Fractional nonlinear partial differential equations are used in many scientific fields to model various processes, although most of these equations lack closed-form solutions. For this reason, methods for approximating solutions that occasionally yield closed-form solutions are crucial for solving these equations. This study introduces a novel technique that combines the residual function and a modified fractional power series with the Elzaki transform to solve various nonlinear problems within the Caputo derivative framework. The accuracy and effectiveness of our approach are validated through analyses of absolute, relative, and residual errors. We utilize the limit principle at zero to identify the coefficients of the series solution terms, while other methods, including variational iteration, homotopy perturbation, and Adomian, depend on integration. In contrast, the residual power series method uses differentiation, and both approaches encounter difficulties in fractional contexts. Furthermore, the effectiveness of our approach in addressing nonlinear problems without relying on Adomian and He polynomials enhances its superiority over various approximate series solution techniques.

Abdus Salam School of Mathematical Sciences Government College University Lahore Pakistan

Department of Computer Engineering Biruni University Istanbul Turkey

Department of Computer Science and Mathematics Lebanese American University Byblos Lebanon

Department of Mathematical Sciences College of Science Princess Nourah bint Abdulrahman University Riyadh Saudi Arabia

Department of Mathematics Art and Science Faculty Siirt University Siirt Istanbul Turkey

Department of Mathematics Mathematics Research Center Near East University Nicosia Mersin Turkey

IT4Innovations VSB Technical University of Ostrava Ostrava Czech Republic

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PLoS One. 2025 Mar 27;20(3):e0321323. doi: 10.1371/journal.pone.0321323. PubMed

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Pratap H, Kumar S, Singh G. Brief History of Fractional Calculus: A Survey. Migration Letters. 2024;21(7):238–43.

Singh AP, Bingi K. Applications of Fractional-Order Calculus in Robotics. Fractal and Fractional. 2024;8(7):403. doi: 10.3390/fractalfract8070403 DOI

Huang G, Qin HY, Chen Q, Shi Z, Jiang S, Huang C. Research on Application of Fractional Calculus Operator in Image Underlying Processing. Fractal and Fractional. 2024;8(1):37. doi: 10.3390/fractalfract8010037 DOI

Xu C, Farman M. Qualitative and Ulam–Hyres stability analysis of fractional order cancer-immune model. Chaos, Solitons & Fractals. 2023;177:114277. doi: 10.1016/j.chaos.2023.114277 DOI

Djaouti AM, Khan ZA, Liaqat MI, Al-Quran A. A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives. Mathematics. 2024;12(11):1654. doi: 10.3390/math12111654 DOI

Liaqat MI, Etemad S, Rezapour S, Park C. A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients. Aims Math. 2022;7(9):16917–48. doi: 10.3934/math.2022929 DOI

Mohammed Djaouti A, Khan ZA, Liaqat MI, Al-Quran A. Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the Lp Space with the Framework of the Ψ-Caputo Derivative. Mathematics. 2024;12(7):1037. doi: 10.3390/math12071037 DOI

Djitte SM, Fall MM, Weth T. A fractional Hadamard formula and applications. Calculus of variations and partial differential equations. 2021;60:1–31. doi: 10.1007/s00526-021-02094-3 DOI

Khirsariya SR, Rao SB. Solution of fractional sawada-kotera-ito equation using caputo and atangana–baleanu derivatives. Mathematical Methods in the Applied Sciences. 2023;46(15):16072–91. doi: 10.1002/mma.9438 DOI

Khirsariya S, Rao S, Chauhan J. Semi-analytic solution of time-fractional Korteweg-de Vries equation using fractional residual power series method. Results in Nonlinear Analysis. 2022;5(3):222–34. doi: 10.53006/rna.1024308 DOI

Khirsariya SR, Rao SB, Chauhan JP. A novel hybrid technique to obtain the solution of generalized fractional-order differential equations. Mathematics and Computers in Simulation. 2023;205:272–90. doi: 10.1016/j.matcom.2022.10.013 DOI

Khirsariya SR, Chauhan JP, Rao SB. A robust computational analysis of residual power series involving general transform to solve fractional differential equations. Mathematics and Computers in Simulation. 2024;216:168–86. doi: 10.1016/j.matcom.2023.09.007 DOI

Brzeziński DW. Fractional order derivative and integral computation with a small number of discrete input values using Grünwald–Letnikov formula. International Journal of Computational Methods. 2020;17(05):1940006. doi: 10.1142/S0219876219400061 DOI

Luchko Y. Fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. Mathematics. 2022;10(6):849. doi: 10.3390/math10060849 DOI

Sene N, Ndiaye A. On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative. Journal of Mathematics. 2020;2020(1):8815377.

Khurshaid A, Khurshaid H. Comparative Analysis and Definitions of Fractional Derivatives. Journal ISSN. 2023;2766:2276.

Rao A, Vats RK, Yadav S. Analytical solution for time-fractional cold plasma equations via novel computational method. International Journal of Applied and Computational Mathematics. 2024;10(1):2. doi: 10.1007/s40819-023-01639-8 DOI

Yadav S, Vats RK, Rao A. Constructing the fractional series solutions for time-fractional K-dV equation using Laplace residual power series technique. Optical and Quantum Electronics. 2024;56(5):721. doi: 10.1007/s11082-024-06412-9 DOI

Bhrawy AH, Alofi AS. The operational matrix of fractional integration for shifted Chebyshev polynomials. Applied Mathematics Letters. 2013;26(1):25–31. doi: 10.1016/j.aml.2012.01.027 DOI

Shiralashetti SC, Deshi AB. An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear dynamics. 2016;83:293–303. doi: 10.1007/s11071-015-2326-4 DOI

Maitama S, Zhao W. New homotopy analysis transform method for solving multidimensional fractional diffusion equations. Arab Journal of Basic and Applied Sciences. 2020;27(1):27–44. doi: 10.1080/25765299.2019.1706234 DOI

Hasan S, El-Ajou A, Hadid S, Al-Smadi M, Momani S. Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system. Chaos, Solitons & Fractals. 2020;133:109624. doi: 10.1016/j.chaos.2020.109624 DOI

Liaqat MI, Khan A, Alam MA, Pandit MK. A Highly Accurate Technique to Obtain Exact Solutions to Time-Fractional Quantum Mechanics Problems with Zero and Nonzero Trapping Potential. Journal of Mathematics. 2022;2022(1):9999070. doi: 10.1155/2022/9999070 DOI

Jassim HK, Mohammed MG. Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations. International Journal of Nonlinear Analysis and Applications. 2021;12(1):812–20.

Chauhan JP, Khirsariya SR. A semi-analytic method to solve nonlinear differential equations with arbitrary order. Results in Control and Optimization. 2023;12:100267. doi: 10.1016/j.rico.2023.100267 DOI

Khirsariya SR, Rao SB. On the semi-analytic technique to deal with nonlinear fractional differential equations. Journal of Applied Mathematics and Computational Mechanics. 2023;22(1). doi: 10.17512/jamcm.2023.1.02 DOI

Chauhan JP, Khirsariya SR, Hathiwala GS, Biswas Hathiwala M. New analytical technique to solve fractional-order Sharma-Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators. Journal of Applied Analysis. 2024;30(1):1–6. doi: 10.1515/jaa-2023-0043 DOI

Khirsariya S, Rao S, Chauhan J. Solution of fractional modified Kawahara equation: a semi-analytic approach. Mathematics in Applied Sciences and Engineering. 2023;4(4):264–84. doi: 10.5206/mase/16369 DOI

Khirsariya SR, Rao SB, Hathiwala GS. Investigation of fractional diabetes model involving glucose–insulin alliance scheme. International Journal of Dynamics and Control. 2024;12(1):1–4. doi: 10.1007/s40435-023-01293-4 DOI

Zahid M, Ud Din F, Shah K, Abdeljawad T. Fuzzy fixed point approach to study the existence of solution for Volterra type integral equations using fuzzy Sehgal contraction. Plos One. 2024;19(6):e0303642. doi: 10.1371/journal.pone.0303642 PubMed DOI PMC

Elzaki TM. On the connections between Laplace and Elzaki transforms. Advances in Theoretical and Applied mathematics. 2011;6(1):1–11.

Mitra A. A comparative study of elzaki and laplace transforms to solve ordinary differential equations of first and second order. In Journal of Physics: Conference Series. 2021; 1913(1):012147.

Das S, Kumar R. Approximate analytical solutions of fractional gas dynamic equations. Applied Mathematics and Computation. 2011;217(24):9905–9915. doi: 10.1016/j.amc.2011.03.144 DOI

Kumar S, Kocak H, Yıldırım A. A fractional model of gas dynamics equations and its analytical approximate solution using Laplace transform. Zeitschrift für Naturforschung A. 2012;67(6–7):389–96. doi: 10.5560/zna.2012-0038 DOI

Iyiola OS. On the solutions of non-linear time-fractional gas dynamic equations: an analytical approach. Int. J. Pure Appl. Math. 2015;98(4):491–502. doi: 10.12732/ijpam.v98i4.8 DOI

Das S, Kumar R. Approximate analytical solutions of fractional gas dynamic equations. Applied Mathematics and Computation. 2011;217(24):9905–15. doi: 10.1016/j.amc.2011.03.144 DOI

Raja Balachandar S, Krishnaveni K, Kannan K, Venkatesh SG. Analytical solution for fractional gas dynamics equation. National Academy Science Letters. 2019;42:51–7. doi: 10.1007/s40009-018-0662-x DOI

Tamsir M, Srivastava VK. Revisiting the approximate analytical solution of fractional-order gas dynamics equation. Alexandria Engineering Journal. 2016;55(2):867–74. doi: 10.1016/j.aej.2016.02.009 DOI

Bin Jebreen H, Cattani C. Solving Fractional Gas Dynamics Equation Using Müntz–Legendre Polynomials. Symmetry. 2023. Nov 16;15(11):2076. doi: 10.3390/sym15112076 DOI

Almutlak SA, Shah R, Weera W, El-Tantawy SA, El-Sherif LS. Fractional View Analysis of Swift–Hohenberg Equations by an Analytical Method and Some Physical Applications. Fractal and Fractional. 2022;6(9):524. doi: 10.3390/fractalfract6090524 DOI

Nonlaopon K, Alsharif AM, Zidan AM, Khan A, Hamed YS, Shah R. Numerical investigation of fractional-order Swift–Hohenberg equations via a Novel transform. Symmetry. 2021;13(7):1263. doi: 10.3390/sym13071263 DOI

Li W, Pang Y. An iterative method for time-fractional Swift-Hohenberg equation. Advances in Mathematical Physics 2018;2018(1):2405432.

Veeresha P, Prakasha DG, Baleanu D. Analysis of fractional Swift-Hohenberg equation using a novel computational technique. Mathematical Methods in the Applied Sciences. 2020;43(4):1970–87. doi: 10.1002/mma.6022 DOI

Jani HP, Singh TR. Some examples of Swift–Hohenberg equation. Examples and Counterexamples. 2022;2:100090. doi: 10.1016/j.exco.2022.100090 DOI

Pavani K, Raghavendar K. Approximate solutions of time-fractional Swift–Hohenberg equation via natural transform decomposition method. International Journal of Applied and Computational Mathematics. 2023;9(3):29. doi: 10.1007/s40819-023-01493-8 DOI

Mahdy AM. Numerical solutions for model time-fractional Fokker-Plank equation. Numerical Methods for Partial Differential Equations. 2021;37(2):1120–35. doi: 10.1002/num.22570 DOI

Wei JL, Wu GC, Liu BQ, Zhao Z. New semi-analytical solutions of the time-fractional Fokker-Planck equation by the neural network method. Optik. 2022;259:168896. doi: 10.1016/j.ijleo.2022.168896 DOI

Yang Y, Huang Y, Zhou Y. Numerical solutions for time fractional Fokker-Plank equations based on spectral collocation methods. Journal of Computational and Applied Mathematics. 2018;339:389–404. doi: 10.1016/j.cam.2017.04.003 DOI

He C, Chen J, Fang H, He H. Fundamental solution of fractional Kolmogorov–Fokker–Planck equation. Examples and Counterexamples. 2021;1:100031. doi: 10.1016/j.exco.2021.100031 DOI

Yan L. Numerical solutions of fractional Fokker-Plank equations using iterative Laplace transform method. InAbstract and applied analysis. 2013;1: 465160.

Baumann G, Stenger F. Fractional Fokker-Planck Equation. Mathematics. 2017;5(1):12. doi: 10.3390/math5010012 DOI

Khan H, Farooq U, Tchier F, Khan Q, Singh G, Kumam P, Sitthithakerngkiet K. The analytical analysis of fractional order Fokker-Planck equations. AIMS Mathematics. 2022;7(7):11919–41. doi: 10.3934/math.2022665 DOI

Loyinmi AC, Akinfe TK. An algorithm for solving the Burgers–Huxley equation using the Elzaki transform. SN Applied sciences. 2020;2(1):7. doi: 10.1007/s42452-019-1653-3 DOI

Liaqat MI, Akgül A, Bayram M. Series and closed form solution of Caputo time-fractional wave and heat problems with the variable coefficients by a novel approach. Optical and Quantum Electronics. 2024;56(2):203. doi: 10.1007/s11082-023-05751-3 DOI

Liaqat MI, Khan A, Akgül A, Ali MS. A novel numerical technique for fractional ordinary differential equations with proportional delay. Journal of Function Spaces. 2022;2022(1):6333084.