FRACTIONAL ORDER EXPLICIT FINITE DIFFERENCE SCHEME FOR TIME FRACTIONAL RADON DIFFUSION EQUATION IN CHARCOAL MEDIUM

In this paper, we introduce the fractional order radon diffusion equation and develop explicit finite difference scheme for time fractional radon diffusion equation (TFRDE). Also, we discuss the stability and convergence of the scheme, as an application of this scheme, we obtain the numerical solutions of the test problem and it represented graphically.


INTRODUCTION
Fractional calculus belongs to the field of mathematical analysis which deals with the investigation and applications of differentiation and integration to arbitrary non-integer order. At present fractional calculus has been simulated by many applications in physics, engineering, bioscience, 1224 SUNIL DATTATRAYA SADEGAONKAR, RAJKUMAR NAMDEVRAO INGLE applied mathematics etc. [1,2,6,17,18,19,20,21,22,23,24]. The analytical solution of fractional diffusion equation is very difficult to find hence researchers developed the finite difference schemes to find numerical solution [3,4,7,8,9,10,11]. Radon is a colorless, odorless, radioactive gas. It forms naturally from the decay of radioactive elements, such as uranium, which are found in different amounts in soil and rock throughout the world. Radon gas in the soil and rock can move into the air and into underground water and surface water. Radon is present outdoors and indoors. Due to dangerous nature of radon many researchers study the radon transport through soil, air, concrete, activated charcoal, etc., [5,12,13,14,15,16]. The diffusion theory came from the famous physiologist Adolf Fick. He stated that the flux density J is proportional to the gradient of concentration. This gives, where J is the radon flux density is diffusion coefficient, is gradient of radon concentration and D is diffusivity coefficient of radon. Now the change in concentration to change in time and position is stated by the Fick's second law which is the extension of Fick's first law, that gives, where  = 2.1 × 10 -6 s -1 is the decay constant.
In this paper, we develop the time fractional explicit finite difference method for fractional order radon diffusion equation. We consider the following time fractional radon diffusion equation We organize the paper as follows: In section 2, we develop explicit finite difference scheme for time fractional radon diffusion equation (TFRDE). The section 3, is devoted for stability of the solution of the scheme and the convergence of the approximated finite difference scheme is proved in section 4. In the last section we solved some text problems and their solutions are represented graphically by mathematical software Mathematica.

FINITE DIFFERENCE SCHEME
We consider the following time fractional radon diffusion equation Boundary conditions: We introduce the finite difference approximation to discretize the time fractional derivative.

NUMERICAL SOLUTION
In this section, we obtain the approximated solution of time fractional radon diffusion equation with initial and boundary conditions. To obtain the numerical solution of the time fractional radon 1239 FRACTIONAL ORDER EXPLICIT FINITE DIFFERENCE SCHEME diffusion equation (TFRDE) by the finite difference scheme, it is important to use some analytical model. Therefore, we present an example to demonstrate that TFRDE can be applied to simulate behavior of a fractional diffusion equation by using Mathematica Software.
We consider the following, dimensionless time fractional radon diffusion equation with suitable initial and boundary conditions,

CONCLUSION
We successfully developed fractional order explicit finite difference scheme for time fractional radon diffusion equation. Furthermore, we discuss its stability and convergence of the scheme. As an application of this method we obtain the numerical solution of text problems and its solutions is simulated graphically by mathematical software Mathematica.