{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T22:03:21Z","timestamp":1765231401875,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T00:00:00Z","timestamp":1742515200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prince Sattam bin Abdulaziz University","award":["PSAU\/2025\/R\/1446"],"award-info":[{"award-number":["PSAU\/2025\/R\/1446"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The primary objective of this study is to expand the application of analytical and numerical methods for solving nonlinear Systems of Fractional Differential Equations (SFDEs) with Caputo fractional derivatives (CFDs) under initial conditions. Our proposed approach, the Multistage Telescoping Decomposition Elzaki Method (MTDEM), integrates the advantages of the Elzaki transform with the Multistage Telescoping Decomposition Method (MTDM), significantly enhancing the efficiency of the solution process and improving the convergence rate. Additionally, it simplifies computational operations and reduces the computational complexity associated with solving these nonlinear systems. A comprehensive comparison is conducted to highlight the accuracy and computational advantages of our proposed method compared to existing techniques, including the exact solution and the Telescoping Decomposition Method (TDM), through numerical examples that demonstrate the effectiveness of the proposed approach. The flexibility of the MTDEM allows for its application in a wide range of nonlinear SFDEs, making it a valuable tool in various scientific and engineering fields. These systems are widely used in modeling numerous physical, biological, and economic phenomena, such as the dynamics of electrical systems, heat transfer, and population growth models, underscoring the importance of developing accurate and efficient computational methods for their solutions. Through this study, we present a novel contribution to enhancing numerical and analytical techniques, paving the way for broader applications in multiple domains that require precise and reliable solutions for complex fractional systems.<\/jats:p>","DOI":"10.3390\/axioms14040233","type":"journal-article","created":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T07:25:00Z","timestamp":1742541900000},"page":"233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Solutions of Nonlinear Fractional-Order Differential Equation Systems Using a Numerical Technique"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1965-4103","authenticated-orcid":false,"given":"Mohammed","family":"Boukedroun","sequence":"first","affiliation":[{"name":"Department of Mathematics, Khemis Miliana University, Ain Defla 44225, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7011-5389","authenticated-orcid":false,"given":"Souad","family":"Ayadi","sequence":"additional","affiliation":[{"name":"Acoustics and Civil Engineering Laboratory, Department of Material Sciences, Khemis Miliana University, Ain Defla 44225, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-0257-3998","authenticated-orcid":false,"given":"Fouzia","family":"Chita","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khemis Miliana University, Ain Defla 44225, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4519-9506","authenticated-orcid":false,"given":"Meltem","family":"Erden Ege","sequence":"additional","affiliation":[{"name":"Independent Researcher, 35000 Izmir, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3877-2714","authenticated-orcid":false,"given":"Ozgur","family":"Ege","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ege University, 35100 Izmir, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9027-0810","authenticated-orcid":false,"given":"Rajagopalan","family":"Ramaswamy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0898-1221(94)90132-5","article-title":"Solution of physical problems by decomposition","volume":"27","author":"Adomian","year":"1994","journal-title":"Comp. Math. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"63","DOI":"10.12816\/0017358","article-title":"Fractional homotopy perturbation transform method for solving the time-fractional KdV, k(2,2) and Burgers equations","volume":"8","author":"Ziane","year":"2015","journal-title":"Int. J. Open Prob. Comput. Sci. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0020-7462(98)00085-7","article-title":"A coupling method of homotoping technique and perturbation technique for nonlinear problems","volume":"35","author":"He","year":"2000","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_4","first-page":"82","article-title":"Variational iteration Sumudu transform method for solving fractional nonlinear gaz dynamics equation","volume":"1","author":"Rida","year":"2014","journal-title":"Int. J. Res. Stud. Sci. Eng. Technol."},{"key":"ref_5","first-page":"790714","article-title":"Solution of nonlinear partial differential equations by new Laplace variational iteration method","volume":"2014","author":"Hilall","year":"2014","journal-title":"J. Funct. Spaces"},{"key":"ref_6","first-page":"1","article-title":"On the connections between Laplace and Elzaki transforms","volume":"6","author":"Elzaki","year":"2014","journal-title":"Adv. Theor. Appl. Math."},{"key":"ref_7","first-page":"173","article-title":"Coupling of Laplace transform and correction functional for wave equations","volume":"9","author":"Afshan","year":"2013","journal-title":"Word J. Mod. Simul."},{"key":"ref_8","first-page":"1945","article-title":"Analytical solution of the systems of nolinear fractional partial differential equations using conformale Laplace transform iterative method","volume":"10","author":"Gul","year":"2025","journal-title":"Fractal Fract."},{"key":"ref_9","first-page":"515","article-title":"Sumudu decomposition method for nonlinear equations","volume":"7","author":"Kumar","year":"2012","journal-title":"Int. Math. Forum"},{"key":"ref_10","first-page":"165","article-title":"Homotopy perturbation Sumudu transform method for nonlinear equations","volume":"4","author":"Singh","year":"2011","journal-title":"Adv. Theor. Appl. Mach."},{"key":"ref_11","first-page":"301","article-title":"Homotopy analysis Sumudu transform method for nonlinear equations","volume":"4","author":"Rathore","year":"2012","journal-title":"Int. J. Ind. Math."},{"key":"ref_12","first-page":"57","article-title":"The new integral transform Elzaki transform","volume":"7","author":"Elzaki","year":"2011","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_13","first-page":"33","article-title":"Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations","volume":"2","author":"Elzaki","year":"2012","journal-title":"Math. Theor. Mod."},{"key":"ref_14","first-page":"1","article-title":"On the new integral transform \u201cElzaki Transform\u201d fundamental properties investigations and applications","volume":"2","author":"Elzaki","year":"2012","journal-title":"Glob. J. Math. Sci."},{"key":"ref_15","first-page":"13","article-title":"On the Elzaki transform and ordinary differential equation with variable coefficients","volume":"6","author":"Elzaki","year":"2011","journal-title":"Adv. Theor. Appl. Math."},{"key":"ref_16","first-page":"944","article-title":"Homotopy perturbation method and Elzaki transform for solving system of nonlinear partial differential equatios","volume":"24","author":"Elzaki","year":"2013","journal-title":"World Appl. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"451","DOI":"10.12732\/ijam.v32i3.7","article-title":"Adomian polynomial and Elzaki transform method for solving Klein Gordon equations","volume":"32","author":"Ige","year":"2019","journal-title":"Int. J. Appl. Math."},{"key":"ref_18","first-page":"406","article-title":"Homotopy perturbation method combined with ZZ transform to solve some nonlinear fractional differential equations","volume":"17","author":"Riabi","year":"2019","journal-title":"Int. J. Anal. Appl."},{"key":"ref_19","unstructured":"Al-Refai, M., Abu-Dalu, M., and Al-Rawashde, A. (2008, January 19\u201321). Telescoping decomposition method for solving first order nonlinear differential equations. Proceedings of the International Multi-Conference of Engineers and Computer Scientists, (IMECS 2008), Kowloon, Hong Kong."},{"key":"ref_20","first-page":"48","article-title":"An Elzaki transform decomposition algorithm applied to a class of non-linear differential equations","volume":"5","author":"Khalid","year":"2015","journal-title":"J. Nat. Sci. Res."},{"key":"ref_21","first-page":"590","article-title":"Telescoping decomposition method for solving second order nonlinear differential equations","volume":"3","author":"Nibron","year":"2014","journal-title":"Int. J. Eng. Res. Technol."},{"key":"ref_22","first-page":"261","article-title":"Adomian polynomial and Elzaki transform method of solving third order Korteweg-De Vries equatios","volume":"15","author":"Ige","year":"2019","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_23","first-page":"1605","article-title":"ZZ transform method","volume":"4","author":"Zafar","year":"2016","journal-title":"Int. J. Adv. Eng. Glob. Technol."},{"key":"ref_24","first-page":"1355","article-title":"Application of ZZ transform method on some fractional differential equation","volume":"4","author":"Zafar","year":"2016","journal-title":"Int. J. Adv. Eng. Glob. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"34","DOI":"10.26637\/MJM0601\/0005","article-title":"The combined of Homotopy analysis method with new transform for nonlinear partial differential equations","volume":"6","author":"Ziane","year":"2018","journal-title":"Malaya J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Saadeh, R., Qazza, A., and Burqan, A. (2020). A new integral transform: ARA transform and its properties and applications. Symmetry, 12.","DOI":"10.3390\/sym12060925"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5881","DOI":"10.37256\/cm.5420245314","article-title":"Application of Natural Transform Decomposition Method for Solution of Fractional Richards Equation","volume":"5","author":"Raghavendar","year":"2024","journal-title":"Contemp. Math."},{"key":"ref_28","first-page":"30","article-title":"New Double Kharrat-Toma Transform and its Application in Partials Differential Equations","volume":"42","author":"Kharrat","year":"2024","journal-title":"World Appl. Sci. J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"114829","DOI":"10.1016\/j.rser.2024.114829","article-title":"Fixed-time fractional-order sliding mode controller with disturbance observer for U-tube steam generator","volume":"205","author":"Hui","year":"2024","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"113550","DOI":"10.1016\/j.rser.2023.113550","article-title":"Load following control of a PWR with load-dependent parameters and perturbations via fixed-time fractional-order sliding mode and disturbance observer techniques","volume":"184","author":"Hui","year":"2023","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3513","DOI":"10.1002\/asjc.3031","article-title":"Fractional-order sliding mode load following control via disturbance observer for modular high-temperature gas-cooled reactor system with disturbances","volume":"25","author":"Hui","year":"2023","journal-title":"Asian J. Control."},{"key":"ref_32","first-page":"1","article-title":"Multistage telescoping decomposition method for solving fractional differential equations","volume":"43","author":"Bouhassoun","year":"2013","journal-title":"Int. J. Appl. Math."},{"key":"ref_33","first-page":"203","article-title":"Solving systems of nonlinear fractional differential equations using multistage telescoping decomposition method","volume":"13","author":"Chita","year":"2016","journal-title":"Asian J. Math. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chita, F., Ayadi, S., Ege, M., Ege, O., and Ramaswamy, R. (2025). A Novel Approach for Solving Fractional Differential Equations via a Multistage Telescoping Decomposition Method. Fractal Fract., 9.","DOI":"10.3390\/fractalfract9020065"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Diethelm, K. (2010). The Analysis Fractional Differential Equations, Springer.","DOI":"10.1007\/978-3-642-14574-2"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/233\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:57:46Z","timestamp":1760029066000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/233"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,21]]},"references-count":35,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040233"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040233","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,3,21]]}}}