SOLVING SYSTEMS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA TWO DIMENSIONAL LAPLACE TRANSFORMS

The problem related to partial differential equation commonly can be solved by using a special integral transform, thus many authors solved the boundary value problems by using single Laplace transform. Furthermore, two dimensional Laplace transforms in the classical sense for solving linear second order partial differential equations were used by Ditkin [11], Brychkov [13]. The Laplace transform, it can be fairly said, stands first in importance among all integral transforms for which there are many specific examples in which other transforms prove more expedient. The Laplace transform is the most powerful in dealing with both initial boundary value problems and transforms [1-9]. The two dimensional Laplace transforms is a powerful tool in applied mathematics and engineering.


Introduction and Preliminaries
The problem related to partial differential equation commonly can be solved by using a special integral transform, thus many authors solved the boundary value problems by using single Laplace transform. Furthermore, two dimensional Laplace transforms in the classical sense for solving linear second order partial differential equations were used by Ditkin [11], Brychkov [13]. The Laplace transform, it can be fairly said, stands first in importance among all integral transforms for which there are many specific examples in which other transforms prove more expedient. The Laplace transform is the most powerful in dealing with both initial boundary value problems and transforms [1][2][3][4][5][6][7][8][9]. The two dimensional Laplace transforms is a powerful tool in applied mathematics and engineering.
The fractional derivative is one of the most interdisciplinary fields of mathematics, with many applications in physics and engineering and deals with extensions of derivatives and integrals to non-integer orders. It represents a powerful tool in applied mathematics to study a myriad of problems from different fields of science and engineering, with many breakthrough results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, Http://www.granthaalayah.com ©International Journal of Research -GRANTHAALAYAH [407] statistical mechanics, astrophysics, cosmology and bioengineering [13,14,15,16]. Several definitions have been proposed for a fractional derivative. We deal with Caputo fractional derivatives only. In this section, we present the definition of this derivative.
Let us take f an arbitrary integrable function. By ( The Caputo fractional derivative is a regularization in the time origin for the Riemann-Liouville fractional derivative by incorporating the relevant initial conditions. The major utility of the Caputo fractional derivative is caused by the treatment of differential equations of the fractional order for physical applications, where the initial conditions are usually expressed in terms of a given function and its derivatives of integer (not fractional order), even if the governing equation is of fractional order. If care is taken, the results obtained using the Caputo formulation can be recast to the Riemann-Liouville version and vice versa.
Let () ft be a function of t specified for 0 t  . Then the Laplace transform of function () ft is defined by Proof -In order to solve the above system, by introducing we can rewrite the above system of partial fractional integro -differential equations in the following form By applying the Laplace transform of both sides of the above equation term -wise we obtain Now, using the fact that Upon taking the inverse Laplace transform of the above term, yields Finally, by taking the real and imaginary part of the above relation, we finally obtain the solutions of the system in the following forms

 
Also, the two-dimensional convolution of ( , ) f x y and ( , ) g x y is given by

Evaluation of Integrals and Solution to Fractional P.D.E. By Means of the Two Dimensional Laplace Transform
In this section, we evaluate the integrals that their evaluations are not an easy task. But, by means of two dimensional Laplace transform we can evaluate these integrals. Proof. ( The inverse two dimensional Laplace transform of the above relation yields  Sub Ballistic Fractional PDE) The following non -homogeneous partial fractional differential equations [17].
The inverse two-dimensional Laplace transforms yields

 
And the figure is shown as follows. Note that when 01   and 1   , by using the two-dimensional Laplace transform, we may calculate ( , ) U p q as follows,

Solving System of Time Fractional Heat Equations
In this section, the authors considered certain homogeneous system of time fractional heat equations which is a generalization to the problem of thermal effects on fluid flow and hydraulic fracturing from well bores and cavities in low permeability formations [12]. In this work, only the Laplace transformation is considered as it is easily understood and being popular among engineers and scientists.