EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6223 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Novel Maclaurin Series Approach to the Sakiadis Flow Problem and Its Fractal Formulation in Fluid Mechanics A. Alameer1,∗ 1 Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia Abstract. This research paper introduces a method utilizing the Maclaurin series to analyze the standard Sakiadis problem. The technique uses the Maclaurin series to describe fluid-mechanics boundary layers, combining the series solution with diagonal Padé approximants for handling the infinity condition. Additionally, the Hausdorff derivative is used to examine the Sakiadis equation’s fractal formulation. The current approach is consistent with the previous process and follows the correct balance. This study is an important resource for furthering research in this area and provides insightful information. 2020 Mathematics Subject Classifications: 76D10, 34M10, 76M22 Key Words and Phrases: Maclaurin series method (MSM), Sakiadis equation, Padé approxi- mants, Hausdorff derivative 1. Introduction Significant engineering applications have investigated laminar, incompressible boundary- layer flows, for example: the smooth removal of plastic sheets, the cooling of an endless metal plate in a cooling tub, and processes in glass and polymer industries. Notably, a fundamental boundary-layer equation arising in fluid mechanics is the Sakiadis equation. In order to solve such fluid-dynamics problems, a wide variety of numerical and theoretical techniques have been used for the Blasius equation. Howarth reported the first numerical results for the Blasius equation using the Runge–Kutta methodology [1]. Chinese profes- sors Liao [2], Yu et al. [3], and Ji Huan He [4] employed approximate methods to analyze the Blasius equation. Wazwaz [5, 6] utilized Adomian decomposition and the Variational Iteration Method to tackle the Blasius equation. A connection between the Adomian decomposition method and the homotopy method for treating the Blasius equation was examined in [7]. An approximate analytic solution of the Blasius equation was achieved in ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6223 Email address: aamalameer@uhb.edu.sa (A. Alameer) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 2 of 9 [8] using Padé approximants. Ganji et al. [9] used the homotopy perturbation method to obtain an analytical solution for the Blasius equation. Parand et al. [10] tested the sinc– collocation method and Xu et al. [11] studied a fixed-point approach to solving the Blasius equation. A two-point block predictor–corrector methodology was developed in [12] for numerically resolving the well-known Blasius and Sakiadis flows. In [13], a generalized version of the Blasius equation was derived by Makhfi and Bebbouchi. Recently, Mutuk [14] employed neural networks in the field of fluid mechanics and treated the Blasius equa- tion. The development and application of the RCW method to the Blasius problem were provided by Rahmanzadeh and Asadi [15]. However, all of the above works [1–15] and other methods highlighted in [16–27] have their own difficulties, shortcomings, and computational challenges. It is therefore essential to introduce a simple and accurate method for solving differential equations in engineer- ing and science, including for non-mathematicians. The Maclaurin series is an elementary strategy that is widely accessible. This paper implements the Maclaurin series strategy for the third-order equation; the clarity of the procedure and the explicit results render the technique appealing for real-world applications. The fundamental purpose of this work is to propose a novel solution of the Sakiadis equation arising in fluid mechanics via the Maclaurin Series Method (MSM). For Newtonian fluids, the Sakiadis equation is essential to the analysis of laminar boundary-layer problems. The basic difficulty presented by the boundary condition at infinity is addressed here. By combining the series obtained using MSM with diagonal Padé approximants, reliable outcomes are obtained [28]. Ta- ble 1 illustrates the convergence of MSM. Additionally, this article compares the outcomes with previous solutions [5, 6]. MSM provides a more straightforward and effective route than many decomposition and iterative approaches, which frequently entail considerable complexity. Because it avoids assumptions, discretization, over-linearization, and other constrictive techniques that can substantially alter the original problem, the method is especially well suited to fluid-mechanics problems. With MSM, a high degree of precision is maintained while removing the need for laborious numerical techniques. The results are both promising and accurate, and the suggested method performs effectively. 2. Sakiadis Flow and its solution We assume a laminar Newtonian fluid flow with ξ > 0 throughout the domain. Con- sider the boundary layer equations in two-dimensional motions [5, 6]: ∂G ∂x + ∂H ∂y = 0, G ∂G ∂x +H ∂G ∂y = γ ∂2G ∂y2 , G(x, 0) = U0, H(x, 0) = 0, G(x, y) → 0 as y → ∞. (1) A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 3 of 9 Considering the similarity transformation ξ = √ U0 γx y, G = U0g ′(ξ), H = − 1 2 √ U0γ x [g(ξ)− ξg′(ξ)] , (2) the Eq. (1) becomes g′′′(ξ) + 1 2 g(ξ) g′′(ξ) = 0, (3) g(0) = 0, g′(0) = 1, g′(∞) = 0. (4) To illustrate the solution process, we assume that g′′(0) = β is an unknown constant that needs to be further identified. In Eq. (3), setting ξ = 0 yields g′′′(0) + 1 2 g(0) g′′(0) = 0. (5) Using Eq. (4) in Eq. (5), we get g′′′(0) = 0. (6) Differentiating Eq. (3) w.r.t ξ, we obtain giv(ξ) + 1 2 g′(ξ) g′′(ξ) + 1 2 g(ξ) g′′′(ξ) = 0. (7) Setting ξ = 0 in Eq. (7) results in giv(0) + 1 2 g′(0) g′′(0) + 1 2 g(0) g′′′(0) = 0. (8) Using Eq. (4), Eq. (6) into Eq. (8), we get giv(0) = −β 2 . (9) Differentiating Eq. (7) w.r.t. ξ, we obtain gv(ξ) + 1 2 g(ξ) giv(ξ) + g′(ξ) g′′′(ξ) + 1 2 ( g′′(ξ) )2 = 0. (10) Setting ξ = 0 in Eq. (7) and considering the Eq. (4), Eq. (6) and Eq. (9), we have gv(0) + 1 2 g(0) giv(0) + g′(0) g′′′(0) + 1 2 ( g′′(0) )2 = 0, (11) gv(0) = −β2 2 . (12) By a similar operation via Mathematica, we can obtain A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 4 of 9 g6(0) = 3β 4 , g7(0) = 11β2 4 , (13) g8(0) = β 8 ( −15 + 22β2 ) , ... ... The Maclaurin series solution can be expressed as g(ξ) = g(0) + g′(0) ξ 1! + g′′(0) ξ2 2! + g′′′(0) ξ3 3! + giv(0) ξ4 4! + gv(0) ξ5 5! + gvi(0) ξ6 6! + g7(0) ξ7 7! + g8(0) ξ8 8! + g9(0) ξ9 9! + g10(0) ξ10 10! + . . . , (14) g(ξ) = ξ + β ξ2 2 − β ξ4 48 − β2 ξ5 240 + β ξ6 960 + 11β2 ξ7 20160 + β ( −15 + 22β2 ) ξ8 322560 − 129β2 ξ9 2903040 − 15 16 ( −7β + 50β3 ) ξ10 10! − 3 16 ( −587β2 + 250β4 ) ξ11 11! + 15 32 ( −63β + 1408β3 ) ξ12 12! + 3 32 ( −9385β2 + 18598β4 ) ξ13 13! + 3 64 ( 3465β − 199988β3 + 37196β5 ) ξ14 14! − 3 64 ( −174645β2 + 1017244β4 ) ξ15 15! − 3 128 ( 45045β − 6081294β3 + 5089516β5 ) ξ16 16! − 3 128 ( 3753435β2 − 51909900β4 + 5089516β6 ) ξ17 17! . (15) A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 5 of 9 Table 1: MSM β = g′′(0) solution comparison with ADM, VIM Padé approximants Present method ADM [5] VIM [6] [2/2] 0.5773502692 0.5773502692 0.577350693 [3/3] 0.5163977795 0.5163977795 0.5163977793 [4/4] 0.5227030798 0.5227030798 0.5227030796 [5/5] 1.267686100 - - [6/6] 0.5217102130 – 0.5217102130 [7/7] 0.5026354150 – 0.5026354150 [8/8] Complex numbers – Complex numbers [9/9] Complex numbers – Complex numbers [10/10] 0.4672639966 – 0.4672639966 [11/11] 0.5176098151 – 0.5176098151 ξ g ′ ( ξ) Figure 1: MSM g′ VIM g′ ADM g′ A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 6 of 9 ξ g (ξ ) Figure 2: Solution of g via MSM Solution of g via VIM Solution of g via ADM Figure 3: Streamlines of g. A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 7 of 9 3. Discussion This paper presents the Sakiadis equation series solution using a novel method from the Maclaurin series (MSM). By adding a similarity variable, Sakiadis developed continuity and momentum equations for a continuous, incompressible laminar flow of fluid. MSM validity is shown in Table 1 and Figs. 1 and 2. Figure 3 reveals streamlines displaying the actual Sakiadis equation phenomena. Streamlines display particle trajectories traveling along the flow paths and provide insight into complex flow characteristics. Figure 3 is a form of communication designed to show the direction in which the fluid particle travels in the flow of fluid at a given level. The following fractal formulation for Eq. (3) may be established in the context of fractal derivative [29, 30]: d dξα ( d dξα ( dg dξα )) + 1 2 g(ξα) d dξα ( dg dξα ) = 0, (16) where dg dξα is the fractal derivative defined as: dg dξα = lim ξ→ξ1 g(ξ)− g(ξ1) ξα − ξα1 (17) Or dg dξα = 1 α ξα−1 dg dξ (18) With transformation [29, 30] ξα = s (19) Using Eq. (19), Eq. (16) analogue to Eq. (3) and one can recover easily the same results obtained in Eq. (15). Provided the traditional calculus, this transformation makes the fractal calculus extremely simple. 4. Conclusion This article introduces a novel method for calculating the new solution of the Sakiadis equation in the Maclaurin series. Padé approximants handle the infinity condition. This approach is easy to implement, and the outcomes demonstrate that by reducing the size of the calculation, the explanations can convert extra precise. Comparing the current solution to previous solutions [5, 6] shows that there is excellent agreement between the two. Furthermore, the analysis presented here strengthens trust in the MSM’s efficiency. Moreover, a fractal model of the Sakiadis equation is introduced. In addition to the demonstrated accuracy and simplicity of the Maclaurin Series Method (MSM), future work may explore its application to more complex fluid models. These include non-Newtonian fluids, fractional derivative formulations, and multi-phase flows. The MSM’s adaptability and precision make it a promising tool for solving advanced boundary-layer problems in engineering and applied sciences. A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 8 of 9 References [1] L. Howarth. On the solution of the laminar boundary layer equations. Proc. R. Soc. Lond. A, 164:547–579, 1938. [2] S. J. Liao. An approximate solution technique not depending on small parameters part 2: an application in fluid mechanics. Int. J. Non-Linear Mech., 32:815–822, 1997. [3] L. T. Yu and C. K. Chen. The solution of the blasius equation by the differential transformation method. Math. Comput. Modelling, 28:101–111, 1998. [4] J. H. He. A simple perturbation approach to blasius equation. Appl. Math. Comput., 140:217–222, 2003. [5] A. M. Wazwaz. A study on a boundary-layer equation arising in an incompressible fluid. Appl. Math. Comput., 87:199–204, 1997. [6] A. M. Wazwaz. The variational iterative method for solving two forms of blasius equation on a half-infinite domain. Appl. Math. Comput., 188:485–491, 2007. [7] S. Abbasbandy. A numerical solution of blasius equation by adomiana(tm)s decompo- sition method and comparison with homotopy perturbation method. Chaos, Solitons and Fractals, 31:257–280, 2007. [8] F. Ahmad and W. H. Al-Barakati. An approximate analytic solution of blasius prob- lem. Commun. Nonlinear Sci. Numer. Simul., 14:1021–1024, 2009. [9] D. D. Ganji, H. Babazadeh, F. Noori, M. M. Pirouz, and M. Janipour. An application of homotopy perturbation method for non-linear blasius equation to boundary layer flow over a flat plate. International Journal of Nonlinear Science, 7:399–404, 2009. [10] K. Parand, M. Dehghan, and A. Pirkhedri. Sinc-collocation method for solving the blasius equation. Phys. Lett. A, 373:4060–4065, 2009. [11] D. Xu and X. Guo. Application of fixed-point method to obtain semi-analytical solution to blasius flow and its variation. Appl. Math. Comput., 224:791–802, 2013. [12] Z. A. Majid and P. P. See. Study of predictor corrector block method via multiple shooting to blasius and sakiadis flow. Appl. Math. Comput., 314:469–483, 2017. [13] A. Makhfi and R. Bebbouchi. On the generalized blasius equation. Afrika Matematika, 2020. [14] H. Mutuk. A neural network study of blasius equation. Neural Processing Letters, 2020. [15] M. Rahmanzadeh, Asadi, and T. M. Atashafrooz. The development and application of the rcw method for the solution of the blasius problem. Journal of Applied and Computational Mechanics, 6:105–111, 2020. [16] Y. Khan, A. Hussain, and N. Faraz. Unsteady linear viscoelastic fluid model over a stretching/shrinking sheet in the region of stagnation point flows. Scientia Iranica B, 19:1541–159, 2012. [17] S. M. Hosseini, M. R. Safaei, P. Estellé, and S. H. Jafarnia. Heat transfer of water- based carbon nanotube nanofluids in the shell and tube cooling heat exchangers of the gasoline product of the residue fluid catalytic cracking unit. Journal of Thermal Analysis and Calorimetry, 140:351–362, 2020. A. Alameer / Eur. J. Pure Appl. Math, 18 (4) (2025), 6223 9 of 9 [18] Y. Khan. A novel laplace iterative method for non-linear stretching sheet problem in the presence of mhd and slip condition. Int. J. Numerical Methods for Heat & Fluid Flow, 24:73–85, 2013. [19] E. Hetmaniok, I. Nowak, D. Slota, and R. Witula. Application of the homotopy per- turbation method for the solution of inverse heat conduction problem. International Communications in Heat and Mass Transfer, 39:30–35, 2012. [20] Y. Khan. Two-dimensional boundary layer flow of chemical reaction mhd fluid over a shrinking sheet with suction and injection. Journal of Aerospace Engineering, 27:04014019, 2014. [21] Mojtaba Fardi and Yasir Khan. A novel finite difference-spectral method for frac- tal mobile/immobile transport model based on Caputo–Fabrizio derivative. Chaos, Solitons & Fractals, 143:110573, 2021. [22] Y. Khan, Q. Wu, N. Faraz, and A. Yildirim. The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet. Computers and Mathematics with Applications, 61:3391–3399, 2011. [23] Shreen El-Sapa and Noura S. Alsedais. Effect of slippage on a translational motion of two interacting non-concentric spheres squeezed by couple stress fluid. Indian Journal of Pure and Applied Mathematics, 2024. [24] Y. Khan. Magnetohydrodynamic flow of linear visco-elastic fluid model above a shrinking/stretching sheet: A series solution. Scientia Iranica. Transaction B, Me- chanical Engineering, 24:2466–2472, 2017. [25] Magdy A. Ezzat and Shreen El-Sapa. State space approach to magnetohydrodynamic flow of perfectly conducting micropolar fluid with stretch. Int. J. Numer. Meth. Fluids, 70:114–134, 2012. [26] Y. Khan. A series solution of the boundary value problem arising in the application of the fluid mechanics. Int. J. Numerical Methods for Heat & Fluid Flow, 28:2480–2490, 2018. [27] Zdeněk Šmarda and Yasir Khan. An efficient computational approach to solving sin- gular initial value problems for lane–emden type equations. Journal of Computational and Applied Mathematics, 290:65–73, 2015. [28] George A. Baker. Essentials of Padé Approximants. Academic Press, London, 1975. [29] W. Chen. Time-space fabric underlying anomalous diffusion. Chaos, Solitons & Fractals, 28:923–929, 2006. [30] Y. Liang, N. Su, and W. Chen. A time-space hausdorff derivative model for anomalous transform in porous media. Fract. Calc. Appl. Anal., 22:1517–1536, 2019.