Fractional Laplace transform for matrix valued functions with applications

Abstract We extend several fractional Laplace transform results to matrix-valued functions in this paper, which we will utilize to obtain some useful and valuable theorems, like results on the set of the piecewise continuous functions with conformable exponential order and the low and conditions to obtain the fractional Laplace transform of matrix-valued functions. As an application, we apply the obtained theorems to solve certain fractional initial value problems for vector-valued and matrix-valued functions and the solution will be victor or matrix and it will exact. Also, we establish a fractional system transfer matrix and the conformable fractional Laplace transform of the conformable exponential matrix function, which will give us directly the exact solution for a precisely type of fractional initial value problems.


Introduction
Fractional derivative emergence dates back to the time of calculus. In 1695, L'Hospital wondered about the meaning of d n f dx n if n ¼ 1 2 , since then, researchers have been attempting to define a fractional derivative. Some of which are: Riemann-Liouville (Miller & Ross, 1993), Caputo (Caputo & Fabrizio, 2015;Miller & Ross, 1993), Caputo and Fabrizio (2015) and Atangana, Baleanu, and Alsaedi (2015) definitions … etc.
In 2014, a new definition of fractional derivative called Conformable derivative was introduced by Khalil, Al Horani, Yousef, and Sababheh (2014).
Given a function y : ½0; 1Þ ! R Then the "Conformable fractional derivative" of y of order h is defined by yðs þ s 1Àh Þ À yðsÞ for all s > 0, h 2 ð0, 1Þ: If y is hÀdifferentiable in some ð0, aÞ, a > 0, and lim t!0 þ y ðhÞ ðsÞ exist, then define y ðhÞ ð0Þ ¼ lim s!0 þ y ðhÞ ðsÞ, and the Conformable fractional integral is defined as where the integral is the usual Riemann improper integral. Most of the definitions give numerical solution to the problems using computer code. However, the Conformable fractional derivative is a natural definition which gives us simple and easy solutions to the problems.
In 2015, Abdeljawad Thabet put forward a definition of Conformable fractional Laplace transform (Abdeljawad, 2015). Now, we extend some results of the Conformable fractional Laplace transform to matrix-valued functions and we obtain certain useful theorems. Therefore, we will use the previous attained theorems to solve the following type of fractional initial value problem for matrix-valued functions where A is a constant matrix and the components of g(s) are members of PE (set of piecewise continuous functions with Conformable exponential order).
Moreover, we provide a fractional system transfer matrix and the Conformable fractional Laplace transform of the Conformable exponential matrix function which will give the solution of the following type of fractional initial value problem for matrix-valued functions where I is the n Â n identity matrix and A is a constant n Â n matrix.
The novel idea behind the current study is to apply the Conformable fractional Laplace transform on a new type of fractional initial value problems for matrix-valued function and obtain its exact solution.
The weaknesses of the current study is just the existence of the Conformable fractional Laplace transform or not (Younis et al., 2022). However, Its strength is in obtaining an exact solution easier without the need of the computer code but the other give an approximate solution also we can use it for the nonlinear case as can be seen in Ilhem, Al Horani, and Khalil (2022).
For further details on Conformable fractional Laplace transform see Abdeljawad (2015)
Let y : 0, 1Þ ! R ½ be a real valued function and 0 < h 1: Then the Conformable fractional Laplace transform of y is defined as provided the integral exists. Let us have as an example for the Conformable fractional Laplace transform of the usual functions in the theorem bellow. Let a, p, c 2 R and 0 < h 1: Then ¼ a n 2 Àa 2 , n > jaj: Proof. Follows by applying Definition 2.1.
w One of the nice results is the relation between the usual and the Conformable fractional Laplace transforms which is given in the theorem below. Let where L is the usual Laplace transform.
w Theorem 2.4. Let y : 0, 1Þ ! R, g : 0, 1Þ ! R ½ ½ and let k, l, a 2 R and 0 < h 1. Then (2) L h e Àa s h h yðsÞ n o ðnÞ ¼ y h ðn þ aÞ, n > jaj: where y h and g h are the Conformable fractional Laplace transform of the functions y and g respectively, y Ã g is the convolution product of y and g and I h yðsÞ is the Conformable fractional integral.
Proof. See Abdeljawad (2015) and Al-Zhour et al. Definition 2.6. A function is called piecewise continuous on an interval if the interval can be broken into a finite number of subintervals on which the function is continuous on each open subinterval and has a finite limit at the endpoints of each subinterval. PE will be used to designate the class of all piecewise continuous functions of Conformable exponential order in the following sections. Any linear combination of functions in PE is also in PE, according to the next theorem. The same is true for the product of two functions in PE: Theorem 2.7. Let's pretend that y(s) and g(s) are two PE members with jyðsÞj M 1 e a 1 s h h , s ! C 1 and jgðsÞj M 2 e a 2 s h h , s ! C 2 : (1) The function byðsÞ þ cgðsÞ is also a member of PE for any constants b and c. Moreover L h byðsÞ þ cgðsÞ ½ ¼bL h yðsÞ ½ þcL h gðsÞ ½ : (2) PE includes the function hðsÞ ¼ yðsÞgðsÞ as an element.

Proof.
(1) byðsÞ þ cgðsÞ is a piecewise continuous function, as can be seen. Now, let This prove that byðsÞ þ cgðsÞ is of Conformable exponential order.
On the other hand, L h byðsÞ þ cgðsÞ ½ ¼bL h yðsÞ ½ þcL h gðsÞ ½ : (2) hðsÞ ¼ yðsÞgðsÞ is a piecewise continuous function, as can be seen. Now, letting So, for s ! C we have Consequently, h(s) is of Conformable exponential order. w

Solution of fractional initial value problems for matrix-valued functions
All results of Conformable fractional Laplace transform can be extended to vector-valued and matrixvalued functions. In this part, we select some results to be extended. Let PE's members be y 1 ðsÞ, y 2 ðsÞ, ::: y n ðsÞ: Take a look at the following vector-valued function. (2) Allow Y ðhÞ ðsÞ to be continuous, and the entries of Y ð2hÞ ðsÞ to be PE members. Then À Y ðhÞ ð0Þ À nYð0Þ, n > 0: (3) Let the entries of Y(s) be members of PE. Then n, n > 0: Proof.
(1) By using Definition 2.1. and integration by parts, we have (2) Similarly, by applying Definition 2.1. and integration by parts, we get a result. (3) By using result (1) in this theorem, we have By using definition of Conformable fractional integral we get I h Yð0Þ ¼ 0, then we obtain Theorem 3.2 and Theorem 3.3 can be used to solve the following type of fractional initial value problem for matrix-valued functions where A is a constant matrix and the components of g(s) are members of PE: We can write using the above theorems nY h ðnÞ À Y 0 ¼ AY h ðnÞ þ g h ðnÞ: Thus where I is the identity matrix, L h ½gðsÞ ¼ g h ðnÞ and L h ½YðsÞ ¼ Y h ðnÞ: If n is not an eigenvalue of matrix A so matrix ðnI À AÞ is invertible and in this case we get Y h ðnÞ ¼ ðnI À AÞ À1 ðY 0 þ g h ðnÞÞ: (2) To compute YðsÞ ¼ L À1 h ½Y h ðnÞ we have to compute the Conformable fractional inverse Laplace transform of every component of Y h ðnÞ: The above discussion can be illustrated by the next example.

À Á
is the solution of the following fractional initial value problem for matrix-valued functions where I is the n Â n identity matrix and A is a constant n Â n matrix.
Proof. Taking  If n is not an eigenvalue of matrix A so matrix ðnI À AÞ is invertible and in this instance, we conclude that Hence, a result as required.
w As an application, we will solve the following fractional initial value problems for matrix-valued functions in the below examples Example 4.2.
Y h ðsÞ ¼ AYðsÞ, Yð0Þ ¼ I, 0 < h 1, where A ¼ 2 0 0 2 and I ¼ 1 0 0 1 : By Theorem 2.2. we obtain the solution Hence a result as required. By Theorem 4.1. we have By Theorem 2.2. we conclude that the solution is Indeed: We are going to prove that It is clear that where P ¼ ðv 1 , v 2 Þ is the passage matrix and v 1 , v 2 are the eigenvectors corresponding to the eigenvalues k 1 , k 2 respectively and D ¼ diagðk 1 , k 2 Þ: In order to determine the eigenvectors of matrix A we must first determine the eigenvalues k 1 , k 2 by solving the equation ðA À kIÞX ¼ 0, where I is the identity matrix and for some nonzero vector X.
Hence, we get k 1 ¼ 1, k 2 ¼ À1: Also, we find Then, we calculate the inverse matrix of P to obtain P À1 ¼ Hence a result as required.

Conclusion
It is quite complicated to find the exact solution for Riemann-Liouville and Caputo fractional differential equations and initial value problems even in the linear scalar case. More details and information on methods solving fractional initial value problems for Caputo and Riemann-Liouville sense can be founded in Bushnaq et al. (2022); Vinh An, Vu, and Van Hoa (2017) and Hristova, Agarwal, and O'Regan (2020). Since, the formulas for the exact solutions are important tools in fractional models. In this paper, we introduce new exact solution to the fractional initial value problems for matrix-valued functions called Conformable fractional Laplace transform method. Our method was illustrated on two types of fractional initial problems for vector-valued functions and matrix-valued functions as mentioned previously. We conclude that the Conformable fractional Laplace transform is an easy and simple method which gives us exact solution to this kind of problems. It is a known fact that Laplace transform is a famous mathematical tool for linear operators, but it is extremely difficult to deal with nonlinear operators. Our interest future work is to develop our method for solving fractional initial and boundary values problems specially the nonlinear case as can be seen in Ilhem et al. (2022). Moreover, the latest publications on fractal theory can be founded in Ain, Anjum, and He (2021)