Fractional Quadruple Laplace Transform and its Properties

In this paper, we introduce definition for fraction al quadruple Laplace transform of order α,0 < fractional differentiable functions. Some main properties and inversion theorem of fractional quadruple Laplace transform are established. Furthe r, the connection between fractional quadruple Laplace transform and fractional Sumudu transform are presented. KEYWORD: quadruple Laplace transform, Sumudu transform, fractional Difference. INTRODUCTION There are different integral transforms in mathemat ics which are used in astronomy, physics and also in engineering. The integral transforms were vastly applied to obtain the solution of differential equa tions; therefore there are different kinds of integral transforms like Mellin, Laplace, and Fourier and so on. Partial differential equations are considered o ne f the most significant topics in mathematics and others. There are no general methods for solve these equations. However, integral transform method is on e 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 b ee determined and studied by [5, 6]. In this study we focus on qua druple integral determined and studied by [5, 6]. In this study we focus on quadruple integra l transforms. First of all, we start to recall the de finition of quadruple Laplace transform as follows | Volume – 2 | Issue – 5| JulAug 2018 6470 | www.ijtsrd.com | Volume Journal of Trend in Scientific and Development (IJTSRD) International Open Access Journal Monisha. G1, Savitha. S Scholar, Faculty of Mathematics ciences for Women [Autonomous], , Tamil Nadu, India

There are different integral transforms in mathematics which are used in astronomy, physics and also in 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 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, 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 been 6]. In this study we 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. Where 0 < ߙ < 1, And the α-derivative of݃(‫)ݐ‬ is known as See the details in [9,10].

Some Properties of Fractional Quadruple Sumudu Transform
We recall some properties of Fractional Quadruple Sumudu Transform where ߲ ௧ ఈ 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

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 ݂ ଵ ‫,ݔ(‬ ‫,ݕ‬ ‫,ݖ‬ ‫)ݐ‬ and ݂ ଶ ‫,ݔ(‬ ‫,ݕ‬ ‫,ݖ‬ ‫)ݐ‬ be functions of the variables ‫ݔ‬ and ‫,ݐ‬ then Where ܽ ଵ and ܽ ଶ are constants. Proof: We can simply get the proof by applying the definition.

Relationship between Two Variables Delta Function of Order α and Mittag-Leffler Function
The relationship between ‫ܧ‬ ఈ ‫ݔ(‬ + ‫ݕ‬ + ‫ݖ‬ + ‫)ݐ‬ ఈ and ߜ ఈ ‫,ݔ(‬ ‫,ݕ‬ ‫,ݖ‬ ‫)ݐ‬ is clarified by the following theorem

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.