J. Nig. Soc. Phys. Sci. 4 (2022) 818 Journal of the Nigerian Society of Physical Sciences Time-Fractional Differential Equations with an Approximate Solution Lamees K. Alzaki∗, Hassan Kamil Jassim Department of Mathematics, Faculty of Education for Pure Science, University of Thi-Qar, Nasiriyah, Iraq. Abstract This paper shows how to use the fractional Sumudu homotopy perturbation technique (SHP) with the Caputo fractional operator (CF) to solve time fractional linear and nonlinear partial differential equations. The Sumudu transform (ST) and the homotopy perturbation technique (HP) are combined in this approach. In the Caputo definition, the fractional derivative is defined. In general, the method is straightforward to execute and yields good results. There are some examples offered to demonstrate the technique’s validity and use. DOI:10.46481/jnsps.2022.818 Keywords: Fractional differential equations, Homotopy perturbation method, Sumudu transform Article History : Received: 16 May 2022 Received in revised form: 05 July 2022 Accepted for publication: 26 July 2022 Published: 18 August 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: O. J. Oluwadare 1. Introduction For characterizing nonlocal structures, fractional calculus has arisen as a new mathematical technique. Due to their nu- merous applications in physics and engineering, fractional dif- ferential equations have attracted a lot of attention during the last decade [1-4]. Many authors have investigated fractional calculus theory, and current study is centered on proving the important benefits of fractional calculus over classical calculus [5, 6]. In several instances, the fractional calculus gave the best outcomes than the conventional method. There is a significant literature dealing with the subject area of approximating solutions to fractional differential equations ∗Corresponding author tel. no: +9647819374140 Email addresses: lkalzaki@utq.edu.iq (Lamees K. Alzaki ), hassankamil@utq.edu.iq (Hassan Kamil Jassim) using various approaches known as perturbation methods. The perturbation techniques have certain disadvantages; for exam- ple, the approximate solution requires a succession of smaller parameters, which is challenging because the majority of non- linear problems do not have any small values. Despite proper selections of smaller parameters might occasionally lead to an optimal solution, in most circumstances, bad choices have ma- jor consequences in the solutions. As a result, an analytical technique that does not need a smaller parameter in the equa- tion representing the phenomena is preferred. He [7] was the first to propose the homotopy perturbation technique (HPM). Many writers investigated the HPM to analyze linear and non- linear equations encountered in numerous scientific and techni- cal sectors. In general, higher performance of the fractional calculus is demonstrated by reduced error levels created during an estimat- ing procedure. Various approximation and methodologies, like 1 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 2 the fractional Adomian decomposition method (FADM) [8-10], fractional homotopy method (FHPM) [11, 12, 13], fractional function decomposition method [14, 15], fractional variational iteration method (FVIM) [16-18], fractional reduce differential transform method (FRDTM) [18, 19, 20, 21], fractional differ- ential transform method [22, 23, 24], fractional Laplace vari- ational iteration method [25-32], fractional Laplace homotopy perturbation method (FLHPM) [33], fractional Laplace decom- position method (FLDM) [34, 35], fractional Sumudu homo- topy analysis method [36], fractional Sumudu variational iter- ation method (FVIM) [37, 38], fractional Sumudu decomposi- tion method (FSDM) [39-41], fractional natural decomposition method (FNDM) [42, 43], fractional Sumudu homotopy per- turbation method (FSHPM) [44, 45], energy balance method (EBM) [46], power series methods (PSM) [47], have been used in latest years to analyze partial differential equations within Caputo sense.Moreover, many studies provide functional dif- ferential equations using various types of transforms, such as the Mellin transform [48]. Also, the Lagrange interpolation is used to estimate the delay argument [49]. Furthermore, the lo- cal truncation errors and stability polynomials are calculated [50-52]. Our objective is to develop and use the Sumudu homotopy perturbation methodology, which combines the Sumudu trans- form with the homotopy perturbation method to solve nonlinear fractional differential equations. 2. Preliminaries This section goes through some of the fractional calculus definitions and notation that will be utilized in this study [4, 45, 46, 47, 53, 54, 55]. Definition 2.1. The Riemann Liouville fractional integral op- erator of order δ ≥ 0 of a function ϕ(µ) ∈ Cϑ,ϑ ≥ −1 is given by the form Iδϕ(µ) =  1 Γ(δ) ∫ µ 0 (µ−τ)δ−1ϕ(τ)dτ, δ > 0,µ > 0 I0ϕ(µ) = ϕ(µ), δ = 0 (1) where Γ(·) is the well-known Gamma function. Properties of the operator Iδare as follows: For ϕ ∈ Cϑ,ϑ ≥ −1,δ,γ ≥ 0, then 1. IδIγϕ(µ) = Iδ+γϕ(µ) 2. IδIγϕ(µ) = IγIδϕ(µ) 3. Iδµm = Γ(m + 1) Γ(δ + m + 1) µδ+m Definition 2.2. In the Caputo interpretation, the fractional deriva- tive of ϕ(µ) is given as Dδϕ(µ) = Im−δDmϕ(µ) = 1 Γ(m −δ) ∫ µ 0 (µ−τ)m−δ−1ϕ(m)(τ)dτ, (2) for m − 1 < α ≤ m, m ∈ N,µ > 0 and ϕ ∈ Cm −1. The fundamental properties of the operator Dδ are given as fol- lows: 1. DδIδϕ(µ) = ϕ(µ) 2. DδIδϕ(µ) = ϕ(µ) − ∑n−1 k=0 ϕ (k)(0) µk k! Definition 2.3. For δ > 0, the gamma function Γ(·), is defined as follows: Γ(δ) = ∫ ∞ 0 xδ−1e−xd x (3) Definition 2.4. By considering Eδ with δ > 0, the definition of the Mittag–Leffler function given as the following: Eδ(z) = ∞∑ m=0 zδ Γ(mδ + 1) (4) Some special cases of the Mittag-Leffler function Eδ(z) 1. E0(z) = 1 1−z , |z| < 1, 2. E1(z) = ez, 3. E2(z) = cosh √ z, z ∈ C, 4. E2(−z2) = cosz, z ∈ C Definition 2.5. The Sumudu transform is identified based on a collection of functions A = {ϕ(τ) : ∃M,ω1,ω2 > 0, with |ϕ(τ)| ≤ Me |τ| ω j , ifτ ∈ (−1) j × [0,∞)} as determined by the formula S[ϕ(ω)] = G(ω) = ∫ ∞ 0 e−τϕ(ωτ)dτ, ω ∈ (−ω1,ω2) Some properties of Sumudu Transform 1. S[k] = k, k constant 2. S [ τmδ Γ(mδ+1) ] = ωmδ Definition 2.6. The Caputo fractional derivative for the Sumudu transform is given as the following S [ Dµδτ ϕ(µ,τ) ] = ω−µδS [ ϕ(µ,τ) ] − m−1∑ k=0 ω(−µδ+k)ϕ(k)(µ, 0), m − 1 < mδ < m (5) 3. Fractional Sumudu Homotopy Perturbation Method (FSHPM) Consider this generic fractional nonlinear PDEs: Dmδτ ϕi(µ,τ) + R [ ϕ(µ,τ) ] + N [ ϕ(µ,τ) ] = ð(µ,τ), m − 1 < mδ ≤ m (6) with ϕ(µ, 0) = f (µ), (7) in which Dmδτ ϕ(µ,τ) is the Caputo fractional derivative of the function ϕ(µ,τ), R refers for the linear differential operator, 2 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 3 while N refers for a nonlinear system differential operator, and ð(µ,τ) is the origin term. Using the ST to both sides of (6), we get S[Dµδτ ϕ(µ,τ)] + S[R [ ϕ(µ,τ) ] ] + S[N [ ϕ(µ,τ) ] ] = S[ð(µ,τ)] (8) We obtain by utilizing the ST’s property S[ϕ(µ,τ)] = −ωδ m−1∑ k=0 ϕ(k)(µ, 0) + ωδS [ ð(µ,τ) ] −ωδS [ R[ϕ(µ,τ)] + N[ϕ(µ,τ)] ] (9) Operating the inverse Sumudu transform on both sides of (9), we get ϕ(µ,τ) = S−1 [ ωδ m−1∑ k=0 ϕ(k)(µ, 0) ] + S−1 [ ωδS [ ð(µ,τ) ]] −S−1 [ ωδS [ R[ϕ(µ,τ)] + N[ϕ(µ,τ)] ]] (10) We implement the HPM: ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ), (11) Thus the nonlinear term may be decomposed as follows: N[ϕ(µ,τ)] = ∞∑ n=0 pn Hn(ϕ0,ϕ1, . . . ,ϕn), (12) where Hn(ϕ0,ϕ1, . . . ,ϕn) = 1 n! ∂n ∂pn [ N [ ∞∑ i=0 piϕi(µ,τ) ]] p=0 When we substitute (11) and (12) into (10), we obtain ∞∑ n=0 pnϕn(µ,τ) = S −1 [ ωδ m−1∑ k=0 ϕ(k)(µ, 0) ] + S−1 [ ωδS [ ð(µ,τ) ]] −pS−1 ( ωδS [ R [ ∞∑ n=0 pnϕn(µ,τ) ] + ∞∑ n=0 pn Hn ]) We get the following set of equations by comparing the terms with comparable powers of p: P0 : ϕ0(µ,τ) = S −1 [ ωδ m−1∑ k=0 ϕ(k)(µ, 0) ] +S−1 [ ωδS [ ð(µ,τ) ]] P1 : ϕ1(µ,τ) = −S −1 [ ωδS [ R[ϕ0(µ,τ)] + H0 ]] P2 : ϕ2(µ,τ) = −S −1 [ ωδS [ R[ϕ1(µ,τ)] + H1 ]] ... Pn : ϕn(µ,τ) = −S −1 [ ωδS [ R[ϕn−1(µ,τ)] + Hn−1 ]] , n ≥ 1 (13) Consequently, we use truncated series to estimate the analytical result ϕ(µ,τ) ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) (14) 4. Applications This section will put the proposed method for solving time fractional partial differential equations into application. 4.1. Example Firstly, examine the time-fractional Cauchy reaction–diffusion equation shown below Dδτϕ(µ,τ) = ϕµµ(µ,τ) −ϕ(µ,τ), 0 < δ ≤ 1 (15) with ϕ(µ, 0) = e−µ + µ (16) For the (15), applying the Sumudu transform ST on both sides of it, we achieve S[ϕ(µ,τ)] = ϕ(µ, 0) + ωδS [ ϕµµ(µ,τ) −ϕ(µ,τ) ] (17) Using the inverse ST to (17), we obtain S[ϕ(µ,τ)] = e−µ + µ + S−1 ( ωδS [ ϕµµ(µ,τ) −ϕ(µ,τ) ]) (18) According to the HPM, and substituting ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) in (18), we have ∞∑ n=0 pnϕn(µ,τ) = e −µ + µ +pS−1 ( ωδS [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn(µ,τ) ] − ∞∑ n=0 pnϕn(µ,τ) ]) (19) We get the following set of equations by comparing the terms with comparable powers of p: P0 : ϕ0(µ,τ) = e −µ + µ P1 : ϕ1(µ,τ) = S −1 ( ωδS [ ∂2ϕ0(µ,τ) ∂µ2 −ϕ0(µ,τ) ]) = S−1 ( ωδS[−µ] ) = − µτδ Γ(δ + 1) , P2 : ϕ2(µ,τ) = S −1 ( ωδS [ ∂2ϕ1(µ,τ) ∂µ2 −ϕ1(µ,τ) ]) = S−1 ( ωδS [ − µτδ Γ(δ + 1) ]) = − µτ2δ Γ(2δ + 1) , 3 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 4 P3 : ϕ3(µ,τ) = S −1 ( ωδS [ ∂2ϕ2(µ,τ) ∂µ2 −ϕ2(µ,τ) ]) = S−1 ( ωδS [ − µτ2δ Γ(2δ + 1) ]) = − µτ3δ Γ(3δ + 1) , P4 : ϕ4(µ,τ) = S −1 ( ωδS [ ∂2ϕ3(µ,τ) ∂µ2 −ϕ3(µ,τ) ]) = S−1 ( ωδS [ − µτ3δ Γ(3δ + 1) ]) = − µτ4δ Γ(4δ + 1) , ... Pn : ϕn(µ,τ) = S −1 ( ωδS [ ∂2ϕn−1(µ,τ) ∂µ2 −ϕn−1(µ,τ) ]) = (−1)n µτnδ Γ(nδ + 1) Hence, the outcome of (15) is provided by ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) = e−µ + µ [ 1 − τδ Γ(δ + 1) + τ2δ Γ(2δ + 1) + τ3δ Γ(3δ + 1) . . . +(−1)n τnδ Γ(nδ + 1) ] = e−µ + µEδ(τ δ) (20) For δ = 1, the (20) is close to the form ϕ(µ,τ) = e−µ + µe−τ, which is the precise solution of (15) for δ = 1. The outcome is same as with FSDJM [53]. 4.2. Example We determine the most important time fractional Cauchy reaction-diffusion equation: Dδτϕ(µ,τ) = ϕµµ(µ,τ) − (1 + 4µ 2)ϕ(µ,τ), 0 < δ ≤ 1 (21) with ϕ(µ, 0) = eµ 2 (22) We obtain by utilizing the ST’s property on both sides of (21), we get S[ϕ(µ,τ)] = ϕ(µ, 0) + ωδS [ ϕµµ(µ,τ) −(1 + 4µ2)ϕ(µ,τ) ] (23) Using the inverse ST to (23), we have S[ϕ(µ,τ)] = eµ2 + S−1 ( ωδS [ ϕµµ(µ,τ) −(1 + 4µ2)ϕ(µ,τ) ]) (24) According to the HPM, and substituting ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) in (24), we have ∞∑ n=0 pnϕn(µ,τ) = e µ2 + pS−1 ( ωδS [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn(µ,τ) ] − (1 + 4µ2) ∞∑ n=0 pnϕn(µ,τ) ]) (25) We get the following set of equations by comparing the terms with comparable powers of p: P0 : ϕ0(µ,τ) = e µ2 P1 : ϕ1(µ,τ) = S −1 ( ωδS [ ∂2ϕ0(µ,τ) ∂µ2 −(1 + 4µ2)ϕ0(µ,τ) ]) = τδ Γ(δ + 1) eµ 2 , P2 : ϕ2(µ,τ) = S −1 ( ωδS [ ∂2ϕ1(µ,τ) ∂µ2 − (1 + 4µ2)ϕ1(µ,τ) ]) = τ2δ Γ(2δ + 1) eµ 2 , P3 : ϕ3(µ,τ) = S −1 ( ωδS [ ∂2ϕ2(µ,τ) ∂µ2 − (1 + 4µ2)ϕ2(µ,τ) ]) = τ3δ Γ(3δ + 1) eµ 2 , P4 : ϕ4(µ,τ) = S −1 ( ωδS [ ∂2ϕ3(µ,τ) ∂µ2 − (1 + 4µ2)ϕ3(µ,τ) ]) = τ4δ Γ(4δ + 1) eµ 2 , ... Pn : ϕn(µ,τ) = S −1 ( ωδS [ ∂2ϕn−1(µ,τ) ∂µ2 − (1 + 4µ2)ϕn−1(µ,τ) ]) = τnδ Γ(nδ + 1) eµ 2 Hence, the outcome of (21) is provided by ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) = eµ 2 [ 1 + τδ Γ(δ + 1) + τ2δ Γ(2δ + 1) + τ3δ Γ(3δ + 1) · · · + 4 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 5 τnδ Γ(nδ + 1) ] = eµ 2 Eδ(τ δ) (26) For δ = 1, the (26) is close to the form ϕ(µ,τ) = eµ 2 +τ, which is the precise solution of (21) for δ = 1. The outcome is same as with FSDJM [53]. 4.3. Example Consider the nonlinear fractional Cauchy reaction–diffusion equation is given as the following Dδτϕ(µ,τ) = ϕµµ(µ,τ) −ϕµ(µ,τ) + ϕ(µ,τ)ϕµµ(µ,τ) −ϕ2(µ,τ) + ϕ(µ,τ), 0 < δ ≤ 1 (27) w.r.t initial condition ϕ(µ, 0) = eµ (28) Operating the ST on both sides of (27), and employing ST’s differential property, we obtain S[ϕ(µ,τ)] = ϕ(µ, 0) + ωδS [ ϕµµ(µ,τ) −ϕµ(µ,τ) +ϕ(µ,τ)ϕµµ(µ,τ) −ϕ 2(µ,τ) + ϕ(µ,τ) ] (29) We obtain by applying the inverse Sumudu transform on both sides of (29) ϕ(µ,τ) = eµ + S−1 ( ωδS [ ϕµµ(µ,τ) −ϕµ(µ,τ) + ϕ(µ,τ)ϕµµ(µ,τ) −ϕ 2(µ,τ) + ϕ(µ,τ) ]) (30) According to the HPM, and substituting ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) ϕϕµµ = ∞∑ n=0 pn Hn ϕ2 = ∞∑ n=0 pnGn in (30), we have ∞∑ n=0 pnϕn(µ,τ) = eµ + pS−1 ( ωδS [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn ] − ∂ ∂µ [ ∞∑ n=0 pnϕn ] + ∞∑ n=0 pn Hn − ∞∑ n=0 pnGn − ∞∑ n=0 pnϕn ]) (31) We get the following set of equations by comparing the terms with comparable powers of p: P0 : ϕ0(µ,τ) = e µ, P1 : ϕ1(µ,τ) = S −1 ( ωδS [ ∂2ϕ0 ∂µ2 − ∂ϕ0 ∂µ + H0 − G0 +ϕ0 ]) = τδ Γ(δ + 1) eµ, P2 : ϕ2(µ,τ) = S −1 ( ωδS [ ∂2ϕ1 ∂µ2 − ∂ϕ1 ∂µ + H1 − G1 +ϕ1 ]) = τ2δ Γ(2δ + 1) eµ, P3 : ϕ3(µ,τ) = S −1 ( ωδS [ ∂2ϕ2 ∂µ2 − ∂ϕ2 ∂µ + H2 − G2 +ϕ2 ]) = τ3δ Γ(3δ + 1) eµ, P4 : ϕ4(µ,τ) = S −1 ( ωδS [ ∂2ϕ3 ∂µ2 − ∂ϕ3 ∂µ + H3 − G3 +ϕ3 ]) = τ4δ Γ(4δ + 1) eµ, (32) ... Pn : ϕn(µ,τ) = S −1 ( ωδS [ ∂2ϕn−1 ∂µ2 − ∂ϕn−1 ∂µ +Hn−1 − Gn−1 + ϕn−1 ]) = τnδ Γ(nδ + 1) eµ Hence, the outcome of (27) is provided by ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) = eµ [ 1 + τδ Γ(δ + 1) + τ2δ Γ(2δ + 1) + · · · + τnδ Γ(nδ + 1) ] = eµEδ(τ δ) (33) For δ = 1, the (33) is close to the form ϕ(µ,τ) = eµ+τ, which is the precise solution of (27) for δ = 1. The outcome is same as with FSDJM [53]. 4.4. Example Assume the coupled fractional Burger’s equations shown below where 0 < δ ≤ 1, 0 < γ ≤ 1 Dδτϕ(µ,τ) −ϕµµ − 2ϕϕµ + (ϕψ)µ = 0, Dδτψ(µ,τ) −ψµµ − 2ψψµ + (ϕψ)µ = 0 (34) with ϕ(µ, 0) = eµ ψ(µ, 0) = eµ (35) 5 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 6 Operating the ST on both sides of (34), and employing ST’s differential property, we obtain S [ ϕ(µ,τ) ] = ϕ(µ, 0) + ωδS [ ϕµµ + 2ϕϕµ − (ϕψ)µ ] , S [ ψ(µ,τ) ] = ψ(µ, 0) + ωγS [ ψµµ + 2ψψµ − (ϕψ)µ ] (36) Implementing with the inverse ST to (36), we have ϕ(µ,τ) = eµ + S−1 ( ωδS [ ϕµµ + 2ϕϕµ − (ϕψ)µ ]) , ψ(µ,τ) = eµ + S−1 ( ωγS [ ψµµ + 2ψψµ − (ϕψ)µ ]) (37) Assume that ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) (38) ψ = ∞∑ n=0 pnψn (39) ϕϕµ = ∞∑ n=0 pn Hn (40) ψψµ = ∞∑ n=0 pn Kn (41) (ϕψ)µ = ∞∑ n=0 pnGn (42) By applying the HPM, and substituting (38)-(42) in (37), we get ∞∑ n=0 pnϕn(µ,τ) = e µ + pS−1 ( ωδS [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn ] +2 ∞∑ n=0 pn Hn − ∞∑ n=0 pnGn ]) , ∞∑ n=0 pnψn(µ,τ) = e µ + pS−1 ( ωγS [ ∂2 ∂µ2 [ ∞∑ n=0 pnψn ] +2 ∞∑ n=0 pn Kn − ∞∑ n=0 pnGn ]) (43) We get the following set of equations by comparing the terms with comparable powers of p: P0 : ϕ0(µ,τ) = e µ, : ψ0(µ,τ) = e µ P1 : ϕ1(µ,τ) = S −1 ( ωδS [ ∂2ϕ0 ∂µ2 + 2H0 − G0 ]) : ψ1(µ,τ) = S −1 ( ωγS [ ∂2ψ0 ∂µ2 + 2K0 − G0 ]) P2 : ϕ2(µ,τ) = S −1 ( ωδS [ ∂2ϕ1 ∂µ2 + 2H1 − G1 ]) : ψ2(µ,τ) = S −1 ( ωγS [ ∂2ψ1 ∂µ2 + 2K1 − G1 ]) P3 : ϕ3(µ,τ) = S −1 ( ωδS [ ∂2ϕ2 ∂µ2 + 2H2 − G2 ]) : ψ3(µ,τ) = S −1 ( ωγS [ ∂2ψ2 ∂µ2 + 2K2 − G2 ]) ... Then, we have P0 : ϕ0(µ,τ) = e µ, : ψ0(µ,τ) = e µ P1 : ϕ1(µ,τ) = S −1 ( ωδS [ eµ + 2e2µ − 2e2µ ]) : ψ1(µ,τ) = S −1 ( ωγS [ eµ + 2e2µ − 2e2µ ]) = S−1(ωδeµ) = τδ Γ(δ + 1) eµ = S−1(ωγeµ) = τγ Γ(γ + 1) eµ P2 : ϕ2(µ,τ) = S −1 ( ωδS [ τδ Γ(δ + 1) eµ +2 τδ Γ(δ + 1) e2µ − 2 τγ Γ(γ + 1) e2µ ]) : ψ2(µ,τ) = S −1 ( ωγS [ τγ Γ(γ + 1) eµ +2 τγ Γ(γ + 1) e2µ − 2 τδ Γ(δ + 1) e2µ ]) (44) = S−1 ( ω2δeµ + 2ω2δe2µ − 2ωδ+γe2µ ) = S−1 ( ω2γeµ + 2ω2γe2µ − 2ωδ+γe2µ ) = τ2δ Γ(2δ + 1) eµ + 2 τ2δ Γ(2δ + 1) e2µ − 2 τδ+γ Γ(δ + γ + 1) e2µ = τ2γ Γ(2γ + 1) eµ + 2 τ2γ Γ(2γ + 1) e2µ − 2 τδ+γ Γ(δ + γ + 1) e2µ ... Thus, the outcome of (34) is given by ϕ(µ,τ) = eµ [ 1 − τδ Γ(δ + 1) + τ2δ Γ(2δ + 1) . . . ] +e2µ [ 2 τ2δ Γ(2δ + 1) − 2 τδ+γ Γ(δ + γ + 1) . . . ] ψ(µ,τ) = eµ [ 1 − τγ Γ(γ + 1) + τ2γ Γ(2γ + 1) . . . ] +e2µ [ 2 τ2γ Γ(2γ + 1) − 2 τδ+γ Γ(δ + γ + 1) . . . ] 6 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 7 (45) Setting δ = γ in (45), we obtain ϕ(µ,τ) = eµ [ 1 − τδ Γ(δ + 1) + τ2δ Γ(2δ + 1) . . . ] = Eδ(−τ δ)eµ ψ(µ,τ) = eµ [ 1 − τγ Γ(γ + 1) + τ2γ Γ(2γ + 1) . . . ] = Eγ(−τ γ)eµ (46) The (46) is close to the form ϕ(µ,τ) = ψ(µ,τ) = eµ−τ for δ = γ = 1 , which is the precise solution of (34) for δ = γ = 1. 5. Conclusion The Sumudu homotopy perturbation approach was effec- tively used in this work to discover approximate solutions to time-fractional partial differential equations. The analytical ap- proach generates a convergence analysis that fast converges to the optimal solution. The simplicity and high precision of the analytical method are clearly illustrated, solving equations in- cludes linear and nonlinear fractional PDEs and a nonlinear sys- tem of fractional PDEs. References [1] G. Aguilar, Y. Martinez, T. Jimenez, C. Fraga, E. Jimenez, O. Peregrino, Homotopy perturbation transform method for nonlinear differential equa- tions involving to fractional operator with exponential kernel, Advances in Difference Equations 68 (2017) 1. [2] L. Debnath, A brief historical introduction to fractional calculus, Int. J. Math. Educ. Sci. Technol. 35 (2004) 487. [3] N. I. Mahmudov, M. Awadalla, K. Abuassba, Nonlinear sequential frac- tional differential equations with nonlocal boundary conditions, Advances in Difference Equations 319 (2017) 1. [4] I. Podlubny, Fractional Differential Equations: An Introduction to Frac- tional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 198. Elsevier, Amsterdam (1998). [5] R. Almeida, N. R. Bastos, Modeling some real phenomena by fractional differential equations, Math.Methods Appl. Sci. 39 (2016) 4846. [6] R. Almeida, A. B. Malinowska, M. T. Monteiro Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Methods Appl. Sci. 41 (2018) 336. [7] J.-H. He, Homotopy perturbation technique, Computer Methods in Ap- plied Mechanics and Engineering 178 (1999) 257. [8] S. A. El-Wakil, A. Elhanbaly, M. A. Abdou, Adomian decomposition method for solving fractional nonlinear partial differential equations, Ap- plied Mathematics and Computation 182 (2006) 313. [9] M. M. El-Borai, W. G. El-Sayed, A. M. Jawad,Adomian decomposition method for solving fractional differential equations, International Re- search Journal of Engineering and Technology 2 (2015) 296. [10] Z. P. Fan, H. K. Jassim, R. K. Rainna, and X. J. Yang, Adomian De- composition Method for Three-Dimensional Diffusion Model in Fractal Heat Transfer Involving Local Fractional Derivatives, Thermal Science 19 (2015) S137. [11] A. Yildirım, A. K. Akbay, Homotopy perturbation method for numerical solutions of coupled Burger’s equations with time-space fractional deriva- tives. International Journal of Numerical Methods for Heat and Fluid Flow 20 (2010) 897. [12] J. Singh, D. Kumar, R. Swroop, Numerical solution of time- and space- fractional coupled Burger’s equations via homotopy algorithm, Alexan- dria Engineering Journal 55 (2016) 1753. [13] S. Momani, A. Yildirim A, Analytical approximate solutions of the frac- tional convection diffusion equation with nonlinear source term by homo- topy perturbation method, Int. J. Comput. Math. 87 (2010) 1057. [14] S. Q. Wang, Y. J. Yang, and H. K. Jassim, Local Fractional Function De- composition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative, Abstract and Applied Analysis, 2014 (2014) 1. [15] S. P. Yan, H. Jafari, and H. K. Jassim, Local Fractional Adomian De- composition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators,Advances in Mathematical Physics 2014 (2014) 1. [16] S. Xu, X. Ling, Y. Zhao, H. K. Jassim, A Novel Schedule for Solving the Two-Dimensional Diffusion in Fractal Heat Transfer, Thermal Science 19 (2015) S99. [17] H. K. Jassim, W.A. Shahab, Fractional variational iteration method to solve one dimensional second order hyperbolic telegraph equations, Jour- nal of Physics: Conference Series 1032 (2018) 1. [18] H. Jafari, H. K. Jassim, J. Vahidi,Reduced Differential Transform and Variational Iteration Methods for 3D Diffusion Model in Fractal Heat Transfer within Local Fractional Operators, Thermal Science 22 (2018) S301. [19] H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan and F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Advances in Mechanical En- gineering 8 (2016) 1. [20] M. S. Rawashdeh, A reliable method for the space-time fractional Burgers and time-fractional Cahn-Allen equations via the FRDTM, Advances in Difference Equations 99 (2017) 1. [21] H. K. Jassim, J. Vahidi, A New Technique of Reduce Differential Trans- form Method to Solve Local Fractional PDEs in Mathematical Physics, International Journal of Nonlinear Analysis and Applications 12 (2021) 37. [22] H. Jafari, H. K. Jassim, F. Tchier, D. Baleanu, On the Approximate So- lutions of Local Fractional Differential Equations with Local Fractional Operator, Entropy 18 (2016) 1. [23] D. Baleanu, H. K. Jassim, Approximate Analytical Solutions of Goursat Problem within Local Fractional Operators, Journal of Nonlinear Science and Applications 9 (2016) 4829. [24] H. K. Jassim, J. Vahidi, V. M. Ariyan, Solving Laplace Equation within Local Fractional Operators by Using Local Fractional Differential Trans- form and Laplace Variational Iteration Methods, Nonlinear Dynamics, and Systems Theory 20 (2020) 388. [25] H. K. Jassim, New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators, Journal of Mathematics 2015 (2015) 1. [26] H. K. Jassim, C. Ünlü, S. P. Moshokoa, C. M. Khalique, Local Frac- tional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators, Math- ematical Problems in Engineering, 2015 (2015) 1. [27] H. K. Jassim, The Approximate Solutions of Three-Dimensional Diffu- sion and Wave Equations within Local Fractional Derivative Operator, Abstract and Applied Analysis 2016 (2016) 2913539. [28] H. Jafari, H. K. Jassim, On the Existence and Uniqueness of Solutions for Local differential equations, Entropy 18 (2016) 1. [29] D. Baleanu, H. K. Jassim, H. Khan, A Modification Fractional Varia- tional Iteration Method for solving Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operators, Thermal Science 22 (2018) S165. [30] H. K. Jassim, D. Baleanu, A novel approach for Korteweg-de Vries equa- tion of fractional order, Journal of Applied Computational Mechanics 5 (2019) 192. [31] D. Baleanu, H. K. Jassim, M. Al Qurashi,Solving Helmholtz Equation with Local Fractional Derivative Operators, Fractal and Fractional 3 (2019) 1. [32] D. Baleanu, H. K. Jassim, Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings, Fractal and Fractional 3 (2019) 1. [33] D. Baleanu, H. K. Jassim, A Modification Fractional Homotopy Pertur- bation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets, Fractal and Fractional 3 (2019) 1. [34] H. K. Jassim, Analytical Approximate Solutions for Local Fractional 7 Alzaki & Jassim / J. Nig. Soc. Phys. Sci. 4 (2022) 818 8 Wave Equations, Mathematical Methods in the Applied Sciences 43 (2020) 939. [35] J. Singh, H. K. Jassim, D. Kumar, An efficient computational technique for local fractional Fokker-Planck equation, Physica A: Statistical Me- chanics and its Applications, 555(124525) (2020) 1. [36] H. K. Jassim, M. G. Mohammed, H. A. Eaued, , A Modification Frac- tional Homotopy Analysis Method for Solving Partial Differential Equa- tions Arising in Mathematical Physics, IOP Conf. Series: Materials Sci- ence and Engineering 928 (2020) 042021. [37] H. A. Eaued, H. K. Jassim, M. G. Mohammed, A Novel Method for the Analytical Solution of Partial Differential Equations Arising in Mathe- matical Physics, IOP Conf. Series: Materials Science and Engineering, 928 (2020) 042037. [38] H. K. Jassim, S. A. Khafif, SVIM for solving Burger’s and coupled Burger’s equations of fractional order, Progress in Fractional Differen- tiation and Applications 7 (2021) 1. [39] H. K. Jassim, H. A. Kadhim, Fractional Sumudu decomposition method for solving PDEs of fractional order, Journal of Applied and Computa- tional Mechanics 7 (2021) 302-311. [40] D. Baleanu, H. K. Jassim, Exact Solution of Two-dimensional Fractional Partial Differential Equations,Fractal Fractional 4 (2020) 1. [41] H. K. Jassim, M. A. Shareef, On approximate solutions for fractional system of differential equations with Caputo-Fabrizio fractional operator, Journal of Mathematics and Computer science 23 (2021) 58. [42] H. Jafari, H. K. Jassim, D. Baleanu, Y. M. Chu, SOn the approximate solutions for a system of coupled Korteweg-de Vries equations with local fractional derivative, Fractals 29 (2021) 1. [43] H. K. Jassim, M. G. Mohammed, Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations, International Journal of Nonlinear Analysis and Applications 12 (2021) 37. [44] H. K. Jassim, A new approach to find approximate solutions of Burger’s and coupled Burger’s equations of fractional order, TWMS Journal of Applied and Engineering Mathematics 11 (2021) 415. [45] L. K. Alzaki, H. K. Jassim, The approximate analytical solutions of non- linear fractional ordinary differential equations, International Journal of Nonlinear Analysis and Applications 12 (2021) 527. [46] I. Mehdipour, D.D.Ganji, M. Mozaffari, Application of the energy bal- ance method to nonlinear vibrating equations, Current Applied Physics 10 (2010) 104. [47] H. M. Hasson, Power series method for solving nonlinear Volterra integro-differential equations of the second kind, Engineering and Tech- nology Journal 28 (2010) 4816. [48] S. E. Fadugba, Solution of Fractional Order Equations in the Domain of the Mellin, Transform. Journal of the Nigerian Society of Physical Sci- ences 1 (2019) 138. [49] V. J. Shaalini, S. E. Fadugba, A New Multi-Step Method for Solving De- lay Differential Equations using Lagrange Interpolation, Journal of the Nigerian Society of Physical Sciences 3 (2010) 159. [50] J. Kuboye, O. R. Elusakin, O. F. Quadri, Numerical Algorithms for Direct Solution of Fourth Order Ordinary Differential Equations, Journal of the Nigerian Society of Physical Sciences, 2(4)(2020), 218–227. [51] J. O. Kuboye, O. F. Quadri, O. R. Elusakin, Solving third order ordinary differential equations directly using hybrid numerical models, Journal of the Nigerian Society of Physical Sciences 2 (2020), 69. [52] Y.Ibrahim, Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Map- pings in Banach Spaces, Journal of the Nigerian Society of Physical Sci- ences 1 (2019) 35. [53] K. Wang, S. Liu, A new Sumudu transform iterative method for time frac- tional Cauchy reaction diffusion equation, Advances in Difference Equa- tions 5 (865) (2016) 1. [54] A.K. Alomari, Homotopy-Sumudu transforms for solving system of frac- tional partial differential equations,Advances in Difference Equations, No. 1 (2020) 222. [55] H. Eltayeb, A. Kilicmai, A note on the Sumudu transform and DEs, Applied Math. Sci. 4 (2010) 1089. 8