{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T16:57:16Z","timestamp":1781110636673,"version":"3.54.1"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T00:00:00Z","timestamp":1649030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The space-fractional stochastic approximate long water wave equation (SFSALWWE) is considered in this work. The Riccati equation method is used to get analytical solutions of the SFSALWWE. This equation has never been examined with stochastic term and fractional space at the same time. In general, the noise term that preserves the symmetry reduces the domain of instability. To check the impact of Brownian motion on these solutions, we use a MATLAB package to plot 3D and 2D graphs for some analytical fractional stochastic solutions.<\/jats:p>","DOI":"10.3390\/sym14040740","type":"journal-article","created":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T05:50:43Z","timestamp":1649051443000},"page":"740","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Impact of Brownian Motion on the Analytical Solutions of the Space-Fractional Stochastic Approximate Long Water Wave Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Farah M.","family":"Al-Askar","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, Collage of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11671, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1402-7584","authenticated-orcid":false,"given":"Wael W.","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 81411, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4030-9083","authenticated-orcid":false,"given":"Mohammad","family":"Alshammari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 81411, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,4]]},"reference":[{"key":"ref_1","unstructured":"Pr\u00e9v\u00f6t, C., and Rxoxckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3547","DOI":"10.1137\/140981952","article-title":"Fast diffusion limit for reaction-diffusion systems with stochastic Neumann boundary conditions","volume":"48","author":"Mohammed","year":"2016","journal-title":"SIAM J. Math. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1142\/S0219493702000443","article-title":"Conceptual stochastic climate models","volume":"2","author":"Imkeller","year":"2002","journal-title":"Stoch. Dynam."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2724","DOI":"10.1016\/j.na.2009.01.105","article-title":"Existence and uniqueness for p-type fractional neutral differential equations","volume":"71","author":"Zhou","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.1016\/j.na.2009.09.018","article-title":"Smoothness and stability of the solutions for nonlinear fractional differential equations","volume":"72","author":"Deng","year":"2010","journal-title":"Nonlinear Anal."},{"key":"ref_6","first-page":"2197247","article-title":"Numerical Methods for Fractional-Order Fornberg-Whitham Equations in the Sense of Atangana-Baleanu Derivative","volume":"2021","author":"Iqbal","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.matcom.2021.03.041","article-title":"Pattern formation induced by fractional cross-diffusion in a 3-species food chain model with harvesting","volume":"188","author":"Iqbal","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"9479","DOI":"10.3934\/math.2022526","article-title":"On degree theory for non-monotone type fractional order delay differential equations","volume":"7","author":"Shah","year":"2021","journal-title":"AIMS Math."},{"key":"ref_9","first-page":"4640467","article-title":"On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations","volume":"2021","author":"Kamal","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physleta.2006.06.024","article-title":"The Adomian decomposition method for solving partial differential equations of fractal order in finite domains","volume":"359","author":"Gaber","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1016\/j.physleta.2011.01.029","article-title":"Fractional sub-equation method and its applications to nonlinear fractional PDEs","volume":"375","author":"Zhang","year":"2011","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.physleta.2011.10.056","article-title":"The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics","volume":"376","author":"Guo","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.physleta.2007.07.051","article-title":"The (G\u2032G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics","volume":"372","author":"Wang","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.mcm.2003.12.010","article-title":"A sine-cosine method for handling nonlinear wave equations","volume":"40","author":"Wazwaz","year":"2004","journal-title":"Math. Comput. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2140","DOI":"10.1002\/mma.6925","article-title":"Approximate solutions for stochastic time-fractional reaction\u2013diffusion equations with multiplicative noise","volume":"44","author":"Mohammed","year":"2021","journal-title":"Chin. Ann. Math. Methods Appl. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W. (2020). Modulation Equation for the Stochastic Swift\u2013Hohenberg Equation with Cubic and Quintic Nonlinearities on the Real Line. Mathematics, 6.","DOI":"10.3390\/math7121217"},{"key":"ref_17","first-page":"72","article-title":"The eyp(-\u03d5(\u03c2))-expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation","volume":"5","author":"Khan","year":"2014","journal-title":"Int. J. Dyn. Syst. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1088\/0031-8949\/54\/6\/003","article-title":"The tanh method. I. Exact solutions of nonlinear evolution and wave equations","volume":"54","author":"Malfliet","year":"1996","journal-title":"Phys. Scr."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1186\/s13662-015-0452-4","article-title":"A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application","volume":"1","author":"Yang","year":"2015","journal-title":"Adv. Diff. Equation"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"636802","DOI":"10.1155\/2013\/636802","article-title":"The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation","volume":"2013","author":"Bulut","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1016\/j.asej.2013.01.006","article-title":"Application of the simplest equation method to some time-fractional partial differential equations","volume":"4","author":"Taghizadeh","year":"2013","journal-title":"Ain Shams Eng. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1186\/1687-1847-2014-135","article-title":"The modified Kudryashov method for solving some fractional-order nonlinear equations","volume":"2014","author":"Ege","year":"2014","journal-title":"Adv. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Al-Askar, F.M., Mohammed, W.W., Albalahi, A.M., and El-Morshedy, M. (2022). The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6030156"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Al-Askar, F.M., Mohammed, W.W., Albalahi, A.M., and El-Morshedy, M. (2022). The Impact of the Wiener process on the analytical solutions of the stochastic (2+ 1)-dimensional breaking soliton equation by using tanh\u2013coth method. Mathematics, 10.","DOI":"10.3390\/math10050817"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1515\/phys-2022-0002","article-title":"The exact solutions of the stochastic fractional-space Allen\u2014Cahn equation","volume":"20","author":"Albosaily","year":"2022","journal-title":"Open Phys."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Iqbal, N., and Botmart, T. (2022). Additive Noise Effects on the Stabilization of Fractional-Space Diffusion Equation Solutions. Mathematics, 10.","DOI":"10.3390\/math10010130"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Bazighifan, O., Al-Sawalha, M.M., Almatroud, A.O., and Aly, E.S. (2021). The Influence of Noise on the Exact Solutions of the Stochastic Fractional-Space Chiral Nonlinear Schr\u00f6dinger Equation. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040262"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2014.01.002","article-title":"A new definition of fractional derivative","volume":"264","author":"Khalil","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_29","first-page":"321","article-title":"Application of the (G\u2032\/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations","volume":"206","author":"Wang","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_30","first-page":"1965","article-title":"The improved (G\u2032\/G)-expansion method and its applications to the Broer\u2013Kaup equations and approximate long water wave equations","volume":"216","author":"Guo","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1142\/S0129183103004760","article-title":"Generalized extended tanh-function method to construct new explicit exact solutions for the approximate equations for long water waves","volume":"14","author":"Chen","year":"2003","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_32","first-page":"77","article-title":"Application of two different algorithms to the approximate long water wave equation with conformable fractional derivative","volume":"25","author":"Kaplan","year":"2018","journal-title":"Arab. J. Basic Appl. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.joes.2018.10.004","article-title":"New analytic solutions of the space-time fractional Broer\u2013Kaup and approximate long water wave equations","volume":"3","author":"Yaslan","year":"2018","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1108\/HFF-04-2013-0126","article-title":"New travelling wave solutions for coupled fractional variant Boussinesq equation and approximate long water wave equation","volume":"25","author":"Yan","year":"2015","journal-title":"Int. J. Num. Meth. Heat Fluid Flow"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"696","DOI":"10.1016\/j.ijleo.2016.10.116","article-title":"New exact solution for space-time fractional differential equations via (G\u2032\/G)-expansion method","volume":"130","author":"Guner","year":"2016","journal-title":"Optik"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.joes.2017.07.001","article-title":"New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water","volume":"2","author":"Khater","year":"2017","journal-title":"J. Ocean. Eng. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/740\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:49:33Z","timestamp":1760136573000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/740"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,4]]},"references-count":36,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040740"],"URL":"https:\/\/doi.org\/10.3390\/sym14040740","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,4]]}}}