Int. J. Anal. Appl. (2023), 21:65 Large Fractional Linear Type Differential Equations Ma’mon Abu Hammad1,∗, Iqbal Jebril1, Roshdi Khalil2 1Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11942, Jordan 2Department of Mathematics, The University of Jordan, Amman, Jordan ∗Corresponding author: m.abuhammad@zuj.edu.jo Abstract. This paper aims to handle some types of fractional differential equations with a fractional- order values β > 1. In particular, we propose a novel analytical solution called an atomic solution for certain fractional linear type differential equations as well as for some other types of partial differential equations with fractional-order values exceeding one. Some examples are provided to validate our findings. 1. Introduction One of the most principal connections between pure and applied mathematics is differential equa- tions in their three main types; ordinary, partial and fractional differential equations. A large number of applications that arise naturally in many fields of science and engineering might be described by fractional differential equations. Their solutions and investigations provide a remarkable growth to several mathematical approaches, see e.g. [1–5]. In 2014, R. Khalil introduced in [8] a newly scheme for dealing differential equations in their two types; ordinary and fractional ones. Such novel scheme relies on the theory of tensor product of Banach spaces, which can be employed for the aim of obtaining the so-called atomic solutions for or- dinary/fractional differential equations. It is commonly known that most of fractional (ordinary/partial) differential equations, which were treated in literature, are handled with fractional-order values between 0 and 1. Hence obtained many interesting results can be. By taking this notion into account. From Received: Apr. 19, 2023. 2020 Mathematics Subject Classification. 26A33. Key words and phrases. fractional calculus; fractional linear type equations atomic. https://doi.org/10.28924/2291-8639-21-2023-65 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-65 2 Int. J. Anal. Appl. (2023), 21:65 this point of view, the object of this paper is to study α−fractional (ordinary/partial) differential equa- tions for fractional-order value greater than one, i.e. β > 1. Accordingly, many notable theoretical results are proved and then used to some given fractional differential equations. The rest of this paper is arrange as follows: Section 2 recalls some basic facts and preliminaries related to the fractional calculus and atomic solution. Section 3 provides the main results of this work. Section 4 illustrates some applications of the derived theoretical results, followed by Section 5 that summarizes the whole conclusions of this work. 2. Preliminaries In this section, some needed basic facts and preliminaries in connection to the fractional calculus and the so-called atomic solution are recalled. In particular, the definition of the so-called α-conformable fractional derivative Dα is introduced , [8]. So, for x ∈ E ⊆ (0,∞), we have: Dαf (x)= lim �→0 f (x + �x1−α)− f (x) � . If the above limit exists, then Dαf (x) is called the α-conformable fractional derivative of f at x. As a result, if f is α-differentiable on (0, r) for some r > 0 and lim �→0+ Dαf (x) exists, then we define Dαf (0)= lim x→0 Dαf (x). For α ∈ (0,1] and f , g are α-differentiable at a point t, one can easily see that the conformable fractional derivative satisfies the following properties: (i) Dα(bf +cg)= bDα(f )+cDα(g), for all b,c ∈R, (ii) Dα(λ)=0, for all constant functions f (t)= λ, (iii) Dα(f g)= f Dα(g)+gDα(f ), (iv) Dα(f g )= gDα(f )−f Dα(g) g2 ,g(t) 6=0. In this regard, we list below the conformable fractional derivatives for certain functions: (i) Dα(tp)= ptp−α, (ii) Dα(sin(1 α tα))= cos(1 α tα), (iii) Dα(cos(1 α tα))=−sin(1 α tα), (iv) Dα(e 1 α tα)= e 1 α tα. On letting α = 1 in the above derivatives, we get their corresponding classical rules for ordinary derivatives. Further, one should notice that a function could be α-conformable differentiable at a point but it is not differentiable at that point. For example, if one takes f (t) = 2 √ t, then D 1 2(f )(0) = 1, while for the classical fractional derivative, D1(f )(0) does not exist. For more overview about fractional calculus and its applications, we refer to the references [6–12]. Many differential equations might be transformed to their corresponding fractional-order forms. These equations can have many applications in many branches of science and engineering, see [13–15]. Int. J. Anal. Appl. (2023), 21:65 3 It is commonly known that the main technique used to solve many partial differential equations is using Fourier series. From this point of view, fractional Fourier series was introduced in [16]. Such a concept proved to be very fruitful in solving fractional partial differential equations. But sometimes it is not possible to use separation of variables technique to deal with these equations. Here comes the concept of atomic solution, which would help us to face this problem. This concept coupled with how we can use it in solving fractional (ordinary/partial) differential equations will be provided in the following content. Definition 2.1. (Atomic solution) Let X and Y be two Banach spaces and X∗ be the dual of X. Assume x ∈ X and y ∈ Y . The operator T : X∗ → Y defined by: T(x∗)= x∗(x)y, is a bounded one rank linear operator. Then, we write x ⊗y for T . Such operators are called atoms, which are among the main ingredients in the theory of tensor products. Besides, the atoms are used in theory of best approximation in Banach spaces. In what follow, we recall one of the most known results that we shall need in our investigation. This result was extensively utilized in many research papers, see [17–20]. Lemma 2.1. If the sum of two atoms is an atom, then either the first components are dependent or the second ones are dependent. In other words, if x1⊗y1+x2⊗y2 = x3⊗y3 , then either x1 = x2 = x3. 3. Main Results For n < β < n+1, the β-fractional derivatives of the function f : [0,∞)→R was defined in [8] as Dβf (x)= lim �→0 f [β]−1(x + �x[β]−α)− f [β]−1(x) � . This turned out to be equivalent to the following assertion: Dβ(f (x))= x[β]−βf [β](x). (3.1) It is worth mentioning that if f is (n+1)-differentiable and n < β < n+1, then we have: Dβ(f (x))= xn+1−αf (n+1)(x)= xn+1−αDn+1x f (x), for n < β < n +1. Now, if one lets 1 < β < 2, then β = 1+ α, where 0 < α < 1. Hence, Dβf = D1+αf . Similarly, if n < β < n+1, then β = n+α, where 0 < α < 1, and so, Dβf = Dn+αf , for 0 < α < 1. Proposition 3.1. Let n < β < n+1 and α = β −n. Then we have: Dβf (x) = DαDnf (x)= Dαf (n)(x) = x1−αf (n+1)(x). 4 Int. J. Anal. Appl. (2023), 21:65 Proof. To prove this result, it should be noted that Dβf (x) = Dn+αf (x) = x(n+1)−βf (n+1)(x). But, from identity (3.1), one can write: Dβf (x) = x(n+1)−(n+α)f (n+1)(x) = x1−αf (n+1)(x). � Definition 3.1. An equation of the form: anD n+αy +an−1D n−1+αy + · · ·+a1Dαy +a0x1−αy =0 (3.2) is called an α-fractional linear equation of order n with 0 < α < 1. In the following content, we state and prove the main result of this section, which would be about the general solution of an α-fractional linear type equation. Theorem 3.1. Let 0 < α < 1 and anD n+αy +an−1D n−1+αy + · · ·+a1Dαy +a0x1−αy =0 (3.3) be an α-fractional linear type equation of order n. Let r0, r1, r2, · · · , rn, rn+1 be the roots of the equation an+1r n+1 +anr n + · · ·+a1r +a0 =0. Then by assuming no repetition of the roots, the general solution of (3.3) has the form: y = c0y0 + · · ·+cnyn +cn+1yn+1, where yk = erkx. Proof. By Proposition 3.1, equation (3.2) can be written in the form: anD α(Dny)+ · · ·+a1Dαy +a0x1−αy =0. This is just of the form: anx 1−αy(n+1) + · · ·+a1x1−αy 8 +a0x1−αy =0. (3.4) Thus (3.3) becomes: an+1y (n+1) +any (n) + · · ·+a1y 8 +a0y =0. Actually, the above equation represents an ordinary linear differential equation. So, the associated characteristic equation is of the form: an+1r n+1 +anr n + · · ·+a1r +a0 =0. (3.5) Int. J. Anal. Appl. (2023), 21:65 5 Solving the above equation to get the roots r0, r1, · · · , rn, rn+1. Consequently, if all the roots are distinct and real, then there are (n+1) solutions that have the forms: y0 = e r0x,y1 = e r1x, · · · ,yn+1 = ern+1x. Therefore, the general solution of the α-fractional linear type equation (3.3) is: yg = c0y0 + · · ·+cnyn +cn+1yn+1. � Remark 3.1. If a root is repeated in 3.5, then as is known in the theory of ordinary differential equations, some of the solutions are multiplied by some factors. 4. Applications In this part, we intend to provide a solution of a fractional ordinary differential equation, and the atomic solution of a fractional partial differential equation. Example 4.1. Let us have the following fractional differential equation: y( 3 2 ) +2y( 1 2 ) −3 √ xy =0. (4.1) Hence, we have, α = 1 2 and n =1. In other words, (3.4) becomes: D 1 2y ′ +2 √ xy ′ −3 √ xy =0. So, we have: √ xy ′′ +2 √ xy ′ −3 √ xy =0. This is equivalent to say: y ′′ +2y ′ −3y =0. (4.2) The characteristic equation of (4.1) is then of the form: r2 +2r −3=0, with the roots r1 =−3 and r2 =1. Hence, we obtain: yg = c1e −3x +c2e x. (4.3) One can easily check that (4.3) is the general solution of (4.1). In the following content, we aim to find the atomic solution of a certain fractional partial differential equation of order α > 3 2 . 6 Int. J. Anal. Appl. (2023), 21:65 Example 4.2. For x > 0 and y > 0, consider the following fractional partial differential equation: D 3 2 x D 3 2 y u +D 1 2 x D 1 2 y u = √ xyu, (4.4) with the following initial conditions u(0,0)=1, ∂u ∂x (0,0)=1, ∂u ∂y (0,0)=1. (4.5) If one wants to solve the above problem with the use of our proposed approach, we firs assume: U(x,y)= P(x)Q(y). Now, conditions (4.5) yield the following condition on P and Q: P ′(0)= P(0)=1 and Q′(0)= Q(0)=1. (4.6) This means: P 3 2(x)Q 3 2(y)+P 1 2(x)Q 1 2(y)= √ xyP(x)Q(y). From the analysis of Dn+αu, we get: √ xP ′′(x) √ yQ′′(y)+ √ xP ′(x) √ yQ′(y)= √ xyP(x)Q(y). Hence, we have: P ′′(x)Q′′(y)+P ′(x)Q′(y)= P(x)Q(y). (4.7) With the use of Lemma 2.1, we have two cases: Case (a): P ′′(x)= P ′(x)= P(x). Case (b): Q′′(x)= Q′(x)= Q(x). Now for Case (a), we have the following three situations: (1) P ′′(x)= P ′(x). (2) P ′′(x)= P(x). (3) P ′(x)= P(x). Observe that situation 1 can be solved to get P(x) = c1 + c2ex. But from conditions (4.6, we get P(x) = ex. In the same regard, solving both situations 2 and 3 gives also the same solution P(x)= ex. Now substituting this result in (4.5), we get: exQ′′(y)+exQ′(y)= exQ(y). Immediately, we have: Q′′(y)+Q′(y)−Q(y)=0. So, we have r2+ r −1=0 with the roots r1 = −1+ √ 5 2 and r2 = −1− √ 5 2 . Now, from conditions (4.6), we get: Q(y)= (1+ √ 5)er1y − √ 5er2y. Int. J. Anal. Appl. (2023), 21:65 7 Therefore, our proposed atomic solutions has the form: U(x,y)= P(x)Q(y)= ex((1+ √ 5)er1y − √ 5er2y). Since equation (4.6) is symmetric in P and Q, then Case (b) gives the following atomic solution: U(x,y)= ((1+ √ 5)er1x − √ 5er2x)ey. (4.8) However, in order to see how the atomic solution (4.8) of equation (4.4) looks like, we plot such solution in Figure 4.2. Figure 1. The atomic solution (4.8) of equation (4.4). 5. Conclusions This paper has successfully introduced a new analytical method for handling the study of a class of α-fractional (ordinary/partial) differential equations with fractional-order value greater than one via atomic solutions method. The theory of tensor product of Banach spaces coupled with some properties of atoms operators have been utilized for achieving such a notion. Acknowledgment: This research was supported by Al-Zaytoonah University of Jordan fund. Project number 32 / 17 / 2022-2023. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] I.M. Batiha, S. Alshorm, A. Ouannas, S. Momani, O.Y. Ababneh, M. Albdareen, Modified Three-Point Frac- tional Formulas with Richardson Extrapolation, Mathematics. 10 (2022), 3489. https://doi.org/10.3390/ math10193489. [2] I.M. Batiha, S. Alshorm, I.H. Jebril, M.A. Hammad, A Brief Review about Fractional Calculus, Int. J. Open Probl. Comput. 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Math. 15 (2022), 106-125. https://doi.org/10.3390/sym14122583 https://doi.org/10.3390/axioms12030306 https://doi.org/10.3390/axioms12030306 https://doi.org/10.18576/pfda/080305 https://doi.org/10.1016/j.cam.2014.10.016 https://doi.org/10.1016/j.cam.2014.10.016 https://doi.org/10.1155/2018/6928130 https://doi.org/10.1155/2018/6928130 https://doi.org/10.1016/j.cam.2014.01.002 https://doi.org/10.1016/j.matcom.2011.03.006 https://doi.org/10.3390/math10010006 https://doi.org/10.28919/jmcs/3806 https://doi.org/10.1109/JEEIT.2019.8717456 https://doi.org/10.1109/JEEIT.2019.8717456 https://doi.org/10.1016/j.cam.2015.04.049 https://doi.org/10.1017/s0305004100065713 https://doi.org/10.1017/s0305004100065713 1. Introduction 2. Preliminaries 3. Main Results 4. Applications 5. Conclusions References