakkal Namakkal, Tamil Nadu, India ABSTRACT In this paper, we introduce definition for fractional quadruple Laplace transform of order α,0 < α ≤ 1, for fractional differentiable functions. Some main properties and inversion theorem of fractional quadruple Laplace transform are established. Further, the connection between fractional quadruple Laplace transform andd fractional Sumudu transform are presented. KEYWORD: quadruple Laplace transform, Sumudu transform, fractional Difference. INTRODUCTION There are different integral transforms in mathematics which are used in astronomy, physics and also in engineering. The integral transforms were vastly applied to obtain the solution of differential equations; therefore there are different kinds of integral transforms like Mellin, Laplace, and Fourier and so on. Partial differential equations are considered one of the most ost significant topics in mathematics and others. There are no general methods for solve these equations. However, integral transform method is one of the most familiar methods in order to get the solution of partial differential equations [1, 2]. In [3, 9] quadruple Laplace transform and Sumudu transforms were used to solve wave and Poisson equations. Moreover the relation between them and their applications to differential equations have bee been determined and studied by [5, 6]. In this study we focus on quadruple druple integral determined and studied by [5, 6]. In this study we focus on quadruple integral transforms. First of all, we start to recall the definition of quadruple Laplace transform as follows follows. Definition: Let be a continuous function of four variables; then the quadruple Laplace transform of , , , is defined by (, ), * , , , , (1) ∞ ∞ ∞ ∞ & , , , 9999 Where , , , 0 variables, and , , , and , (, ), *are Laplace %!"∞ !"∞ :!"∞ 1 1 $ 1 $ 1 $ # # # 2 2 2 2 "∞ '!"∞ %"∞ :"∞ & , (, ), * $9;$9;$9;9 '"∞ is the inverse Laplace transform. Fractional Derivative via Fractional Difference Definition: let +, be a continuous function, but not necessarily differentiable function, then the forward operator -(h) (h) is defined as follows -h +, +, . / , Where / 0 denote a constant discretization span. Moreover, the fractional difference of +, is known as ∆ +, - 1 / +, ∞ 6 2 11 3 5 8 +, + . 6 1 7 / 7 34 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5| Jul-Aug Aug 2018 Page: 2404 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Where 0 < 6 < 1, And the α-derivative of+(,) is known as △ +(,) +( ) (,) = lim A↓ ℎ See the details in [9, 10]. Fractional quadruple Sumudu Transform Recently, in [13] the author defined quadruple sumudu transform of the function depended on two variables. Analogously, fractional quadruple sumudu transform was defined and some properties were given as the following Modified Fractional Riemann-Liouville Derivative The author in [10] proposed an alternative definition of the Riemann-Liouville derivative Definition: let +(,) be a continuous function, but not necessarily differentiable function, then Let us presume than +(,) = D , where D is a constant, thus α-derivative of the function +(,) , 6 ≤ 0, $ DΓH (1 − 6)t isEF D = G 0, K,ℎ)*. On the other hand, when +(,) ≠ D hence +(,) = +(0) + N+(,) − +(0)O, and fractional derivative of the function +(,) will become known as +( ) (,) = EF +(0) + EF N+(,) − +(0)O, at any negative α, (α < 0) one has EF N+(,) − +(0)O F 1 = (, − P) Γ(−α) H +(P)9P, 6 < 0, While for positive α, we will put EF N+(,) − +(0)O = EF +(,) = EF N+( H) O. When 7 < 6 < 7 + 1, we place +( ) (,) = (+( 3) (,))(3) , 7 ≤ 6 < 7 + 1, 7 ≥ 1. Integral with Respect to (RS)T The next lemma show the solution of fractional differential equation 9 = +(,)(9,) , , ≥ 0, (0) = 0 By integration with respect to (9,) (2) Lemma: If +(,) is a continuous function, so the solution of (2) is defined as the following H (,) = +(P)(9P) , (0) = 0 H = 6 (, − P) +(P)9P, 0 < 6 < 1 ∞ ∞ ∞ ∞ = [ (−( + + + ,) )(, (, ), *,)(9) (9) (9) (9,) Where, (, ), * ∈ ℂ, , , , , > 0 and 3 [ () = 2 34 Γ(αm + 1) is the Mittag-Leffler function. ∞ Some Properties of Fractional Quadruple Sumudu Transform We recall some properties of Fractional Quadruple Sumudu Transform V W X (^, _, `, 9,)Y = Z W (^, _(, `), 9*), V W X ( − ^, − _, − `, , − 9)Y = [ (−(^ + _ + ` + 9) )Z W (^, _(, `), 9*), V W XaF (^, _, `, 9,)Y Z W (, (, ), *) − Γ(1 + 6)(0, , , ) (4) = b where aF is denoted to fractional partial derivative of order α (see [10]). Main Results The main results in this work are present in the following sections Quadruple Laplace Transform of Fractional order Definition 6: If (, , , ,) is a function where , , , , > 0, then Quadruple Laplace Transform of Fractional order of (, , , ,) is defined as W X(, , , ,)Y = W (, (, ), *) (5) ∞ ∞ ∞ ∞ = [ (−( + ( + ) H Definition: [14] The fractional quadruple sumudu transform of function (, ,) is known as V W X (, , , ,)Y = Z W (, (, ), *), (3) For more results and varies views on fractional calculus, see for example [15, 16, 17, 18, 19, 20, 21] +*,) )(, , , ,)(9) (9) (9) (9,) Where , (, ), * ∈ ℂ and [ () is the Mittag-Leffler function. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5| Jul-Aug 2018 Page: 2405 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Corollary 10.1 By using the Mittag-Leffler property then we can rewrite the formula (5) as the following: W X(, , , ,)Y = W (, (, ), *) = [ (−() )[ (−( )[ (−()) ) ∞ ∞ ∞ ∞ [ (−(*,) )(, , , ,)(9) (9) (9) (9,) (6) Remark 8: In particular case, fractional quadruple Laplace transform (5) turns to quadruple Laplace transform (1) when α = 1. Some properties of fractional quadruple Laplace transform In this section, various properties of fractional quadruple Laplace transform are discussed and proved such as linearity property, change of scale property and so on. 1. Linearity property Let H (, , , ,) and d (, , , ,) be functions of the variables and ,, then W X^ H H (, , , ,) + ^d d (, , , ,)Y =^H W XH (, , , ,)Y + ^d W Xd (, , , ,)Y(7) Where ^H and ^d are constants. Proof: We can simply get the proof by applying the definition. 2. Changing of scale property W X(, If , , ,)Y = W (, (, ), *) W X(^, _, `, 9,)Y 1 ( ) * = W 5 $ , , , $ 8 ^ _ ` 9 ^ _ ` 9 ∞ ∞ hence ∞ = [ (−( + ( + ) + *,) )(^, _, `, 9,) (9) (9) (9) (9,) (8) We set f = ^ , g = _, h = ` and 7 = 9, in the above equality (8), therefore we obtain W X(^, _, `, 9,)Y ∞ ∞ ∞ ∞ 1 f (g )h [ i− i + + ^ _ ` 9 ^ _ ` *7 + $ 8 j$ (f, g, h, 7)(9) (9) (9) (9,) 9 = 1 ( ) * W 5 $ , , , $ 8 ^ _ ` 9 ^ _ ` 9 3. Shifting property W X(, Let , , ,)Y = W (, (, ), *) then W X[ (−( + ( + ) + *,) )(, , , ,)Y = W ( + ^, ( + _, ) + `, * + 9) Proof: W X[ (−(^ + _ + ` + 9,) )(, , , ,)} ∞ ∞ ∞ ∞ = [ (−(^ + _ + ` + 9,) )[ (−( + ( + ) + *,) ) (, , , ,)(9) (9) (9) (9,) By using the equality X[ (k( + + + ,) ) =[ (k ) [ (k )[ (k )[ (k, ) Which implies that W X[ (−(^ + _ + ` + 9,) )(, , , ,) = [ (−((^ + ) + (_ + () + (` ∞ ∞ ∞ ∞ + ))(9 + *),) ) (, , , ,)(9) (9) (9) (9,) Hence W = Whenever^ and _ are constants. Proof: X(^, _, `, 9,)Y ∞ = W ( X[ (−( + ( + ) + *,) )(, , , ,)Y + ^, ( + _, ) + `, * + 9) 4. Multiplication by , W X(, , , ,)Y = W (, (, ), *) ∞ ∞ ∞ ∞ = [ (−() )[ (−(() )[ (−()) ) [ (−(*,) )(, , , ,)(9) (9) (9) (9,) then ad X , (, , , ,)Y = a a( a) a* W X(, , , ,)Y Proof: ∞ ∞ ∞ W X ∞ , (, , , ,)Y = [ (−() )[ (−(() )[ (−()) ) [ (−(*,) )(, , , ,)(9) (9) (9) (9,) By using the fact E& ([ (−* )) = [ (−* ), then W X - , (, , , ,)Y @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5| Jul-Aug 2018 Page: 2406 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 a a Proof: We obtain the definition of fractional [ (−() ) [ (−(() ) quadruple Laplace transform and fractional quadruple a( a convolution above, then we obtain a a W X( (9,) [ (−()) [ (−(*,) )(, , , ,)(9) (9) (9) ∗∗∗∗ +)(, , , ,)Y a) a* ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ W a = [ (−() )[ (−(()=) [ (−() )[ (−(() )[ (−()) ) a a( a) a* [ (−(*,) )( [ (−()) )[ (−(*,) )(, , , ,)(9) (9) (9) (9,) ∞ ∞ ∞ ∞ ∗∗∗∗ +)(9) (9) (9) (9,) ad = [ (−() )[ (−(() )[ (−()) )(9,) W X(, = , , ,)Y a a( a) a* F [ (−(*,) ) s ( − P, − p, − q, , Remark: All results above are suitable for quadruple Laplace transform when α = 1. − r)+(P, p, q, r)(9P) (9p) (9q) (9r) t (9) ∞ ∞ ∞ = ∞ Theorem: If the fractional quadruple Laplace transform of the function (, , , ,) is W X(, , , ,)Y = W (, (, ), *), and fractional quadruple sumudu transform of the function (, , , ,) is V W X (, , , ,)Y = Z W (, (, ), *),then 1 1 1 1 1 Z W (, (, ), *) = W i $ , , , $ j ( ) * ( ) * Proof: Z W (, (, ), *) ∞ ∞ ∞ ∞ = [ (− )[ (− )[ (− )[ (−, ) (, (, ), *,)(9) (9) (9) (9,) (9) (9) (9,) let f = − P, g = − p, h = − q, 7 = , − r taking the limit from 0 to ∞, it gives ∞ ∞ ∞ and ∞ = [ (− (f + P) ) [ (−( (g + p) )[ (−) (h + q) )[ (−* (7 F + r) ) (f, g, h, 7)+(P, p, q, r)(9P) (9p) (9q) (9r) (9f) (9g) (9h) (97) ∞ ∞ ∞ ∞ = [ (− f )[ (−( g )[ (−) h ) By using change of variables f → , g → (, h → ) [ (−* 7 )(f, g, h, 7)(9f) (9g) (9h) (97) 7→9 and F ∞ ∞ ∞ ∞ 1 f g h [ (− P )[ (−( p )[ (−) q ) i = [ m− i j n [ m− i j n [ m− j n ) ( ) * ( [ (−* r )+(P, p, q, r)(9P) (9p) (9q) (9r) 7 W X(, [ i− 5 8 j (f, g, h, 7)(9f) (9g) (9h) (97) , , ,)Y W X(, , , ,)Y * 1 1 1 1 1 Z W (, (, ), *) = W i , , , j Inversion formula of Quadruple Fractional ( ) * ( ) * Laplace’s transform The Convolution Theorem of the Fractional Firstly, we will set up definition of fractional delta function of two variables as follows Double Laplace Transform Theorem: The double convolution of order α of functions(, , , ,) and +(, , , ,) can be defined as the expression ((, , , ,) ∗∗∗∗ +(, , , ,)) F = ( − P, − p, − q, , − r)+(P, p, q, r) (9P) (9p) (9q) (9r) , therefore one has the equality W X( ∗∗∗∗ +)(, , , ,)Y = W X(, , , ,)Y W X(, , , ,)Y Definition: Two variables delta function r ( − ^, − _, − `, , − 9) of fractional order α, 0 < α ≤1, can be defined as next formula +(, , , ,) r ( − ^, − _, − `, , u u u u − 9)(9) (9) (9) (9,) = 6 W +(^, _, `, 9)(9) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5| Jul-Aug 2018 Page: 2407 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 [ ( (ℎ − ) )[ ( (b − () ) In special case, we have +(, , , ,) r (, , , ,)(9) (9) (9) (9,) ℝ ℝ ℝ ℝ [ ( (w − )) [ (, ( − *) (9) (9() (9)) (9*) u u u u W = 6 +(0,0,0,0) Example: we can obtain fractional quadruple Laplace transform of function r ( − ^, − _, − `, , − 9) as follows W Xr ( − ^, − _, − `, , − 9)Y ∞ ∞ ∞ ∞ = [ (−( + ( + ) + *,) ) r ( − ^, − _, − `, , − 9)(9) (9) (9) (9,) = 6 W [ (−(^ + ( + ) + *,) )(10) In particular, we have W Xr (, , , ,)Y = 6 d Relationship between Two Variables Delta Function of Order α and Mittag-Leffler Function The relationship between [ ( + + + ,) and r (, , , ,) is clarified by the following theorem = (9) (9) (9) (9,) ℝ ℝ ℝ ℝ 6W [ ((−f) )[ ((−g) ) (v )W ℝ ℝ ℝ ℝ [ ((−h) )[ ((−,7) )(9f) (9g) (9h) (97) = r (, , , ,)(9) (9) (9) (9,) ℝ ℝ ℝ ℝ d =6 Note that one has as well 6W [ ((−ℎ) [ ((−b) [ ((−w) (v )W ℝ ℝ ℝ ℝ [ ((−,) (9ℎ) (9b) (9w) (9) = r (, , , ,) Inversion Theorem of Quadruple Fractional Theorem: The following formula holds Laplace Transform 6W Theorem: Here we recall the fractional quadruple [ ((−ℎ) [ ((−b) [ ((−w) W Laplace transform (5) for convenience (v ) W X(, W (, ℝ ℝ ℝ ℝ , , ,)Y = (, ), *)(12) [ ((−,) (9ℎ) (9b) (9w) (9) ∞ ∞ ∞ ∞ = r (, , , ,)(11) = [ (−( + ( + ) Wherev satisfy the equivalence [ (((v ) ) =1, and it is called period ofthe Mittag-Leffler function. Proof: We test that (11) agreement with 6 d = [ ((ℎ) [ ((b) [ ((w) ℝ ℝ ℝ ℝ [ ((,) (9) (9) (9) (9,) We replace r (, , , ,)in above equality by (11) to get 6W 6 d = (9) (9) (9) (9,) (v )W ℝ ℝ ℝ ℝ [ ((ℎ) )[ ((b) )[ ((w) ) ℝ ℝ ℝ ℝ [ ((,) )[ ((−) [ ((−() [ ((−)) [ ((−*,) (9) (9() (9)) (9*) = (9) (9) (9) (9,) ℝ ℝ ℝ ℝ 6 (v )W W + *,) )(, , , ,)9 9 9 9, And its inverse formula define as (, , , ,) ∞ ∞ ∞ ∞ 1 = [ (( + ( + ) (v ) W + *,) ) W (, (, ), *)(9) (9() ( 9)) (9*) (13) Proof: Substituting (12) into (13) and using the formula (11), (9) respectively, we obtain in turn (, , , ,) = ∞ 1 (v ) !"∞ !"∞ !"∞ !"∞ W [ () "∞ "∞ "∞ "∞ [ (() [ ()) [ (*,) (9) (9() (9)) (9*) ∞ ∞ ∞ [ (−(f + (g + )h + *7) (f, g, h, 7)(9f) (9g) (9h) (97) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5| Jul-Aug 2018 Page: 2408 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 1 = (y, q, z, p) (9y) (9q) (v )W References 1. D. G. Duffy, Transform methods for solving partial differential equations (CRC press, 2004) (9z) (9p) 2. T. A. Estrin and T. J. Higgins, Journal of the Franklin Institute 252, 153-167(1951) ∞ ∞ ∞ ∞ !"∞ !"∞ !"∞ !"∞ [ ( ( − f) ) "∞ "∞ "∞ "∞ [ (( ( − g) )[ () ( − h) )[ (* (* − 7) ) (9) (9() (9)) (9*) 1 (v )W = (y, q, z, p) (v )W 6W ∞ ∞ ∞ ∞ r (f − , g − , h − , 7 − *)(9y) (9q) (9z) (9p) 1 = W (y, q, z, p) r (f 6 ∞ ∞ ∞ ∞ − , g − , h − , 7 − *)(9y) (9q) (9z) (9p) = (, , , ,) Conclusion In this present work, fractional quadruple Laplace transform and its inverse are defined, and several properties of fractional quadruple transform have been discussed which are consistent with quadruple Laplace transform when α = 1. More over convolution theorem is presented. 3. A. Kilicman and H. Gadain, Journal of Franklin Institute 347,848-862(2010) 4. A. Kilicman and H. Eltayeb, Applied Mathematical Sciences 4, 109-118(2010) 5. H. Eltayeband A. 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