EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 397-402 ISSN 1307-5543 – ejpam.com Published by New York Business Global Atomic Solution of Poisson Type Equation Rouba Shatnawi Abstract. In this paper we find a certain solution of Poisson type fractional differential equation using theory of tensor product of Banach spaces. 2020 Mathematics Subject Classifications: 46M05, 46Bxx, 35J05, 44-XX Key Words and Phrases: Tensor product, Banach spaces, Fractional Poisson equation, Con- formable derivative 1. Introduction In [5], a new definition, which is called conformable fractional derivative was intro- duced. For a given a function f : [0,∞) → R. The conformable fractional derivative of f of order α for α ∈ (0, 1], is defined by: Dα(f)(t) = lim ε→0 f(t+ εt1−α)− f(t) ε . If the conformable fractional derivative of f of order α exists, then we simply say f is α−differentiable. For α ∈ (0, 1] and for f, g which α−differentiable functions at a point t, the conformable derivative satisfies: 1. Dα(af + bg) = aDα(f) + bDα(g), for all a, b ∈ R. 2. Dα(λ) = 0, for all constant functions f(t) = λ. 3. Dα(fg) = fDα(g) + gDα(f). 4. Dα(fg ) = gDα(f)−fDα(g) g2 . We list here the fractional derivatives of certain functions: 1. Dα(c) = 0, where c is constant. 2. Dα(ect) = ct1−αect, c ∈ R 3. Dα(cos bt) = −bt1−α sin bt, b ∈ R. 4. Dα(sin bt) = bt1−α cos bt, b ∈ R. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4318 Email address: roubak35@gmail.com (R. Shatnawi) https://www.ejpam.com 397 © 2022 EJPAM All rights reserved. R. Shatnawi / Eur. J. Pure Appl. Math, 15 (2) (2022), 397-402 398 5. Dα( 1α t α) = α( 1α t α−α) = 1. 6. Dα(sin 1 α t α) = cos 1 α t α. 7. Dα(cos 1 α t α) = − sin 1 α t α. 8. Dα(e 1 α tα) = e 1 α tα . 2. Atomic solution Let X and Y be Banach spaces and X∗ denotes the dual of X. For x ∈ X and y ∈ Y define T(x,y) : X ∗ → Y as: T(x,y)(x ∗) = ⟨x∗, x⟩ y. We denote T(x,y) by x⊗ y and we call x⊗ y an atom [6]. Atoms are used in theory of best approximation in Banach spaces, see [3]. One of the known results, see [4], that we need in our paper is that: If the sum of two atoms is an atom, then either the first components are dependent or the second ones are dependent. Let us write D2αf to mean DαDαf . Further we write f (α), f (2α) to denote Dαf , D2αf respectively. If u is a function of two variables, say x, y , we write Dα xu for the partial α−derivative of u with respect to x, and by D2α x u we mean Dα xD α xu. Similarly for derivatives with respect to y. Our main object in this paper is to find an atomic solution of the Poisson type fractional differential equation: D2α x u(x, y) +Dα xD β yu(x, y) = f(x, y), 0 < α, β < 1, (1) where f(x, y) is a given function. 3. Procedure In order to find a certain atomic solution, assume f(x, y) = A(x)B(y), is given. Now, put u(x, y) = P (x)Q(y) (2) and assume that P (0) = 1, P (α)(0) = 1, and Q(0) = 1. R. Shatnawi / Eur. J. Pure Appl. Math, 15 (2) (2022), 397-402 399 Substitute u(x, y) = P (x)Q(y) in (1) to get: P (2α)(x)Q(y) + P (α)(x)Q(β)(y) = A(x)B(y) (3) This can be written in tensor product form as: P (2α)(x)⊗Q(y) + P (α)(x)⊗Q(β)(y) = A(x)⊗B(y) (4) Now, we have a situation where the sum of two atoms is an atom. Hence, we have two cases: Case (i) P (2α)(x) = P (α)(x) = A(x). Consider P (2α)(x) = P (α)(x) this is a 2α− order linear differential equation [ 7], so the corresponding auxiliary equation is r2 − r = 0 Hence r = 0, 1,and P (x) = c1 + c2e xα/α Since P (α)(0) = 1,we have c2 = 1 and since P (0) = 1, then c1 = 0 So P (α)(x) = c2e xα/α Hence P (x) = ex α/α. (5) Thus A(x) must equal to ex α/α in order to an atomic solution to be exist. Substitute in (3) to get ex α/αQ(y) + ex α/α(x)Q(β)(y) = ex α/αB(y). Hence Q(y) +Q(β)(y) = B(y). (6) This is a linear fractional differential equation of order β. Hence, using result in [1], we multiply equation (6) by the integrating factor R. Shatnawi / Eur. J. Pure Appl. Math, 15 (2) (2022), 397-402 400 µ(y) = eIβ(1) = e y∫ a 1 t1−β dt = ey β/β. to get ey β/βQ(y) + ey β/βQ(β)(y) = ey β/βB(y), Dβ[ey β/βQ(y)] = ey β/βB(y). Hence Q(y) = e−yβ/βIβ[e yβ/βB(y)]. So Q(y) = e−yβ/β y∫ et β/βB(t) t1−β dt. (7) Equations (5) and (7) give that: u(x, y) = ex α/α−yβ/β y∫ 0 ey β/βB(y) t1−β dt. (8) This is the atomic solution for case (i). Case (ii) : Q(β)(y) = Q(y) = B(y). Consider Q(β)(y) = Q(y). Using conformable derivative properties, we get Q(y) = aey β/β Apply Q(0) = 1 to get Q(y) = ey β/β. (9) Consequently, if we want to get an atomic solution, B(y) must equal to ey β/β. Substitute in (3) to get ey β/βP (2α)(x) + ey β/βP (α)(x) = ey β/βA(x). Hence P (2α)(x) + P (α)(x) = A(x). (10) REFERENCES 401 The homogeneous equation P (2α)(x) + P (α)(x) = 0 is solved to give Ph(x) = b1e −xα/α + b2 According to [2], the particular solution of (10) is: Pp(x) = −P1(x) ∫ A(x)P2(x) W [P1, P2](x)x2−2α dx+ P2(x) ∫ A(x)P1(x) W [P1, P2](x)x2−2α dx, where P1(x) = e−xα/α, P2(x) = 1,W [P1, P2] = −xα−1 α e−xα/α. Simplifying to get Pp(x) = e−xα/α ∫ A(x) (x α−1 α e−xα/α)x2−2α dx− ∫ A(x)e−xα/α (x α−1 α e−xα/α)x2−2α dx, Pp(x) = αe−xα/α ∫ A(x) x1−αe−xα/α dx− α ∫ A(x) x1−α dx. So P (x) = αe−xα/α ∫ A(x) x1−αe−xα/α dx− α ∫ A(x) x1−α dx+ b1e −xα/α + b2. (11) Hence, by (9) and (11) u(x, y) = αey β/β−xα/α ∫ A(x) x1−αe−xα/α dx−αey β/β ∫ A(x) x1−α dx+ b1e yβ/β−xα/α+ey β/βb2. (12) This is the atomic solution for case (ii). References [1] M Al-Horani, R Khalil, and I Aldarawi. Fractional cauchy euler di¤ erential equation. Journal of Computational Analysis & Applications, 28(2), 2020. [2] Mohammed Al Horani, M Abu Hammad, and Roshdi Khalil. Variation of parameters for local fractional nonhomogenous linear differential equations. J. Math. Computer Sci, 16:147–153, 2016. [3] W Deeb and R Khalil. Best approximation in l (x, y). In Mathematical Proceed- ings of the Cambridge Philosophical Society, volume 104, pages 527–531. Cambridge University Press, 1988. [4] Roshdi Khalil. Isometries of lp* lp. Tam. J. Math., 16:77–85, 1985. REFERENCES 402 [5] Roshdi Khalil, Mohammed Al Horani, Abdelrahman Yousef, and Mohammad Sabab- heh. A new definition of fractional derivative. Journal of computational and applied mathematics, 264:65–70, 2014. [6] W. A. Light and Elliott Ward Cheney. Approximation theory in tensor product spaces. 1985.