SOME RESULTS AND APPLICATIONS ON CONFORMABLE FRACTIONAL KAMAL TRANSFORM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we derive conformable fractional Kamal transform from the classical Kamal transform and we present their properties. We discuss the conformable fractional kamal transform of some functions and relationship with conformable fractional Laplace transform. Moreover we solve the general analytical solution of a general conformable Bernoulli’s fractional differential equation by this new transform and Adomain polynomial method. Also we use this method to find the solution of linear and nonlinear conformable fractional differential equations.


INTRODUCTION
The theory of fractional derivation has known great importance in mathematical research these last decades. The definition of fractional derivative don't have a standard form. But the most used definitions are those from the integration. Caputo definition, Riemann-Liouville, Hadamard, Grunwald [22,23,24]. Here, these fractional derivatives do not provide some properties of algebra of derivative. To overcome these difficulties, Khalil et al. [18], came up with an idea that extends the limit definition of the derivative. He derived some results of fractional derivative by using his new definition of fractional derivative. Almeida et al [7]. introduced different definition of the fractional derivative. He also discussed some important results of fractional derivative by using his definition if fractional derivative. Definition 1.1. [18] If φ : [0, ∞) → R be a function and ∀ α ∈ (0, 1), then the conformable fractional derivative of φ of order α is defined as If D α (φ (t)) and lim µ→0 + φ (α) (t) is exist in (0, c), where c > 0, then α-derivative is defined as Definition 1.2. [18] Let φ : [0, ∞) → R and α ∈ (n, n + 1] be an n-differentiable at t, where t > 0. Then conformable fractional derivative of φ is defined as where n ∈ N and α is the smallest integer number greater than or equal to [18] Let , 0 ≤ γ ≤ t and φ be a function defined on (γ,t], then New α-fractional integral is defined by Remark 1.4. [18] The most useful result is that where α ∈ (n, n + 1] and φ is an (n + 1)-differentiable function at t > 0.
Recently introduced Kamal transform by Abdelilah Kamal and H. Sedeeg [2] . It is defined for functions of exponential order in the set A by: where M is a constant but finite number, λ 1 , λ 2 are finite or infinite. The Kamal transform is defined by the integral equation and it is denoted by K(.) The most useful rules of classical Kamal transform are [2] (1) Shifting property: (4) Convolution property: If the Kamal transform of functions φ (t) and ψ(t) are Φ(v) and Ψ(v) respectively, then the convolution of their where φ , ψ : [0, ∞) → R are given functions, φ * ψ is the convolution product of φ and ψ, The conformable fractional Laplace transform (CFLT) for a given function φ : [0, ∞) → R at t > 0 is defined by [1,11] In particular, if α = 1, then Eq. (1.10) is convert to the definition of the fractional Laplace transform: Note: The relationship between Kamal transform G(v) = K{φ (t)} and Laplace transform 12) and also between conformable fractional laplace transform (CFLT) L α {φ (t)} = Φ α (s) and Laplace transform Φ(s) = L{φ (t)} is given by [ The most important results for the conformable fractional laplace transform (CFLT) are [1,11].

PROPERTIES OF CONFORMABLE FRACTIONAL KAMAL TRANSFORM (CFKT)
In this section, we introduce the definition of conformable fractional kamal transform (CFKT) and derive some properties and rules of this fractional transform for some functions and a relationship between conformable fractional kamal transform (CFKT) and conformable fractional laplace transform (CFLT) are determinate which are important role for solving conformable fractional linear and nonlinear differential equations (CFDEs).
Definition 2.1. Consider the conformable fractional kamal transform for functions of exponential order in the set A by where M is a constant but finite number, λ 1 , λ 2 are finite or infinite, then the conformable fractional kamal transform (CFKT) of φ can be defined as where d α t = t α−1 dt, 0 < α ≤ 1 and provided the integral exists.
Proof By using the Definition 2.1, we have be a given function and 0 < α ≤ 1, then where G α (v) and Φ α (s) are the conformable fractional kamal transform (CFKT) and CFLTs, respectively.
Proof By Using Definition 2.1, we have Putting u = t α α , then du = t α−1 dt, and substitute in above equation, then by using equations (1.5) and (1.6), we have Hence proved.
Proof By Applying Theorem 2.3 and equation (1.14), we have Hence the proof is complete the Theorem 2.4.
Proof Proof is follows by using induction and Theorem 2.4.
Note: In the following example we solve the fractional kamal transform for certain functions.
Example 2.6. Consider the a, c ∈ R, then by using Theorem 2.4, we have Where Γ() denotes to the gamma function.

APPLICATIONS OF CONFORMABLE FRACTIONAL KAMAL TRANSFORM (CFKT)
In this section, we solve the general analytical solution of the generalized conformable Bernoulli's fractional differential equations by using conformable fractional kamal transform (CFKT) and ADM. The ADM in [3,4,25,26,27] that used a very spontaneous method and has been successfully applied to solve nonlinear ordinary and fractional differential equations of various kinds. The ADM calculates the solutions of nonlinear equations as infinite series solution determined. Each term of these series is a generalized polynomial called Adomian's polynomial. The convergence, the order of convergence and the principle and convergence of ADM has been studied by Cherruault, Babolian and Biazar, Jiao et al. respectively [10,9,30].
We solve the general solutions of some linear and nonlinear conformable fractional differential equations (CFDEs) to new approach and to verify that this method can be applied successfully for finding the general solutions of many other linear and nonlinear conformable fractional differential equations (CFDEs).
where P(t), Q(t) and R(x) are α-differentiable functions and n ∈ N. Clearly this equation is a nonlinear if n = 0 or 1.
Applying K α on both sides of equation (3.1), we get On taking the inverse K −1 α of both sides of equation (3.2), we get An infinite general solution y(t) of equation (3.1) as

Now equation (3.3) can be rewritten as
where A k and B k are the Adomian polynomials of nonlinear functions φ (y) = y n and ψ(y) = y, respectively.

(3.5)
By applying K α of both sides of above equation (3.5), we have Taking inverse transform K −1 α of both sides of equation (3.6), we get the solution of this problem as By using ADM, then the equation (3.7) can be rewritten as where {A k } ∞ k=0 are the Adomian polynomials representing to φ (y) = e −y(t) . Then the recursive relation as follows Thus, the general recursive relation is given by By using this recursive relation, we have Then the general solution of equation (3.5) is series form and with the help of we obtain (see Example 3.3. Consider the following non-linear CFDE This equation is directly solved by using equation (1.1) to convert into the ordinary differential equation as Now it is solve by separable, we have This problem can also solved by using our technique as in above. Comparing equation (3.8) with equation (3.1), we have P(t) = 0, Q(t) = R(t) = 1 and n = 2. Then by applying K α on both sides of equation (3.8), we get Taking inverse K −1 α of both sides of equation (3.9), we get the solution of this problem as By using ADM, then the equation (3.10) can be rewritten as where {A k } ∞ k=0 are the Adomian polynomials of function φ (y) = y 2 . Then the recursive relation as follows: Thus, the general recursive relation is given by By using this recursive relation, we have    By applying K α of both sides of above equation (3.12), we have Taking inverse transform K −1 α of both sides of equation (3.13), we get the solution of this problem as (3.14) By using ADM, then the equation (3.14) can be rewritten as where {A k } ∞ k=0 are the Adomian polynomials representing to φ (y) = y 2 . Then the recursive relation as follows Thus, the general recursive relation is given by By using this recursive relation, we have Then the approximate solution of equation (3.12) is series form and with the help of we obtain (see Figure 3) y(t) = y 0 (t) + y 1 (t) + y 2 (t) + y 3 (t) + · · · = ∞ ∑ n=0 y n (t),  By applying K α on both sides of equation (3.15), we obtain

CONCLUSION
In this paper we derived some important results of the conformable fractional kamal transform (CFKT) which are main roles for solving conformable linear and nonlinear fractional differential equations. Also we have discussed the general analytical solution of the generalized conformable Bernoulli fractional differential equation based on the conformable fractional kamal transform (CFKT) and adomain decomposition method (ADM). Moreover we solved conformable linear and nonlinear fractional differential equations with help of this conformable fractional kamal transform (CFKT) and ADM we gives most appropriate solution.