Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

Recently, there is great interest to develop new types of fractional derivatives of nonsingular kernel. Motivated by applications, Caputo and Fabrizio were the first to introduce such types of fractional derivatives with nonlocal and nonsingular kernel [1]. ,e Caputo–Fabrizio derivative is connected with a variety of applications (see [2–6]). Stability analysis of fractional differential equations without inputs was studied in [7], where exponential stability is obtained for the Caputo–Fabrizio derivative. Since their kernels are nonlocal, fractional derivatives preserve memories, and therefore, they have been used to model several (SIR) epidemic models (see [8–14]). Several analytical techniques have been implemented to study various fractional equations with fractional derivatives without singular kernels, such as the Laplace transform, reduction to initial value problems with integer derivatives, maximum principles, and fixed point theorems (see [15–21]), just to mention a few out of many in the literature.,e following definitions are required to state our problem.


Introduction
Recently, there is great interest to develop new types of fractional derivatives of nonsingular kernel. Motivated by applications, Caputo and Fabrizio were the first to introduce such types of fractional derivatives with nonlocal and nonsingular kernel [1]. e Caputo-Fabrizio derivative is connected with a variety of applications (see [2][3][4][5][6]). Stability analysis of fractional differential equations without inputs was studied in [7], where exponential stability is obtained for the Caputo-Fabrizio derivative. Since their kernels are nonlocal, fractional derivatives preserve memories, and therefore, they have been used to model several (SIR) epidemic models (see [8][9][10][11][12][13][14]). Several analytical techniques have been implemented to study various fractional equations with fractional derivatives without singular kernels, such as the Laplace transform, reduction to initial value problems with integer derivatives, maximum principles, and fixed point theorems (see [15][16][17][18][19][20][21]), just to mention a few out of many in the literature. e following definitions are required to state our problem.

Definition 1.
A function f is said to be absolutely continuous on [a, b], if there exists a function f ′ ∈ L 1 (a, b) such that In the following, we will use the notation AC ([a, b]) to denote the space of absolutely continuous functions on [a, b].
en, we define the following space.
In this paper, we consider the multiterm fractional initial value problem: is the Caputo-Fabrizio fractional derivative of Caputo sense. Here, we assume that y, f ∈ M 1 , so their Laplace transforms are well defined.
Definition 4 (see [1]). For 0 < α < 1, t > 0 and f ∈ M 1 , the Caputo-Fabrizio fractional derivative of Caputo sense is defined by where B(α) > 0 is a normalization function satisfying For the corresponding fractional integral and more properties of the derivative, we refer the readers to [1,5,[22][23][24]. In [15,17], the fractional initial value problems were transformed to equivalent initial value problems with integer derivatives. However, the technique is not valid for the multiterm initial value problems. e single term of the problem with n � 1 was discussed in [23], and the solution of the problem was obtained in a closed form. In this paper, we apply the Laplace transform to analyze the solutions of the fractional initial value problems (3) and (4). is paper is organized as follows: in Section 2, we present some preliminary results about the Caputo-Fabrizio fractional derivative and derive necessary and sufficient conditions to guarantee the existence of solutions to problems (3) and (4). We also obtain the exact solutions in closed forms using the Laplace transform. In Section 3, we present two examples to illustrate the validity of our results. Finally, we close up with some concluding remarks in Section 4.

Main Results
We start with the definition and main results concerning the Caputo-Fabrizio derivative. en, we present necessary and sufficient conditions for the solution of problems (3) and (4).
and * denotes to the convolution of two functions.
where P n (s) and Q m (s) are polynomials with degrees n and m, respectively, and are with real coefficients. If n < m, then the inverse Laplace transform of F(s) exists and can be evaluated using partial fractions.

Lemma 1.
For f ∈ M 1 , the following hold: (1) For any function f ∈ AC([0, T]), CFC D α 0 f is the convolution product of a continuous function and L 1 function, that is continuous e following holds for multiterm fractional initial value problems (3) and (4). (1) Let y ∈ C([0, +∞)) be a solution to multiterm fractional initial value problems (3) and (4) with (2) Let y 0 ≠ 0, f ∈ M 1 , c n+1 y(0) + f(0) � 0, and c n+1 ≠ n k�1 c k . en, applying the Laplace transform 2 Journal of Mathematics to equation (6) and using the convolution result, we have where F(s) � L(f)(t). e above equation yields We have where Q n− 1 (s) � n k�1 c k n j�1,j ≠ k (s + μ j ) and P n (s) � n k�1 (s + μ k ) are polynomials of degrees n − 1 and n, respectively. Substituting in equation (11), we have (13) or L(y)(s) � y 0 Q n− 1 (s) sQ n− 1 (s) − c n+1 P n (s) Now, the leading coefficient of the polynomial Q n− 1 (s) is n k�1 c k , and the leading coefficient of the polynomial P n (s) is 1. If c n+1 ≠ n k�1 c k , then sQ n− 1 (s) − c n+1 P n (s) is a polynomial of degree n. Let , Using the uniqueness result of the inverse Laplace transform, we have which completes the proof. (3) Let a solution y ∈ M 1 exists for y 0 ≠ 0 and f ∈ M 2 .
Starting from equation (14), let us define Define H 2 admits an inverse Laplace transform since sQ n− 1 (s) − c n+1 P n (s) is of degree n − 1. Moreover, since f ∈ M 2 , also s 2 F(s) admits an inverse Laplace transform, thus also g(s) (by the convolution rule). One can rewrite equation (14) as However, since sQ n− 1 (s) − c n+1 P n (s) is of degree n − 1 and y ≠ 0, the right-hand side does not admit an inverse Laplace transform while the left-hand side does, which is a contradiction.

Lemma 4. Let y(t) be a possible solution to multiterm fractional initial value problems (3) and (4), with
and H 3 is defined in equation (22).

Journal of Mathematics
Since c n+1 � n k�1 c k , we have sQ n− 1 (s) − c n+1 P n (s) is a polynomial of degree n − 1. Let is well defined. en, equation (14) yields If f ∈ M 2 , then L(y)(t) in equation (23) can be written as Applying the inverse Laplace transform, we have We summarize the obtained results in the following main theorem. □ Theorem 1. Consider multiterm fractional initial value problems (3) and (4).

Illustrative Examples
We discuss two main examples. e first one is a two-term problem where it holds that c n+1 � n k�1 c k . We show the existence of a solution for a specific case by imposing extra conditions. e second example is a three-term problem where it holds that c n+1 ≠ n k�1 c k . So, the existence of a unique solution is guaranteed. We present the solution in a closed form and discuss several special cases. Example 1. Consider the two-term fractional initial value problem: Since c 1 + c 2 � c 3 , then the problem has no solution for y(0) ≠ 0. For y(0) � f(0) � 0 and s 2 L(f)(t) has Laplace inverse, the problem admits a solution. To verify, let us find the solution given by equation (25). We have where (28) us, and the solution As a special case, let us consider f(t) � t 2 . en, f(0) � 0, L(f)(t) � (2/s 3 ), and s 2 L(f)(t) � (2/s) have Laplace inverse. us, For μ 1 � (2/3) and Example 2. As a second example, we consider the threeterm initial value problem: Since c 1 � c 2 � c 3 � c 4 � 1 and c 1 + c 2 + c 3 ≠ c 4 , the problem has a unique solution given by equation (17) provided that y(0) + f(0) � 0. We have 4 Journal of Mathematics Let then with and the solution is given by For f(t) � t, we have y(0) � − f(0) � 0, and thus For f(t) � 1, we have y(0) � − f(0) � − 1, and thus  , h 1 (t) � A 0 e − t + A 1 e − Δ 1 t +A 2 e − Δ 2 t , and h 2 (t) � h 1 (t) − 1.

Concluding Remarks
We obtained the solutions of a class of multiterm fractional initial value problems in closed forms using the Laplace transform. We have also discussed several necessary and sufficient conditions to guarantee the existence of solutions to the problem. Whether the results are extendable to wider classes of multiterm initial value problems or systems of fractional equations is left for future work.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.

Authors' Contributions
e first author has initiated the idea and obtained the analytical results. e second author did the numerical examples, shared the discussion, and approved the final draft of the manuscript.