{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T00:16:03Z","timestamp":1776903363481,"version":"3.51.2"},"reference-count":43,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T00:00:00Z","timestamp":1689552000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Umm Al-Qura University","award":["23UQU4282396DSR006"],"award-info":[{"award-number":["23UQU4282396DSR006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein\u2013Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach\u2019s reliability, capability, and efficiency.<\/jats:p>","DOI":"10.3390\/axioms12070694","type":"journal-article","created":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T01:06:36Z","timestamp":1689555996000},"page":"694","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method"],"prefix":"10.3390","volume":"12","author":[{"given":"Haneen","family":"Khresat","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7470-8162","authenticated-orcid":false,"given":"Ahmad","family":"El-Ajou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8955-5552","authenticated-orcid":false,"given":"Shrideh","family":"Al-Omari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7761-4196","authenticated-orcid":false,"given":"Sharifah E.","family":"Alhazmi","sequence":"additional","affiliation":[{"name":"Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca 21955, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Moa\u2019ath N.","family":"Oqielat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,17]]},"reference":[{"key":"ref_1","unstructured":"Tom, A., Craig, F., and Ivor, G.-G. (2004). The History of Differential Equations, EMS Press."},{"key":"ref_2","unstructured":"Davis, H.T. (2010). Introduction to Nonlinear Differential and Integral Equations (F), Dover Publications, Inc."},{"key":"ref_3","unstructured":"Nagle, R.K., Saff, E.B., and Snider, A.D. (2011). Fundamentals of Differential Equations and Boundary Value Problems, Pearson Education."},{"key":"ref_4","unstructured":"Olver, P.J. (1984). Trends and Applications of Pure Mathematics to Mechanics, Springer."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1080\/00207160412331336026","article-title":"The tanh function method for solving some important non-linear partial differential equations","volume":"82","author":"Evans","year":"2005","journal-title":"Int. J. Comput. Math."},{"key":"ref_6","first-page":"73","article-title":"Homotopy perturbation method: A new nonlinear analytical technique","volume":"135","author":"He","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_7","first-page":"853","article-title":"Properties of analytic solution and numerical solution of multi-pantograph equation","volume":"155","author":"Liu","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_8","unstructured":"El-Ajou, A., Odibat, Z., Momani, S., and Alawneh, A. (2010). Construction of analytical solutions to fractional differential equations using homotopy analysis method. IAENG Int. J. Appl. Math., 40."},{"key":"ref_9","unstructured":"Nishimoto, K. (1990). Fractional Calculus and Its Applications, College of Engineering, Nihon University."},{"key":"ref_10","first-page":"3","article-title":"Exact solution of some linear fractional differential equations by Laplace transform","volume":"16","author":"Kazem","year":"2013","journal-title":"Int. J. Nonlinear Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0898-1221(95)00031-S","article-title":"The exact solution of certain differential equations of fractional order by using operational calculus","volume":"29","author":"Luchko","year":"1995","journal-title":"Comput. Math. Appl."},{"key":"ref_12","first-page":"165","article-title":"The modified decomposition method for analytic treatment of differential equations","volume":"173","author":"Wazwaz","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1016\/j.camwa.2009.07.006","article-title":"A new operational matrix for solving fractional-order differential equations","volume":"59","author":"Saadatmandi","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.camwa.2008.09.045","article-title":"Analytical solution of a fractional diffusion equation by variational iteration method","volume":"57","author":"Das","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.jcp.2014.08.004","article-title":"Approximate analytical solution of the nonlinear fractional KdV-Burgers equation a new iterative algorithm","volume":"293","author":"Arqub","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Oqielat, M.N., Eriqat, T., Ogilat, O., El-Ajou, A., Alhazmi, S.E., and Al-Omari, S. (2023). Laplace-Residual Power Series Method for Solving Time-Fractional Reaction\u2013Diffusion Model. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040309"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"112394","DOI":"10.1016\/j.chaos.2022.112394","article-title":"Efficient numerical techniques for computing the Riesz fractional-order reac-tion\u2013diffusion models arising in biology","volume":"161","author":"Alqhtani","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/j.physa.2017.09.014","article-title":"A numerical solution for a variable-order reaction\u2013diffusion model by using fractional derivatives with non-local and non-singular kernel","volume":"491","author":"Torres","year":"2018","journal-title":"Phys. A Stat. Mech. its Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4093","DOI":"10.1007\/s00521-019-04350-2","article-title":"Numerical simulation of fractional-order reaction\u2013diffusion equations with the Riesz and Caputo derivatives","volume":"32","author":"Owolabi","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_20","first-page":"19","article-title":"Rational solutions to the cylindrical nonlinear Schr\u00f6dinger equation: Rogue waves, breathers, and Jacobi breathers solutions","volume":"13","author":"Matoog","year":"2022","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"23736","DOI":"10.1038\/s41598-021-02997-3","article-title":"Entropy generation and induced magnetic field in pseudoplastic nanofluid flow near a stagnant point","volume":"11","author":"Hou","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"110763","DOI":"10.1016\/j.chaos.2021.110763","article-title":"Fractional order biological snap oscillator: Analysis and control","volume":"145","author":"Trikha","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"104130","DOI":"10.1016\/j.rinp.2021.104130","article-title":"Chaos control and Penta-compound combination anti-synchronization on a novel fractional chaotic system with analysis and application","volume":"24","author":"Mahmoud","year":"2021","journal-title":"Results Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103105","DOI":"10.1063\/5.0109427","article-title":"On the analytical and numerical approximations to the forced damped Gardner Kawahara equation and modeling the nonlinear structures in a collisional plasma","volume":"34","author":"Alyousef","year":"2022","journal-title":"Phys. Fluids"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3713","DOI":"10.1007\/s00500-022-06885-4","article-title":"Multi-step reproducing kernel algorithm for solving Caputo\u2013Fabrizio fractional stiff models arising in electric circuits","volume":"26","author":"Hasan","year":"2022","journal-title":"Soft Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"e1021","DOI":"10.1002\/cmm4.1021","article-title":"Two novel computational techniques for fractional Gardner and Cahn-Hilliard equations","volume":"1","author":"Prakasha","year":"2019","journal-title":"Comput. Math. Methods"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1016\/j.asej.2014.03.014","article-title":"Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations","volume":"5","author":"Iyiola","year":"2014","journal-title":"Ain Shams Eng. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6717","DOI":"10.1002\/mma.5186","article-title":"Exact solutions for some time-fractional evolution equations using Lie group theory","volume":"41","author":"Bira","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1080\/16583655.2019.1640446","article-title":"Theory and application for the time fractional Gardner equation with Mittag-Leffler kernel","volume":"13","author":"Korpinar","year":"2019","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1750029","DOI":"10.1142\/S0219876217500293","article-title":"Numerical Multistep Approach for Solving Fractional Partial Differential Equations","volume":"14","author":"Freihat","year":"2017","journal-title":"Int. J. Comput. Methods"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"109957","DOI":"10.1016\/j.chaos.2020.109957","article-title":"A New Attractive Analytic Approach for Solutions of Linear and Nonlinear Neutral Fractional Pantograph Equations","volume":"138","author":"Eriqat","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1140\/epjp\/s13360-020-01061-9","article-title":"Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach","volume":"136","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"525250","DOI":"10.3389\/fphy.2021.525250","article-title":"A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients","volume":"9","year":"2021","journal-title":"Front. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1016\/j.camwa.2016.03.026","article-title":"On the time-fractional Navier\u2013Stokes equations","volume":"73","author":"Yong","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"23","DOI":"10.5391\/IJFIS.2022.22.1.23","article-title":"A New Approach to Solving Fuzzy Quadratic Riccati Differential Equations","volume":"22","author":"Oqielat","year":"2022","journal-title":"Int. J. Fuzzy Log. Intell. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1186\/s42787-020-00099-z","article-title":"Two efficient methods for solving fractional Lane\u2013Emden equations with conformable fractional derivative","volume":"28","author":"Adyan","year":"2020","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_37","first-page":"207","article-title":"Numerical solutions of Time-fractional nonlinear water wave partial differential equation via Caputo fractional derivative: An effective analytical method and some applications","volume":"21","author":"Oqielat","year":"2022","journal-title":"Appl. Comput. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1007\/s40435-022-01001-8","article-title":"Construction of fractional series solutions to nonlinear fractional reaction\u2013diffusion for bacteria growth model via Laplace residual power series method","volume":"11","author":"Oqielat","year":"2022","journal-title":"Int. J. Dyn. Control"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s40435-022-01020-5","article-title":"Revisited Fisher\u2019s equation and logistic system model: A new fractional approach and some modifications","volume":"11","author":"Eriqat","year":"2022","journal-title":"Int. J. Dyn. Control"},{"key":"ref_40","unstructured":"Tenenbaum, M., and Pollard, H. (1985). Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences, Courier Corporation."},{"key":"ref_41","unstructured":"Zill, D.G., and Shanahan, P.D. (2013). A First Course in Complex Analysis with Applications, Jones & Bartlett Learning."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.physleta.2006.02.048","article-title":"Analytical approach to linear fractional partial differential equations arising in fluid mechanics","volume":"355","author":"Momani","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1016\/j.camwa.2006.12.037","article-title":"Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations","volume":"54","author":"Momani","year":"2007","journal-title":"Comput. Math. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/7\/694\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:13:05Z","timestamp":1760127185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/7\/694"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,17]]},"references-count":43,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["axioms12070694"],"URL":"https:\/\/doi.org\/10.3390\/axioms12070694","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,17]]}}}