On Katugampola Laplace transform

The aim of this article is to introduce a new form for the Laplace transform. This new definition will be considered as one of the generalizations of the usual (classical) Laplace transform. We employ the new ”Katugampola derivative”, which obeys classical properties and define Katugampola Laplace transform. We obtain some properties of this transform and find the relation between the Katugampola Laplace transform and the usual Laplace one.


Introduction
The derivative of non-integral order "Fractional derivative" is an interesting research topic since it is a generalization of the classical integer calculus. Several types of fractional derivatives were introduced and studied by Riemann-Liouville, Caputo, Hadamard, Weyl, and Grünwald-Letnikov; for more details one can see [4,6,7,8]. Unfortunately all these fractional derivatives fail to satisfy some basic properties of the classical integer calculus like product rule, quotient rule, chain rule, Roll's theorem, mean-value theorem and composition of two functions. Also, those fractional derivatives inherit non-locality and most of them propose that the derivative of a constant is not zero. Those inconsistencies lead to some difficulties in the applications of fractional derivatives in physics, engineering and real world problems.
To overcome all the difficulties raised, Khalil et al. [5] introduced and investigated the so called conformable fractional derivative and also, Katugampola [3] introduced and studied a similar type of derivative, later called Katugampola derivative and is defined as follows Definition 1.1 [3] Let f : [0, ∞) → R and t>0 . Then, the Katugampola derivative of f of order α is defined by for t>0 and α ∈ (0, 1] . If f is α−differentiable in some (0, a) , a>0 and lim t→ 0

Definition 1.2[3]
Let α ∈ (n, n + 1], for some n ∈ N and f be an n−differentiable at t > 0. Then the α−fractional derivative of f is defined by if the limit exists. Note that Katugampola derivative satisfies product rule, quotient rule, chain rule,. . . etc. and it is consistent in its properties with the classical calculus of integer order. In addition, we have the following theorem.

Katugampola Laplace Transform
Salim, T.O., et al [9] have introduced a new definition of Katugampola Fourier transform which finds very interesting reputation between mathematicians. Following the same procedure, they continue their work and define a new Laplace transform called Katugampola Laplace transform. Some basic properties of this transform are given here. Abdeljawad [1] gave the definition of conformable Laplace transform.
In this section, we introduce and study the relation between Katugampola Laplace transform and the usual Laplace transform. The Katugampola Laplace transform of some functions are established and then we obtain a convolution formula for this transform. Remember here the usual Laplace transform to the function f , Definition 2.1 Let α ∈ (n, n + 1] for some n ∈ N and f (t) be a real valued function on [0, ∞) . The Katugampola Laplace transform of f(t) of order α is defined as

Theorem 2.2
Let α ∈ (n, n + 1], for some n ∈ N and f (x) be a real valued function on [0, ∞) . Then, Proof. By using Definition 2.1 and Theorem 1.3, we have

Now by using integration by parts, we get
Proof. The proof is directly obtained by letting n = 0 in Theorem 2.2.

Lemma 2.4
Let α ∈ k−1 k 1 , k ∈ N and u (x, t) be kα−differentiable real valued function. Then, Proof. We can prove this theorem by mathematical induction on k.
which is true by Corollary 2.3 . Now, assume that the theorem is true for a particular value of k, say r. Then, we have Now, we need to prove that the Theorem is true for r + 1, that is By using Theorem 2.4 and the assumption, we have Therefore the theorem is true for every positive integral value of k.
In the following Lemma, we present the relation between the Katugampola Laplace transform and usual Laplace transform.
Proof. By setting y = t α−n α−n , t = ((α − n) y) Let us now present the Katugampola Laplace transform for some selected functions.

Theorem 2.6
Let α ∈ (n, n + 1], n ∈ N. We have the following transformations where Bessel function [2] of order n denoted by J n (t) is defined by where L n (t) = e t n! d n dt n t n e t du is a Laguerre polynomial.
where δ(t) is a Delta function. Proof. We give the proofs of some transformations, where the rest of the proofs follows by using Definition 2.1, the substitution y = t α−n α−n , and then integration.
where H (t − a) is the Unit Step (or Heaviside's unit) function. We next turn to obtain some important properties of the Katugampola Laplace transform. Lemma 2. 8 Katugampola Laplace transform L α {f(x),κ} is liner. That is

Proof.
The proof is trivial.

Proof. Starting with
putting u = at, so that t = u a and du = a dt (as t : where m = 1, 2, 3, ... . Proof. We can prove this Theorem by mathematical induction on m . where g * h is the Convolutions of function g(t) and h (t) defined as Proof. It is easy to prove the results by using Lemma 2.5, and the definition of Laplace transform.

Conclusions
In this paper, we obtained several results that have close resemblance to the results found in classical calculus. We defined a new kind of fractional Laplace transform. Also we gave some prosperities of this transform which is considered as a generalization to the usual Laplace transform.