International Journal of Analysis and Applications Volume 17, Number 6 (2019), 1019-1033 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-1019 NEW RESULTS ON THE CONFORMABLE FRACTIONAL SUMUDU TRANSFORM: THEORIES AND APPLICATIONS ZEYAD AL-ZHOUR1,∗, FATIMAH ALRAWAJEH2, NOUF AL-MUTAIRI2 AND RAED ALKHASAWNEH3 1Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia 2Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia 3Department of General Courses, College of Applied Studies and Community Service, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia ∗Corresponding author: zalzhour@iau.edu.sa; zeyad1968@yahoo.com Abstract. In this paper, we generalize the formula of Sumudu transform to the conformable fractional order and some interesting and important rules of this transform and conformable fractional Laplace transform are derived and discussed. Moreover, we present the general analytical solution of a singular and nonlinear conformable fractional Poisson- Boltzmann differential equation based on the conformable fractional Sumudu transform. Also, our proposed method is applied successfully for obtaining the general solutions of some linear and nonhomogeneous conformable fractional differential equations. Finally, the results show that our proposed method is an efficient and can be applied for finding the general solutions of the all cases realted to the conformable fractional differential equations. Received 2019-06-28; accepted 2019-08-07; published 2019-11-01. 2010 Mathematics Subject Classification. 26A33, 44A05, 44A10. Key words and phrases. Conformable fractional operator; Sumudu transform; Laplace transform; Poisson-Boltzmann dif- ferential equation; Adomain decomposition method. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 1019 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-1019 Int. J. Anal. Appl. 17 (6) (2019) 1020 1. Introduction and Preliminaries Various types of fractional derivatives were introduced such as: Riemann-Liouville, Caputo, Modified Riemann-Liouville, Hadamard, Grunwald, Letnikov and Riesz operators. The most two popular definitions of fractional derivatives of order α > 0 are ([1], [2], [3], [4], [5], [6], [7]): (i) Riemann-Liouville fractional derivative: RDαf(x) = 1 Γ(n−α) dn dxn ∫ x 0 (x− t)n−α−1f(t)dt. (1.1) (ii) Caputo fractional derivative: cDαf(x) = 1 Γ(n−α) ∫ x 0 (x− t)n−α−1 dn dxn f(t)dt. (1.2) Here Γ(.) denotes to the gamma function. In the last few decades, fractional differentiation has been used applied scientists for solving several fractional differential equations and they proved that the fractional calculus is very useful in several fields of applications with some restrictions such as: Physics (quantum mechanics and thermodynamics), chemistry, biology, economics, engineering, signal and image processing and control theory ([2], [3], [4], [5], [6], [7], [8], [9], [10]). Very recently, the discrepancies between known definitions can be solved in simple way by presenting a new fractional definition which is called the ”Conformable Fractional Derivative ” and defined for a given function f : [0,∞) → R of fractional (ordinary) order α > 0 by Khalil et el. [11] as follows: Dnαf(x) = lim ε→0 f[α]−1(x + εx[α]−α) −f[α]−1(x) ε , n− 1 < α ≤ n, x > 0, (1.3) where [α] is the smallest integer number greater than or equal α and n ∈ N. As a special case, if 0 < α ≤ 1, then we have: Dαf(x) = lim ε→0 f(x + εx1−α) −f(x) ε . (1.4) This definition is very easy for calculating derivatives and solving fractional differential equations com- pared with other fractional definitions such as the definitions of Liouville-Reimman and Caputo fractional derivatives. It has received a lot of attention and many applications have been remodeled using this new definition ([12], [13], [14], [15], [16], [17], [18], [19]). Moreover, it has many interesting advantages which make its easier and flexible more than the definitions of other fractional derivatives ([12], [13], [14], [15], [16], [17], [18], [19]). Some of these adavantages are: (i) It satisfies the all concepts of ordinary calculus such as: quotient, product and chain rules, Rolle’s theorem and mean-value theorem. (ii) A non-differentiable function can be α- differentiable in the conformable sense. (iii) It can be easily extended to generalize many integral transforms such as: Laplace, Mellin, Natural and Sumudu transforms. Int. J. Anal. Appl. 17 (6) (2019) 1021 Also, the conformable fractional integral has been also defined of order α > 0 by: Iαf(x) = ∫ x 0 f(t) tα−1dt. (1.5) In fact, if f(x) is an n -differentiable function at x > 0 and α ∈ (0, 1], n ∈ N, then [11]: (i) Dαf(x) = x1−α d dx f(x), (1.6) (ii) DαIαf(x) = f(x). (1.7) In the literature there are several works on the theory and applications of integral transforms such as the Laplace and Sumudu transforms ([20], [21], [22], [23], [24], [25], [26], [27], [28]) that are widely used in physics, electric circuit theory, astronomy, as well as engineering and sciences. Sumudu transform was introduced by Watugala [20 ] and defined over the following set of functions: A = { f(x) : ∃ M,τ1,τ2 > 0, |f(x)| < Me |x| τj , if x ∈ (−1)j × [0,∞),j = 1, 2 } , (1.8) as follows: S[f(x)] = F(u) = ∫ ∞ 0 e−xf(ux)dx, u ∈ (−τ1,τ2), (1.9) or equivalently: S[f(x)] = F(u) = 1 u ∫ ∞ 0 e− x u f(x)dx, u > 0. (1.10) This transform has many interesting advantages over other integral transforms especially the ”unity” feature which could provide convergence when solvimg differential equations and also used to solve problems without resorting to a new frequency domain while for example, Laplace transform must satisfy the Dirichlet condition which is f(x) must be piecewise continuous which means that it must be single valued but can have a finite number of finite isolated discontinuues for x > 0. Also this transform possesses many interesting advantages which make its visualization easier and some of these advantages can be found in ([20], [21], [22], [23], [24], [25], [26], [27], [28]). There are several methods and techniques were applied for obtaining the analytical and numerical solution of such nonlinear and singular fractional differential equations such as the fractional Laplace transform method [19], generalized Kudryashov method [29], Adomian decomposition method ([30], [31]) and modified Kudryashov method [32]. Note that the fractional order differential equations are now span a half-century or more and play a crucial role in several theoretical and applied sciences such as, but certainly not limited to, theoretical biology and ecology, solid state physics, viscoelasticity, fiber optics, signal processing and electric Int. J. Anal. Appl. 17 (6) (2019) 1022 control theory, stochastic based finance and thermodynamics ([19], 20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]). Finally, the theory of thermal explosions originally proposed by Frank-Kamenetzky ([33], [34]) and revis- ited by Barenblatt et al. [ 35] has been required the solution of the nonlinear Poisson-Boltzmann differential equation which is given and solved numerically by Chambré [36]. The ordinary Poisson-Boltzmann differen- tial equation as follows ([36], [37]): d2y dx2 + β x dy dx = ey (1.11) This equation is also a very useful in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor; in the critical theory of gravitation and combustion or explosion. and to describe the distribution of the electric potential in solution in the direction normal to a charged surface and from this equation, many other equations can been derived with a number of different assumptions. In this paper, we extend the definition of Sumudu transform to fractional order and derive a list of inter- esting rules and properties of this extension which including the rules of the conformable fractional Laplace transform. Also, a very nice relationship between conformable fractional Sumudu and Laplace transforms is derived and proved. Moreover, we give two important and attractive applications for conformable fractional Sumudu transform. These applications are: Firstly, we apply the conforrmable fractional Sumudu transform together with Adomain decomposition method for presenting the general analytical solution of a singular and nonlinear conformable fractional Poisson-Boltzmann differential equation. Secondly, we also apply the conformable fractional Sumudu transform to find the general solutions of some linear and nonhomogeneous conformable fractional differential equations. Finally, the results show that our proposed method is an effi- cient method and applied successfully to find the general solutions of the all cases (Singular, linear, nonlinear, homogeneous and nonhomogeneous) realted to the conformable fractional differential equations. 2. Conformable fractional Laplace and Sumudu transforms In this Section, we introduce (with proofs) a list of important basic rules and properties for the conformable fractional Laplace and Sumudu transforms involving the nice relationship between of these transforms which are playing a central role in the solutions of conformable fractional differential equations. Definition 2.1.: Let f : [0,∞) → R be a given function and 0 < α ≤ 1. Then the conformable fractional Laplace transform of f is defined as: Lα{f(x)} = Fα(s) = ∫ ∞ 0 e−s xα α f(x)xα−1dx, (2.1) provided the integral exists. Int. J. Anal. Appl. 17 (6) (2019) 1023 Theorem 2.1.: Let f : [0,∞) → R be a given function and 0 < α ≤ 1. Then Lα{Dαf(x)} = sFα(s) −f(0), s > 0. (2.2) Proof. By using Definition 2.1 and integration by parts, we have: Lα{Dαf(x)} = ∫ ∞ 0 e−s xα α Dαf(x)xα−1dx = ∫ ∞ 0 e−s xα α x1−αf′(x)xα−1dx = ∫ ∞ 0 e−s xα α f′(x)dx = [ e−s xα α f(x) ]∞ 0 − ∫ ∞ 0 f(x) ( − s α α xα−1 ) e−s xα α dx = lim c→∞ [ e−s xα α f(x) ]c 0 + s ∫ ∞ 0 f(x) xα−1e−s xα α dx = sFα(s) −f(0), which completes the proof of Theorem 2.1. � Theorem 2.2 .: Let f : [0,∞) → R be a function. Then Fα(s) = L { f(αx) 1 α } (s). (2.3) Proof. By using Definition 2.1 and letting v = x α α , we have: Fα(s) = ∫ ∞ 0 e−s xα α f(x)xα−1dx = ∫ ∞ 0 e−svf(αv) 1 α dv = L { f(αv) 1 α } , which completes the proof of Theorem 2.2. � Theorem 2.3.: Let c, a, p ∈ R and 0 < α ≤ 1. Then (i) Lα{c}(s) = c s , s > 0. (2.4) (ii) Lα{xp}(s) = α p α Γ(1 + p α ) s1+ p α , s > 0. (2.5) (iii) Lα { e axα α } (s) = 1 s−a , s > a. (2.6) (iv) Lα { sin( axα α ) } (s) = a s2 + a2 , s > 0. (2.7) (v) Lα { cos( axα α ) } (s) = s s2 + a2 , s > 0. (2.8) (vi) Lα { sinh( axα α ) } (s) = a s2 −a2 , s > |a| . (2.9) (vii) Lα { cosh( axα α ) } (s) = s s2 −a2 , s > |a| . (2.10) Proof. Follows by applying Definition 2.1. � Int. J. Anal. Appl. 17 (6) (2019) 1024 Theorem 2.4. : Let f and g : [0,∞) → R andl let λ, µ, a ∈ R and 0 < α ≤ 1. Then (i) Lα{λf(x) + µg(x)} = λFα(s) + µGα(s),s > 0, (2.11) (ii) Lα { e−a xα α f(x) } = Fα(s + a), s > |a| , (2.12) (iii) Lα{Iαf(x)} = Fα(s) s ,s > 0, (2.13) (iv) Lα { xnα αn f(x) } = (−1)n dn dsn Fα(s),s > 0, (2.14) (v) Lα{(f ∗g)(x)} = Fα(s)Gα(s), s > 0, (2.15) where f ∗g is the convolution product of f and g. Proof. (i) Straightforward (ii) By using Thereom 2.2, we get: Lα { e−a xα α f(x) } = L { e −a α (αx) α. 1 α f(αx) 1 α } = L { e−axf(αx) 1 α } = L { f(αx) 1 α } |s→s+a = Fα(s + a). (iii) By using Theorem 2.1, we have: Lα{DαIαf(x)} = sLα{Iαf(x)}− Iαf(0). Since Iαf(0) = 0, then we obtain: Fα(s) = sLα{Iαf(x)} Lα{Iαf(x)} = Fα(s) s . (vi) By using Theorem 2.2 , we obtain: Lα { xnα αn f(x) } = L { (αx)nα. 1 α αn f(αx) 1 α } = L { xnf(αx) 1 α } = (−1)n dn dsn Fα(s). (v) By using Theorem 2.2, we get: Lα{(f ∗g)(x)} = L { (f ∗g)(αx) 1 α } = L { f(αx) 1 α } L { g(αx) 1 α } = Fα(s) Gα(s). Which completes the proof of Theorem 2.4. � Int. J. Anal. Appl. 17 (6) (2019) 1025 Definition 2.2.: Over the following set of functions: Aα = { f(x) : ∃ M,τ1,τ2 > 0, |f(x)| < Me |xα| ατj , if xα ∈ (−1)j × [0,∞),j = 1, 2 } , (2.16) then the conformable fractional Sumudu transform of f can be generalized by: Sα[f(x)] = Fα(u) = 1 u ∫ ∞ 0 e −xα αu f(x)dαx, (2.17) where dαx = x α−1dx , 0 < α ≤ 1 and provided the integral exists. The relationship between the conformable fractional Sumudu and conformable fractional Laplace trans- forms is given as in the next result. Theorem 2.5.: Let f : [0,∞) → R be a given function and 0 < α ≤ 1.Then Sα[f(x)] = Fα( 1u) u . (2.18) Proof. Applying Definition 2.2, we get: Sα[f(x)] = 1 u ∫ ∞ 0 e− xα αu f(x)dαx. Letting v = x α α ⇒ dv = xα−1dx, then we have: Sα[f(x)] = 1 u ∫ ∞ 0 e− v u f(αv) 1 α dv = 1 u L { f(αv) 1 α } s→ 1 u = Fα( 1u) u , which completes the proof of Theorem 2.5. � Theorem 2.6. : Let f : [0,∞) → R be a given function and 0 < α ≤ 1.Then Sα[Dαf(x)] = Fα(u) u − f(0) u . (2.19) Proof. Using Theorems 2.5 and 2.1, we get: Sα[Dαf(x)] = Lα{Dαf(x)}s→ 1 u u = [sFα(s) −f(0)]s→ 1 u u = Fα( 1u) u2 − f(0) u = Fα(u) u − f(0) u , which completes the proof of Theorem 2.6. � Theorem 2.7.: Let f : [0,∞) → R be an n-differentiable function and 0 < α ≤ 1.Then Sα[Dnαf(x)] = Sα[f(x)] un − f(0) un , 0 < α ≤ 1 and n ∈ N. (2.20) Proof. Follows by using the induction process on n and Theorem 2.6. � Int. J. Anal. Appl. 17 (6) (2019) 1026 Now we introduce the basic rules of conformable fractional Sumudu transform for some certain functions as in the next result. Theorem 2.8.: Let a, c ∈ R and 0 < α ≤ 1.Then we have: (i) Sα[c] = c. (2.21) (ii) Sα [ ea xα α ] = 1 1 −au , u > 1 a . (2.22) (iii) Sα [ sin(a xα α ) ] = au 1 + a2u2 , u > 1 |a| . (2.23) (iv) Sα [ cos(a xα α ) ] = 1 1 + a2u2 , u > 1 |a| . (2.24) (v) Sα [ sinh(a xα α ) ] = au 1 −a2u2 , u > 1 |a| . (2.25) (vi) Sα [ cosh(a xα α ) ] = 1 1 −a2u2 , u > 1 |a| . (2.26) (vii) Sα [ xnα αn ] = Γ(n + 1)un, u > 0. (2.27) Proof. By using Theorems 2.5 and 2.2, we have: (i) Sα[c] = Lα{c}s→ 1 u u = { c s } s→ 1 u u = c. (ii) Sα [ ea xα α ] = Lα { ea xα α } s→ 1 u u = L{eax}s→ 1 u u = { 1 s−a } s→ 1 u u = 1 1 −au . (iii) Sα [ sin(a xα α ) ] = Lα { sin(ax α α ) } s→ 1 u u = L{sin(ax)}s→ 1 u u = { a s2+a2 } s→ 1 u u = au 1 + a2u2 . Similarly we can prove (iv) and then can easy to prove (v) and (vi) based on (iii) and (iv) of Theorem 2.8. (vii) Sα [ xnα αn ] = Lα { xnα αn } s→ 1 u u = Γ(n + 1)un+1 u = Γ(n + 1)un. Which completes the proof of Theorem 2.8. � Int. J. Anal. Appl. 17 (6) (2019) 1027 Theorem 2.9.: Let f and g : [0,∞) → R given functiins and let λ, µ ∈ R and 0 < α ≤ 1.Then we have: (i) Linearly property: Sα{λf(x) + µg(x)} = λFα(u) + µGα(u), (2.28) (ii) Shifting property: Sα [ e−a xα α f(x) ] = Fα( 1u + a) u . (2.29) (iii) Integral property: Sα [Iαf(x)] = Fα( 1 u ). (2.30) (iv) Convolution property: Sα [(f ∗g)(x)] = uFα(u)Gα(u). (2.31) (v) Power product property: Sα [ xnα αn f(x) ] = 1 u [ (−1)n dn dsn Fα(s) ] s→ 1 u . (2.32) Proof. (i) Straightforward by using Definition 2.2. (ii) By applying Theorems 2.5 and 2.2, we have: Sα [ e−a xα α f(x) ] = Lα { e−a xα α f(x) } s→ 1 u u = L { e−axf(αx) 1 α } s→ 1 u u = Fα( 1u + a) u . (iii) By using Theorem 2.5 and Eq.(2.13), we have: Sα [Iαf(x)] = Lα{Iαf(x)}s→ 1 u u = Fα( 1 u ). (iv) By applying Theorem 2.5 and Eq. (2.15), we have: Sα [(f ∗g)(x)] = Lα{f ∗g}s→ 1 u u = [Fα(s)Gα(s)]s→ 1 u u = Fα( 1u)Gα( 1 u ) u = uFα(u)Gα(u). (v) By using Theorem 2.5 and Eq.(2.3), we have: Sα [ xnα αn f(x) ] = Lα { xnα αn f(x) } s→ 1 u u = 1 u L { xnf(αx) 1 α } s→ 1 u = 1 u [ (−1)n dn dsn Fα(s) ] s→ 1 u . Which completes the proof of Theorem 2.9. � Int. J. Anal. Appl. 17 (6) (2019) 1028 Theorem 2.10.: Let y = f(x) be an n-differentiable function and 0 < α ≤ 1. Then Sα [ xα α Dnαy ] = u d du (u Sα[Dnαy]) , n = 1, 2, ... (2.33) Proof. Let dαx = x α−1dx , then we have: Sα[Dnαy] = 1 u ∫ ∞ 0 e− xα αu Dnαy dαx d du (u Sα[Dnαy]) = ∫ ∞ 0 d du e− xα αu Dnαy dαx = 1 u2 ∫ ∞ 0 e− xα αu xα α Dnαy dαx = 1 u ∫ ∞ 0 1 u e− xα αu xα α Dnαy dαx = 1 u Sα [ xα α Dnαy ] . which concludes the result as in Eq. (2.33). � 3. Analytical solution of a singular and nonlinear conformable fractional Poisson-Boltzmann differential equation In this Section, we apply the conformable fractional Sumudu transform togethor with Adomain decompo- sition method to present the general analytical solution of the following singular and nonlinear conformable fractional Poisson Boltzmann differential equation: D2αy + β xα Dαy = ey, y(0) = 0, (3.1) where 0 < α ≤ 1 and β > 0. To solve this problem: Multiply Eq. (3.1) by x α α and take Sαof both sides, then by applying Theorems 2.10 and 2.6, we get: u d du ( Yα(u) u ) + γ ( Yα(u) u ) = Sα ( xα α ey ) , (3.2) where γ = β α . Since u 6= 0 and integrating Eq. (3.2) with respect to z, we get: Yα(u) u = ∫ u 0 1 z Sα [ xα α ey ] dz −γ ∫ u 0 Yα(z) z2 dz (3.3) Yα(u) = u ∫ u 0 1 z Sα [ xα α ey ] dz −uγ ∫ u 0 Yα(z) z2 dz. (3.4) Suppose the solution y(x) and non-linear function ey by the following infinite series: y(x) = ∞∑ n=0 yn(x) , e y = ∞∑ n=0 An(x), (3.5) Int. J. Anal. Appl. 17 (6) (2019) 1029 where An is the Adomain polynomial of e y. Note that: A◦ = e y◦, A1 = y1e y◦, (3.6) A2 = y2e y◦ + 1 2! y21e y◦, ... Substituting Eq. (3.5) into Eq.(3.4), we have: Sα [ ∞∑ n=0 yn(x) ] = u ∫ u 0 1 z Sα [ xα α An ] dz −uγ ∫ u 0 Sα[ ∑∞ n=0 yn(x)] z2 dz. (3.7) By taking S−1α of both sides of Eq (3.7), we obtain: ∞∑ n=0 yn(x) = S−1α [ u ∫ u 0 1 z Sα[ xα α An]dz −uγ ∫ u 0 Sα[ ∑∞ n=0 yn(x)] z2 dz ] . Thus, the general solution of Eq.(3.1) is given by: y(x) = S−1α [ u ∫ u 0 1 z Sα[ xα α An]dz −uγ ∫ u 0 Sα[ ∑∞ n=0 yn(x)] z2 dz ] , (3.8) where y◦ = 0 yn+1 = S−1α [ u ∫ u 0 1 z Sα[ xα α An]dz −uγ ∫ u 0 Sα[yn(x)] z2 dz ] . (3.9) Now, (i) For n = 0, then: y1 = S−1α [ u ∫ u 0 1 z Sα[ xα α A◦]dz −uγ ∫ u 0 Sα[y◦] z2 dz ] = S−1α [ u ∫ u 0 1 z Sα[ xα α ]dz ] = 1 2! x2α α2 . (3.10) (ii) For n = 1, then: y2 = S−1α [ u ∫ u 0 1 z Sα[ xα α A1]dz −uγ ∫ u 0 Sα[y1] z2 dz ] = S−1α [ u ∫ u 0 1 Γ(3)z Sα[ x3α α3 ]dz −uγ ∫ u 0 Sα[x 2α α2 ] Γ(3)z2 dz ] = S−1α [ Γ(4) 3Γ(3) u4 −γu2 ] = 1 4! x4α α4 − γ 2! x2α α2 . (3.11) Int. J. Anal. Appl. 17 (6) (2019) 1030 (iii) For n = 2, then y3 = S−1α [ u ∫ u 0 1 z Sα [ xα α A2 ] dz −uγ ∫ u 0 Sα[y2] z2 dz ] = S−1α [ u ∫ u 0 1 z Sα [ xα α y2e y◦ + 1 2! xα α y21e y◦ ] dz −uγ ∫ u 0 Sα[y2] z2 dz ] = Γ(6) 5Γ(7) x6α α6 − Γ(4) 6Γ(5) γ x4α α4 + Γ(6) 40Γ(7) x6α α6 − γ2 2! x2α α2 − 1 3Γ(5) x4α α4 . (3.12) Now by above discussion, then it is easy to get the power series solution of Eq. (3.1) as follows: y(x) = y◦ + y1 + y2 + y3 + ....... = 1 2! x2α α2 + 1 4! x4α α4 − γ 2! x2α α2 + ..... ((3.13)) 4. Analytical solutions of some linear and nonhomogeneous conformable fractional differential equations In this Section, we apply the conformable fractional Sumudu transform to present the general analytical solutions of some linear and nonhomogeneous conformable fractional differential equations as in the following problems. Problem 4.1.: Consider the following linear and nonhomogeneous conformable fractional differential equation: D3αy + Dαy = xα α , y(0) = 0. (4.1) By applying the conformable fractional Sumudu transform Sα of both sides of Eq. (4.1) and using Theorem 2.7, then we get: Yα(u) u3 + Yα(u) u = Γ(2)u, which implies that: Yα(u) [ 1 + u2 u3 ] = u. Thus, Yα(u) = u4 1 + u2 . (4.2) By taking S−1α of both sides of Eq. (4.2), then we obtain the general solution of Eq. (4.1) as follows: y(x) = S−1α [ u2 − 1 + 1 u2 + 1 ] = x2α 2α2 + cos xα α − 1. (4.3) Int. J. Anal. Appl. 17 (6) (2019) 1031 Problem 4.2.: Consider the following linear and nonhomogeneous conformable fractional differential equation: D2αy + y = e 2xα α , y(0) = 1. (4.4) By applying Sα of both sides of Eq. (4.4) and using Theorems 2.7 and 2.8, we obtain: Yα(u) u2 − 1 u2 + Yα(u) = 1 1 − 2u , which implies that: Yα(u) [ 1 + u2 u2 ] = 1 1 − 2u + 1 u2 . (4.5) By taking S−1α of both sides of Eq. (4.5), then we obtain the general solution of Eq. (4.4) as follows: y(x) = S−1α [ u2 (1 + u2)(1 − 2u) ] + S−1α [ 1 1 + u2 ] = S−1α [ Au + B 1 + u2 ] + S−1α [ c 1 − 2u ] + S−1α [ 1 1 + u2 ] = −2 5 S−1α [ u 1 + u2 ] − 1 5 S−1α [ 1 1 + u2 ] + 1 5 S−1α [ 1 1 − 2u ] + S−1α [ 1 1 + u2 ] = − 2 5 sin xα α + 4 5 cos xα α + 1 5 e2 xα α . (4.6) Problem 4.3.: Consider the following linear and nonhomogeneous conformable fractional differential equation: D2αy −y = cos( xα α ), y(0) = 0. (4.7) By applying Sα of both sides of Eq. (4.7) and using Theorems 2.7 and 2.8, we obtain: Yα(u) u2 −Yα(u) = 1 1 + u2 , which implies that: Yα(u) = u2 (1 −u2)(1 + u2) . (4.8) By taking S−1α of both sides of Eq. (4.8), then we obtain the general solution of Eq. (4.7) as follows: y(x) = S−1α [ u2 (1 −u2)(1 + u2) ] = S−1α [ Au + B 1 −u2 ] + S−1α [ Du + C 1 + u2 ] = 1 2 S−1α 1 1 −u2 − 1 2 S−1α [ 1 1 + u2 ] = 1 2 cosh xα α − 1 2 cos xα α . (4.9) Int. J. Anal. Appl. 17 (6) (2019) 1032 5. Conclusion We have developed the conformable fractional Sumudu transform for presenting the general analytical solu- tion of the singular and nonlinear conformable fractional Poisson -Boltzmann equation and also for presenting the general solutions of some linear and nonhomogeneuos conformable fractional differential equations. In our opinion, it is worth to extend some other conformable fractional transforms such as: The conformable fractional Natural and conformable fractional Mellin trabsforns and using them in many application and comparisons. 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