REDUCTION OF ORDER OF FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper we study the solution of the second order fractional differential equation of the form F(x,y,y(α),y(2α)) = 0, in case either x is missing or in case y is missing.


Introduction
There are many definitions available in the literature for fractional derivatives.The main ones are the Riemann Liouville definition and the Caputo definition, see [7] .
Such definitions have many setbacks such as The Riemann-Liouville derivative does not satisfy D α a (1) = 0 (D α a (1) = 0 for the Caputo derivative), if α is not a natural number.
(ii) All fractional derivatives do not satisfy the known formula of the derivative of the product of two functions: (iii) All fractional derivatives do not satisfy the known formula of the derivative of the quotient of two functions: .
(iv) All fractional derivatives do not satisfy the chain rule: (v) All fractional derivatives do not satisfy: (vi) All fractional derivatives, specially Caputo definition, assumes that the function f is differentiable.
We refer the reader to [7] for more results on Caputo and Riemann -Liouville Definitions.
Recently, the authors in [ 5 ], gave a new definition of fractional derivative which is a natural extension to the usual first derivative.So many papers since then were written, and many equations were solved using such definition.We refer to [1][2][3][4][5][6] and references there in for recent results on conformable fractional derivative.The definition goes as follows: Given a function f : [0, ∞) −→ R. Then for all t > 0, α ∈ (0, 1), let T α is called the conformable fractional derivative of f of order α.
If f is α−differentiable in some (0, b), b > 0, and lim According to this definition, we have the following properties, see [ 5], Further, many functions behave as in the usual derivative.Here are some formulas:

Main Result
Consider a second order fractional differential equation of the form: (1) F(x, y, y (α) , y (2α) ) = 0 , where y (α) is the α− conformable derivative of y with respect to x and α ∈ (0, 1], and y (2α) = D α D α y.Often, equation is not a standard equation in the sense it is not of any type that we can handle.
The object of this paper is to try to solve equation (1) in case either x is missing or y is missing using what we will call fractional reduction of order.
There are two cases to be considered:

Case(i): y is missing
In this case put y (α) = u.Consequently, we get y (2α) = u (α) .This reduces the order of the equation from 2α to order α, which is much easier to handle. Examples: This equation is not a linear equation.However, here y is missing.So put y (α) = u and consequently, y (2α) = u (α) .The equation becomes which is a separable differential equation that can be solved as follows: Since , the equation u (α) = u 2 + 1 can be written as: Thus tan −1 (u) = 1 α x α + a. Consequently, u = tan 1 α x α + a .Replacing u by y (α) and then substituting y (α) by x (1−α) dy dx and integrating we get: Here y is missing.Hence put y (α) = u.Then y (2α) = u (α) .The equation becomes 4x α−1 (cos x) u (α) − (sin x)u 2 − 4 sin x = 0 which is a separable differential equation: Replacing u by y (α) and then substituting y (α) by x (1−α) dy dx , we get:

1 u 2
+ 4 du = sin x cos x dx , which can be solved to get u = 2 tan 2 c 1 − ln | cos x| .