CONFORMABLE FRACTIONAL DERIVATIVE AND ITS APPLICATION TO PARTIAL FRACTIONAL DERIVATIVES

Recently, the study on conformable fractional derivative has been gaining more attention from various researchers. This is due to its importance in numerous practical applications. In this paper, we introduce new definition of conformable fractional derivation to find partial fractional derivatives. The fractional derivatives are taken in the sense of Conformable fractional derivative. This approach is considered as a powerful tool for solving linear or nonlinear equations. The new formulation of fractional calculus is applied to solve many applied problems including fractional partial derivatives. The results obtained through the conformable fractional approach have been evaluated and presented to illustrate the efficiency and good accuracy. It has been observed that the proposed approach is prevailing for the solutions of partial fractional derivatives.


INTRODUCTION
The theory of fractional derivatives is as old as the classical calculus and has been in existence since late 17th century [1]. This form of calculus is classified as a generalized fractional integrals or derivatives [2]. However, the subject just became more popular in recent years due to its wide application in numerous practical applications. Recently, there has been an increasing progress by several researchers to define the fractional derivative [3]. Most of these researchers are focusing on the fractional calculus due to the significant of fractional order derivatives in describing numerous real world physical phenomena [4]. Some of these studies include the application of fractional derivatives in modeling the fluid-dynamics capable of eliminating the deficiency developing from the occurrence of continuum flow of traffic [5]. More so, in would be interesting to know that the idea of a fractional derivative has advanced and playing essential role in the mathematical analysis of applied problems. Phenomena of a complex nature usually depend on more than one variable. These literatures and many more make the area of fractional calculus to becomes more convenient approach for describing the mathematical models of different aspects of physical and dynamical systems [1].
Recently, Khalil et al. [6] define fractional derivative using a new concept that obviously wellsuited with the classical derivative definition. In contrast to the previous definitions, the authors present a definition that possess similar formulas of derivative of product and quotient of two functions with an even easier chain rule [7]. More so, these authors gave a new definition for the Mean value theorem for conformable fractional differentiable functions, Rolle theorem, and conformable fractional integral. Based on the above idea, [8] studied the fractional integrals of higher orders concepts and further presented left and right conformable fractional derivative. These studies discussed above has led to rigorous research on studies related to this new fractional derivative definition was done [9].
More recent literatures tend to studied the fractional calculus for functions with one variable 3029 APPLICATION OF CONFORMABLE FRACTIONAL DERIVATIVE [6,10,11]. More so, [3,10] studied various skills for finding the most accurate solution of fractional derivative using Newton's fractional method by expanding the idea and knowledge of fractional calculus for all continuous functions ( ).
In this paper, we present a new definition of partial fractional derivative and further show that the partial fractional derivative arises geometrically. This idea is extended to show how the partial fractional derivative is been calculated using certain defined rules. We begin with functions of two independent variables.

CONFORMABLE PARTIAL FRACTIONAL DERIVATIVE
Some definitions of fractional calculus have been introduced including partial derivative and partial integral. For fractional derivatives, there are actually many different definitions, and in general, these definitions are not equivalent to each other. Definition 1 [6,11]: Let ( , ) be a function : [0, ∞) → ℝ 2 . Then, the "conformable partial fractional derivative , , " of with order α is defined by Definition 2 [6,11]: If is a continuous function on [ , ∞), > 0. Then,

Property 1.
If ( ) is continuous on (0, ∞) and and are positive real numbers then
Based on the above definitions and the definition of fractional differentiation, we present the following definition which is generalized to partial fractional differentiation.

Definition 3.
For to be the smallest integer that exceeds , the conformable partial fractional derivative operator of order > 0 is defined as Let be any arbitrary real number and be a function with two variables as ( , ), then, or as partial with respect to , hence is a constant.
The proof of this theorem follows from [11].

MIXED FRACTIONAL PARTIAL DERIVATIVES
In this section, we define and denote the -fractional partial derivatives with respect to and likewise -fractional partial derivatives to same function with respect to , knowing that the real numbers and are subject to the following condition; 0 < < 1 and 0 < < 1,

APPLICATION
In order to verify the accuracy of the method, this section present the solutions of some benchmark problems. .

Solution
We fixe as a constant and as product of and sin( ). By theorem 1, we have 3033 APPLICATION OF CONFORMABLE FRACTIONAL DERIVATIVE

I.
Find the explicit solution on the interval 0 ≤ < 1.

II.
Find the explicit solution on the interval ≥ 1.

Solution
In this case, we divide into two cases. For the first case, let = 2 , then, we have

Solution
We begin by finding

CONCLUSION
The objective of this study is to implement the conformable fractional derivative on fractional order partial differential and optimal numerical solution of fractional order partial differential equations with initial conditions. The obtained solution demonstrates the reliability of the proposed method and shows the applicability of this method to solve large amount of non-linear fractional equations. The proposed method is free from assumptions and rounding-off errors. It offers more realistic series solutions whose continuity depends on the fractional derivative. Convergence of approximate solutions is observed as the number of decomposed terms is increased. Consistent accuracy throughout the domain of the problem is also witnessed.